Complex analysis

Complex analysis studies functions of a complex number, especially holomorphic functions. Its distinctive feature is that a single complex derivative controls variation in every direction of the complex plane. The map

f (z) = z^{2}

gives a first picture: straight coordinate lines become curved, yet small perpendicular arrows remain perpendicular wherever

f^{'} (z) \neq = 0

Figure 1.
The complex square bends a grid globally and acts near 1 + i as a rotation through 45 degrees and a scaling by twice the square root of two
. **Details**.

The bottom panels show displacements rather than absolute positions. Multiplication by

f^{'} (1 + i) = 2 + 2 i

predicts the image of each small arrow; the exact image differs by

h^{2}

. The origin is exceptional because

f^{'} (0) = 0

. This distinction between local conformal maps and a critical point is central: complex differentiability imposes strong geometric structure, without making every holomorphic function globally one-to-one.

The remarkable next step is that holomorphic functions are determined inside a circle by their values on its boundary. Precisely, let

R > 0

, let

D = {z : ∣ z - a ∣ < R}

, and suppose

f

is a holomorphic function on an open set containing the closed disc

\overline{D}

. Orient its boundary counterclockwise. The Cauchy integral formula states that, for every

w \in D

f (w) = \frac{1}{2 πi} \oint_{∣ z - a ∣ = R} \frac{f ( z )}{z - w} d z .

(1)

The contour integral combines boundary values with a kernel depending on the requested interior point. It is a reconstruction formula, not merely a trick for evaluating an integral. The hypothesis about the closed disc ensures that all boundary values and the contour integral are well defined. ^[ref]

At the centre, substituting

z = a + R e^{i θ}

cancels the denominator against

d z = i R e^{i θ} d θ

f (a) = \frac{1}{2 π} \int_{0}^{2 π} f (a + R e^{i θ}) d θ .

(2)

Thus the centre value is the angular average of the boundary values. For the polynomial

f (z) = z^{2}

a = 1 + i

and

R = 1/2

, expand the boundary data as

f (a + R e^{i θ}) = 2 i + (1 + i) e^{i θ} + \frac{1}{4} e^{2 i θ} .

(3)

Both oscillating terms have zero average, leaving

f (a) = 2 i

. Away from the centre the weights change: on the unit circle,

f (w) = \frac{1}{2 π} \int_{0}^{2 π} \frac{f ( e ^{i θ} )}{1 - w e ^{- i θ}} d θ, ∣ w ∣ < 1.

(4)

With

f (z) = z^{2}

and

w = 1/2

, expanding the denominator as a geometric series leaves only its second-power term after integration, giving

f (1/2) = 1/4

. This also explains how the Cauchy integral formula recovers an entire Taylor series from boundary data.

When a holomorphic function has isolated singularities, a different boundary calculation detects their residues. Suppose

Γ

is a counterclockwise simple closed contour and

f

is holomorphic on a neighbourhood of it and its interior except at finitely many interior isolated singularities

a_{j}

. The residue theorem gives

\oint_{Γ} f (z) d z = 2 πi \sum_{j} Res (f, a_{j}) .

(5)

A residue is the coefficient of

(z - a_{j})^{- 1}

in the Laurent series. A short proof idea is to deform the contour to small circles around the isolated singularities using the Cauchy integral theorem. Integrating the Laurent series on each circle kills every power except

- 1

. ^[ref]

Here is how a complex pole determines a real improper integral. Set

g (z) = 1/ (1 + z^{2})

and use the contour consisting of the segment from

- R

R

, followed by the upper semicircle

A_{R}

, with

R > 1

. Only the simple pole

i

lies inside, and

Res (g, i) = lim_{z \to i} \frac{z - i}{( z - i ) ( z + i )} = \frac{1}{2 i}, \int_{- R}^{R} \frac{d x}{1 + x ^{2}} + \int_{A_{R}} g (z) d z = π .

(6)

The arc must actually disappear. On

A_{R}

, the reverse triangle inequality gives

∣1 + z^{2} ∣ \geq R^{2} - 1

, and its length is

π R

, so

\int_{A_{R}} g (z) d z \leq \frac{π R}{R ^{2} - 1} ⟶ 0.

(7)

The real integrand is bounded by

x^{- 2}

for

∣ x ∣ \geq 1

, so both tails converge separately. Taking the limit therefore proves the ordinary improper integral

\int_{- \infty}^{\infty} \frac{d x}{1 + x ^{2}} = π .

(8)

For example, at

R = 10

the real segment contributes

2 arctan (10) \approx 2.942255

, and the arc contributes

0.199337

, below the bound

10 π /99 \approx 0.317333

. The usefulness of the residue theorem is that the same strategy works for many rational functions whose real antiderivatives are cumbersome: find the enclosed poles, compute their residues, and control the connecting arc. ^[ref]

Historically, Cauchy's 1825 memoir on integrals with imaginary limits develops complex contour integration and its use in transforming and evaluating definite integrals. The opening sections define complex path integrals, discuss changing the path, and examine what happens near singular points. These are recognizable ancestors of the modern link between holomorphic functions, contours and residues. ^[ref]

Table of contents

Complex singularity

 0  0

A point where a specified complex function fails to be holomorphic, but which is approached by points of its domain. An isolated singularity has a punctured neighbourhood on which the function is holomorphic; a non-isolated singularity has other singularities arbitrarily close to it. The classification as a removable singularity, a pole, or an essential singularity applies to isolated singularities.

Complex integration

 0  0

Integration of a function along a path in the complex plane, expressed through a contour integral.

Power maps of a reciprocal-coordinate hyperbola ( $z = t + ia / t, Re (z^{4}) = [Im (z^{4})]^{2} / (16 a^{2}) - 4 a^{2}$ )

 0  0

Let

a > 0

t > 0

and

z = t + ia / t

. The real and imaginary coordinates obey

X Y = a

, a hyperbola. Put

s = t^{2} - a^{2} / t^{2}

, which is strictly increasing. Then

z^{2} = s + 2 ia

traces a horizontal line, and

z^{4} = s^{2} - 4 a^{2} + 4 ia s

traces a parabola. Eliminating

s

gives the displayed quadratic relation between real part and imaginary part. The value of

t

fixes an orientation and determines which segment is traversed; the full real parabola is obtained as

t

ranges over

(0, \infty)

Nontangential limit ( $∠ lim_{z \to 0} f (z)$ )

 0  0

A boundary limit taken while approaching within a cone that stays away from tangency. At zero in the complex upper half-plane, every fixed cone

∣ Re z ∣ \leq A Im z

is allowed. A nontangential limit need not be a limit along all paths approaching the boundary.

// Destination: analysis.bigb

Stolz region ( $Γ_{A} (ξ) = {z \in D : ∣ z - ξ ∣ < A (1 - ∣ z ∣)}, A > 1$ )

 0  0

A fixed-aperture approach region at a boundary point of the unit disc. Its points approach the boundary point without becoming tangent to the circle. Convergence inside every such fixed region is a nontangential limit; the corresponding supremum defines a non-tangential maximal function.

Polynomial hull ( $K = {z : ∣ p (z) ∣ \leq sup_{K} ∣ p ∣ for every polynomial p}$ )

 0  0

For a compact set

K \subset C^{n}

, its polynomial hull is the set displayed, where

p

ranges over all complex multivariate polynomials. It contains

K

. In one complex dimension, bounded complementary components are filled in: the maximum modulus principle bounds every polynomial inside those components by its boundary values. Polynomial approximation and separation distinguish points outside the hull.

Polynomial hull in one complex variable ( $K$ )

 0  0

For a compact set

K \subset C

, its polynomial hull is

K = {z : ∣ P (z) ∣ \leq sup_{K} ∣ P ∣ for every polynomial P}

. In one complex variable this is

K

together with all bounded components of its complement. The maximum modulus principle proves inclusion of each filled hole. Conversely, polynomial approximation of an exterior pole by the Runge approximation theorem separates an exterior point from the filled compact set. The filled set has connected complement, which makes it the natural set for polynomial approximation of functions whose poles lie in the unbounded complementary component.

Lower half-plane ( ${z : Im z < 0}$ )

 0  0

The lower half-plane is the set of complex numbers with negative imaginary part. It is the reflection of the complex upper half-plane across the real axis.

Holomorphic square root ( $g^{2} = f$ )

 0  0

A holomorphic square root of a nonvanishing holomorphic function

f

is a holomorphic function

g

satisfying

g^{2} = f

. It exists locally. On a connected open domain it exists globally exactly when the winding number of

f \circ γ

around zero is even for every closed path

γ

in the domain. Continuing a local square root once along such a path multiplies it by

(- 1)^{m}

, where

m

is that winding number. Thus the obstruction concerns parity, whereas a holomorphic logarithm needs every such winding number to be zero.

C ∖ {0}

, the function

f (z) = z

has no global holomorphic square root: its value winds once on the unit circle, and the lift shown under monodromy changes sign. In contrast,

f (z) = z^{2}

winds twice there and has global square roots

g (z) = z

and

g (z) = - z

. It still has no holomorphic logarithm on that domain. This distinguishes the two global continuation conditions.

Square root of z squared minus one ( $f (z) = z^{2} - 1$ )

 0  0

The branch points are

\pm 1

; infinity has two unbranched meromorphic choices. On the complex plane slit along

[- 1, 1]

, normalization

f (z) \sim z

at infinity selects a holomorphic square root with boundary values

f (x + i 0) = i 1 - x^{2}

and

f (x - i 0) = - i 1 - x^{2}

for

- 1 < x < 1

. On the exterior real intervals,

f (x) = sgn (x) x^{2} - 1

. A counterclockwise contour integral around the finite branch cut satisfies

\oint d ζ / f (ζ) = 2 πi

: the upper lip runs right to left, the lower lip left to right, and the endpoint arcs contribute

O (ε)

. The opposite global branch reverses these signs.

Holomorphic square root outside all polynomial roots

 0  0

An even-degree polynomial with all roots in

∣ z ∣ < R

admits a holomorphic square root throughout

∣ z ∣ > R

. Factor out its even power of

z

and use the convergent logarithm of each

1 - a_{j} / z

to construct it. Even degree removes the annular winding obstruction. The two choices differ by a global sign.

Monodromy

 1  0

Monodromy records how a germ of a holomorphic function changes after analytic continuation around a closed path. The source point returns to itself, but the continued germ of a holomorphic function need not. For

z (t) = e^{i t}

0 \leq t \leq 2 π

, starting with holomorphic square root value

1

gives

w (t) = e^{i t /2}

and hence

w (2 π) = - 1

; starting with complex logarithm value

0

gives

L (t) = i t

and hence

L (2 π) = 2 πi

. After

m

counterclockwise turns the values are

(- 1)^{m}

and

2 πim

. Thus monodromy exchanges the two square-root branches of a multivalued function, whereas it shifts through infinitely many logarithm branches of a multivalued function. ^[ref]

Figure 1.
One closed loop in the source plane lifts to an open semicircle for the square root and an open vertical segment for the logarithm
. **Details**.

The three panels use the same parameter

t

. They show analytic continuation through local holomorphic patches, including across a chosen branch cut; repeatedly evaluating the principal logarithm would instead reset its imaginary part at the cut. Monodromy makes the obstruction to a global branch of a multivalued function visible and motivates the use of Riemann surfaces as domains on which continued values stay distinct.

Isomonodromic deformation ( $A_{z} - B_{λ} + [A, B] = 0$ )

 0  0

A parameter deformation preserves generalized monodromy data, including Stokes matrices and formal monodromy at irregular points. If normalized sectorial matrices have parameter-independent jumps,

B = Y_{z} Y^{- 1}

agrees across overlaps. Its finite singularities and growth at infinity determine it by complex analytic arguments. Compatibility of

Y_{λ} = A Y

and

Y_{z} = B Y

is precisely the displayed zero-curvature identity, obtained by differentiating both equations and subtracting.

Formal monodromy ( $M_{f} = e^{2 πi Θ}$ )

 0  0

An unramified formal fundamental matrix

Y = G (λ) λ^{Θ} e^{Q (λ)}

acquires the right factor

e^{2 πi Θ}

on a positive turn of the logarithm branch. This is formal monodromy. It can be nontrivial even when the true equation has an entire fundamental matrix and trivial actual monodromy, because Stokes matrices compensate for it.

Formal monodromy correction to an entire-system Stokes product ( $S_{N} \dots S_{1} = M_{f}^{- 1}$ )

 0  0

When Stokes factors act on solution coefficient vectors, a full positive turn gives their ordered product equal to the inverse of formal monodromy for a polynomial system with no finite singularities. The true fundamental matrix is entire, so its actual monodromy is trivial, but its normalized formal branch changes by

M_{f}

. Indeed, write

Y_{j + 1} = Y_{j} S_{j}^{- 1}

; then

Y_{N + 1} = Y_{1} (S_{N} \dots S_{1})^{- 1}

, whereas normalized continuation after the positive turn gives

Y_{N + 1} = Y_{1} M_{f}

. Cancelling the invertible fundamental matrix proves the identity. The uncorrected product is identity only when

M_{f} = I

. For a cubic-exponential two-by-two system with

U = 1, V = - 1, P = 1, R = 2, d = 3/2

, the formal exponent is

θ = 1/2

, so

M_{f} = - I

and the Stokes product is

- I

Monodromy eigenfunction Laurent representation

 0  0

For a linear ordinary differential equation with holomorphic coefficients on a punctured disc, analytic continuation of a basis gives an invertible monodromy matrix. A nonzero eigenvector with eigenvalue

λ = e^{2 πiσ}

selects a solution

w

for which

z^{- σ} w

is single-valued and holomorphic. Its Laurent series gives

w = z^{σ} \sum_{n \in Z} c_{n} z^{n}

. No regular-singular hypothesis is needed; infinitely many negative powers may occur.

Cayley transform between the half-plane and disk ( $C (z) = (z - i) / (z + i)$ )

 0  0

The Cayley transform gives a conformal bijection between an unbounded half-plane and the unit disc. Two conventions are useful. For the right half-plane, use

R (z) = \frac{z - 1}{z + 1}, R^{- 1} (w) = \frac{1 + w}{1 - w}, 1 - ∣ R (z) ∣^{2} = \frac{4 Re z}{∣ z + 1 ∣ ^{2}} .

(1)

For the upper half-plane, use

C (z) = \frac{z - i}{z + i}, C^{- 1} (w) = i \frac{1 + w}{1 - w}, 1 - ∣ C (z) ∣^{2} = \frac{4 Im z}{∣ z + i ∣ ^{2}} .

(2)

Rotating the upper half-plane onto the right half-plane by

z \mapsto - i z

explains the relationship

C (z) = R (- i z)

. The factors of

i

distinguish which half-plane is mapped onto the unit disc. ^[ref]

Figure 1.
The upper half-plane Cayley transform bends a rectangular grid into the unit disc while preserving angles at intersections
. **Details**.

In particular,

C (i) = 0

C (2 i) = 1/3

and

C (1 + i) = (1 - 2 i) /5

. The whole positive imaginary axis becomes the real diameter:

C (i y) = (y - 1) / (y + 1)

for

y > 0

. Horizontal grid lines become circles tangent to the unit circle at

1

when completed by their endpoint at infinity; they are not hyperbolic geodesics.

On the Riemann sphere,

C

maps the extended real line onto the unit circle:

C (- 1) = i, C (0) = - 1, C (1) = - i, C (\infty) = 1.

(3)

Every finite real

x

has

∣ C (x) ∣ = 1

; the remaining point

w = 1

corresponds to infinity. The pole

z = - i

is outside the upper half-plane, and the pole of

C^{- 1}

w = 1

is on the boundary of the unit disc. Neither pole obstructs the interior conformal bijection.

For a short verification, subtract the squared moduli:

∣ z + i ∣^{2} - ∣ z - i ∣^{2} = 4 Im z, Im C^{- 1} (w) = \frac{1 - ∣ w ∣ ^{2}}{∣1 - w ∣ ^{2}} .

(4)

These identities put each interior image in the required domain; solving

w (z + i) = z - i

gives the stated inverse function. Both maps are holomorphic, with

C^{'} (z) = 2 i / (z + i)^{2} \neq = 0

, so the conformal bijection preserves Euclidean angles between tangent directions.

Euclidean distance is not preserved:

i

and

2 i

are one unit apart, whereas

0

and

1/3

are only one third of a unit apart. The striking feature is that

C

does preserve length for the hyperbolic metrics of the Poincaré half-plane model and Poincaré disk model:

d s_{H} = \frac{∣ d z ∣}{Im z}, d s_{D} = \frac{2∣ d w ∣}{1 - ∣ w ∣ ^{2}}, \frac{2∣ C ^{'} ( z ) ∣ ∣ d z ∣}{1 - ∣ C ( z ) ∣ ^{2}} = \frac{∣ d z ∣}{Im z} .

(5)

Consequently it is an isometry for these hyperbolic metrics, including their induced hyperbolic distances. For the same example, the corresponding hyperbolic geodesic segments have equal length:

\int_{1}^{2} \frac{d y}{y} = lo g 2 = \int_{0}^{1/3} \frac{2 d r}{1 - r ^{2}} .

(6)

Thus the Cayley transform lets us choose the most convenient picture: a straight boundary for calculations, or a bounded unit disc for visualization. It also transports automorphisms of the unit disk into upper half-plane automorphisms by function composition

C^{- 1} \circ g \circ C

Right half-plane ( ${z \in C : Re z > 0}$ )

 1  0

The right half-plane consists of complex numbers with positive real part. Its boundary is the imaginary axis. For example,

1 + i

is inside,

i

is on the boundary, and

- 1 + i

is outside. Rotation

z \mapsto i z

takes the right half-plane onto the upper half-plane; the Cayley transform between the half-plane and disk uses

R (z) = (z - 1) / (z + 1)

for this convention.

Holomorphic logarithm ( $e^{g} = f, g^{'} = f^{'} / f$ )

 0  0

A holomorphic logarithm of a nowhere-zero holomorphic function

f

on a domain

U

is a holomorphic function

g

such that

e^{g (z)} = f (z)

throughout

U

. Differentiating with the chain rule gives

g^{'} = f^{'} / f

. For

f (z) = z

, a holomorphic logarithm is a branch of the complex logarithm, and its derivative is

1/ z

The principal logarithm is holomorphic on the slit complex plane

D = C ∖ (- \infty, 0]

, with

Log z = lo g ∣ z ∣ + i Arg z, - π < Arg z < π, (Log z)^{'} = \frac{1}{z} .

(1)

Here

Arg

is the chosen complex argument, and

Im Log z \in (- π, π)

. One may conventionally assign a principal value on the negative real axis, but that axis is excluded from this holomorphic domain. ^[ref] For example,

1 + i = 2 e^{iπ /4}

gives

Log (1 + i) = \frac{1}{2} lo g 2 + iπ /4

and

(Log)^{'} (1 + i) = (1 - i) /2

; all its other complex logarithms differ by

2 πik

k \in Z

There is no holomorphic logarithm of

z

on the punctured complex plane

C ∖ {0}

. Such a function would be a holomorphic primitive of

1/ z

, so its contour integral around every closed path would vanish. On the counterclockwise unit circle, however,

\oint_{∣ z ∣ = 1} \frac{d z}{z} = \int_{0}^{2 π} \frac{i e ^{i t}}{e ^{i t}} d t = 2 πi \neq = 0.

(2)

This is the period obstruction to a holomorphic antiderivative, and also exactly the logarithm change in the monodromy diagram.

Every nowhere-zero holomorphic function

f

on a simply connected domain

U

does admit a holomorphic logarithm. Choose

a \in U

and

b \in C

with

e^{b} = f (a)

, and define

g (z) = b + \int_{a}^{z} \frac{f ^{'} ( ζ )}{f ( ζ )} d ζ .

(3)

The Cauchy integral theorem makes this contour integral independent of the path in

U

. Then

g^{'} = f^{'} / f

and

(f e^{- g})^{'} = e^{- g} (f^{'} - f g^{'}) = 0

; its value at

a

1

, proving

e^{g} = f

. Any two choices on a connected domain differ by a constant

2 πik

, since their difference is continuous and takes values in the discrete set

2 πi Z

. ^[ref]

For a computed example, take

f (z) = 1 + z

on the unit disc, with

a = b = 0

. Expanding

f^{'} / f = 1/ (1 + z)

as a geometric series and integrating gives

g (z) = \sum_{n = 1}^{\infty} \frac{( - 1 ) ^{n - 1} z ^{n}}{n}, g (1/2) = lo g (3/2), e^{g (1/2)} = 3/2.

(4)

The construction also produces holomorphic square roots:

h = e^{g /2}

satisfies

h^{2} = f

, so here

h (1/2) = 3/2

. More generally,

e^{αg}

gives a consistent branch of a multivalued function for complex exponentiation by

α

. A holomorphic logarithm converts multiplicative expressions into additive ones while making the domain restrictions explicit.

Upper half-plane (complex analysis, $H$ )

 0  0

The upper half-plane is the open set

H = {z = x + i y \in C : y > 0} .

(1)

It contains precisely the complex numbers with positive imaginary part:

1 + i

is inside,

1

lies on its boundary, and

1 - i

belongs to the lower half-plane. In the complex plane its boundary is the real line; in the Riemann sphere the boundary also includes infinity. It is a convex set, hence a simply connected domain, and the Cayley transform between the half-plane and disk makes it biholomorphic to the unit disc.

Figure 1.
Boundary points and hyperbolic geodesics correspond under the Cayley transform from the upper half-plane to the unit disc
. **Details**.

Matching colours identify each hyperbolic geodesic and its image. The vertical line through zero maps to a diameter; the vertical line through one maps to an arc ending at

- i

and

1

. A semicircle from

- 1

1

also maps to a diameter, whereas the semicircle from

0

2

maps to a curved arc. Their endpoints lie on the boundary, outside the domains: the endpoints are infinitely far from interior points in hyperbolic distance.

The upper half-plane is useful because its straight boundary and explicit upper half-plane automorphisms make conformal maps and hyperbolic geometry calculable. The picture explains why shortest hyperbolic geodesic routes can look curved in Euclidean coordinates; their lengths are measured with the position-dependent hyperbolic metric.

For historical context, in his letter to Klein of 22 September 1882, Poincaré constructs a quadrilateral bounded by four circles orthogonal to a fundamental circle, and studies transformations generating repeated copies. In modern language, the sides inside that circle are hyperbolic geodesics in a disk model. The letter directly exhibits the connection between transformation groups, geometry and complex functions. ^[ref]

Conformal automorphism of the upper half-plane ( $ϕ (z) = (a z + b) / (cz + d)$ )

 0  0

Every conformal automorphism of the upper half-plane is a Möbius transformation with real coefficients and positive determinant, and conversely every such map is a conformal bijection of the upper half-plane to itself:

f (z) = \frac{a z + b}{cz + d}, a, b, c, d \in R, Δ = a d - b c > 0.

(1)

The real-coefficient and sign conditions matter. The complex Möbius transformation

z \mapsto i z

sends

i

- 1

on the boundary, while the real Möbius transformation

z \mapsto 1/ z

has negative determinant and sends

i

- i

in the lower half-plane. ^[ref]

For two concrete upper half-plane automorphisms, take

T (z) = z + 1, A_{T} = (1011), S (z) = - \frac{1}{z}, A_{S} = (01 - 1 0) .

(2)

Both matrices have determinant one. Here

T (i) = 1 + i

S (i) = i

, and

S (1 + i) = (- 1 + i) /2

. On the extended real line,

T

fixes infinity and

S

exchanges zero and infinity. On the unit disc, the same maps become

C \circ T \circ C^{- 1}

and

C \circ S \circ C^{- 1}

; the second is simply

w \mapsto - w

Multiplying all four coefficients by a nonzero real scalar leaves

f

unchanged. Dividing them by

Δ

normalizes the determinant to one; the two remaining representatives

A

and

- A

give the same map. Thus the group under function composition is the projective special linear group

Aut (H) ≅ PS L_{2} (R) = S L_{2} (R) / {I, - I},

(3)

where

S L_{2} (R)

is the real special linear group of degree two. Composition of the Möbius transformations corresponds to multiplication of their matrices. ^[ref]

On the boundary,

f^{'} (x) = \frac{Δ}{( c x + d ) ^{2}} > 0 (c x + d \neq = 0) .

(4)

f

preserves the cyclic boundary orientation of the extended real line; when

c \neq = 0

, its pole at

- d / c

prevents treating it as a single increasing function across that point on the ordinary real line. Three distinct boundary images determine a Möbius transformation uniquely, but an upper half-plane automorphism can realize only triples with matching cyclic boundary orientation.

A brief proof outline starts with the identities

Im f (z) = \frac{Δ Im z}{∣ cz + d ∣ ^{2}}, f^{- 1} (w) = \frac{d w - b}{- c w + a} .

(5)

They show that

f

and its inverse function preserve the upper half-plane. For completeness of the classification, conjugate an arbitrary conformal bijection by the Cayley transform. Normalize its image of zero with an automorphism of the unit disk, then apply the Schwarz lemma to the resulting map and its inverse function; both inequalities force a rotation. Conjugating back gives the real Möbius transformation above.

Finally,

∣ f^{'} (z) ∣ = Δ/∣ cz + d ∣^{2}

and the displayed imaginary part identity show that

∣ df (z) ∣/ Im f (z) = ∣ d z ∣/ Im z

. Every upper half-plane automorphism is therefore also an isometry of the Poincaré half-plane model. The same formulas describe both holomorphic coordinate symmetries and motions of the hyperbolic plane.

Polylogarithm ( $Li_{s} (z)$ )

 1  0

For

∣ z ∣ < 1

, the polylogarithm is

Li_{s} (z) = \sum_{n \geq 1} z^{n} / n^{s}

; contour formulas analytically continue it to a slit plane.

Polylogarithm series remainder bound ( $∣ R_{N} ∣ \leq ∣ z ∣^{N + 1} / [(N + 1)^{a} (1 - ∣ z ∣)]$ )

 0  0

For

∣ z ∣ < 1

a \geq 0

and integer

N \geq 0

, truncating the defining polylogarithm power series after

N

terms has remainder

\sum_{k > N} \frac{z ^{k}}{k ^{a}} \leq \sum_{k > N} \frac{∣ z ∣ ^{k}}{( N + 1 ) ^{a}} = \frac{∣ z ∣ ^{N + 1}}{( N + 1 ) ^{a} ( 1 - ∣ z ∣ )} .

(1)

The inequality uses monotonicity of

k^{a}

and then the geometric series. The bound applies to negative real

z

as well as positive real

z

, and gives a rigorous absolute truncation tolerance independently of cancellation. It does not bound floating-point rounding error.

Dilogarithm ( $Li_{2} (z)$ )

 0  0

The dilogarithm is the order-two polylogarithm, defined for

∣ z ∣ < 1

Li_{2} (z) = \sum_{n \geq 1} z^{n} / n^{2}

. Its derivative is

- Log (1 - z) / z

, with a removable value at zero; a chosen complex logarithm branch gives its analytic continuation.

Holomorphic function ( $f \in O (U)$ )

 1  0

A complex-valued function

f : U \to C

is holomorphic on an open set

U

when it has a complex derivative at every point of

U

. Requiring a derivative throughout a neighbourhood is much stronger than complex differentiability at a point. It forces locally convergent Taylor series and derivatives of every order, as follows from the Cauchy integral formula and Cauchy derivative formula.

To see the restriction in real coordinates, write

z = x + i y

and

f (z) = u (x, y) + i v (x, y)

. The real and imaginary increment quotients must agree, giving the Cauchy-Riemann equations

u_{x} = v_{y}, u_{y} = - v_{x}, f^{'} = u_{x} + i v_{x} .

(1)

Conversely, if

u, v

have continuous partial derivatives on

U

and satisfy the Cauchy-Riemann equations there, then

f

is a holomorphic function. The equations at a lone point, without a differentiability hypothesis for the real coordinate map, are insufficient. Their role is to force the Jacobian matrix to have the special form

D f = (u_{x} v_{x} u_{y} v_{y}) = (p q - q p), f^{'} = p + i q .

(2)

Such a matrix represents multiplication by a complex number; it has only two free real entries. ^[ref]

For

f (z) = z^{2}

u = x^{2} - y^{2}

and

v = 2 x y

, so

u_{x} = v_{y} = 2 x

and

u_{y} = - v_{x} = - 2 y

. Directly,

f (z_{0} + h) - f (z_{0}) = 2 z_{0} h + h^{2}, f^{'} (z_{0}) = 2 z_{0} .

(3)

z_{0} = 1 + i

, multiplication by

2 + 2 i

scales lengths by its modulus

22

and rotates by its argument

π /4

. A horizontal increment

h = 0.01

gives the exact displacement

0.0201 + 0.02 i

; a vertical increment

h = 0.01 i

gives

- 0.0201 + 0.02 i

. The first-order predictions

0.02 + 0.02 i

and

- 0.02 + 0.02 i

are perpendicular, and the errors have modulus

1 0^{- 4}

At every nonzero

z_{0}

, this local rotation and scaling makes

z^{2}

a local conformal map. At zero, the first-order map vanishes:

r e^{i θ}

becomes

r^{2} e^{2 i θ}

, so the argument doubles and the origin is a critical point. Globally

z

and

- z

have the same square. Local preservation of angles therefore coexists with global folding, a useful warning when using holomorphic functions as coordinate changes.

Weakly holomorphic Banach-valued function ( $φ \circ f holomorphic for every φ \in X^{'}$ )

 0  0

A map from an open subset of

C

into a complex Banach space is weakly holomorphic when every composition with a bounded complex-linear functional is a scalar holomorphic function. Norm holomorphy requires its difference quotients to converge in the Banach norm. Weak holomorphy is initially a condition on scalar observations; the weak holomorphy implies norm holomorphy argument recovers an actual vector derivative. A closed target subspace containing values on a sequence with an interior accumulation point contains every value on the connected domain: annihilate the subspace, apply the scalar identity theorem, and use Hahn-Banach separation.

Weak holomorphy implies norm holomorphy

 0  0

For a weakly holomorphic Banach-valued function, choose a closed disk contained in its domain. Each continuous functional is bounded on the image of that disk by scalar continuity. The Uniform boundedness principle, applied to its evaluation functionals in the bidual, gives a common bound

M

on the norms of the image vectors. Cauchy's formula on the disk's boundary then bounds the scalar difference between difference quotients at increments

h, k

by a constant times

M ∥ φ ∥∣ h - k ∣

, uniformly for sufficiently small increments. Explicitly the difference kernel is

\frac{h - k}{( ζ - z ) ( ζ - z - h ) ( ζ - z - k )} .

(1)

The Hahn-Banach theorem dual norm formula converts this into the same norm bound for the vector difference quotients. They are Cauchy, so completeness gives a norm derivative. This also explains why weak holomorphy suffices without a separately assumed norm-continuity hypothesis.

Holomorphic extension ( $F ∣_{U} = f, F \in O (V), U \subseteq V$ )

 0  0

A holomorphic function

F

on a larger open set

V

whose restriction to

U

is the original holomorphic function

f

. A removable singularity is a point across which such an extension exists. On a connected larger domain, a holomorphic extension is unique by the identity theorem, provided

U

is nonempty.

Holomorphic surjection from the unit disc onto the complex plane ( $F (z) = z^{2} / (1 - z)^{2}$ )

 0  0

The rational function

F (z) = z^{2} / (1 - z)^{2}

is holomorphic on the unit disc and maps it onto the entire complex plane. For any

w \in C

, choose a square root

r

with

Re r \geq 0

and put

z = r / (1 + r)

. The identity

∣1 + r ∣^{2} - ∣ r ∣^{2} = 1 + 2 Re r > 0

gives

∣ z ∣ < 1

, while

F (z) = r^{2} = w

. It is not injective: the two choices

r = \pm i

give distinct points in the disc with image

- 1

A holomorphic bijection in this direction is impossible: its inverse would be a bounded entire function, contradicting the Liouville theorem. Thus a surjection onto the complex plane can exist even when a conformal equivalence cannot.

Conjugate reflection of a holomorphic function ( $f^{*} (z) = \overline{f (\overline{z})}$ )

 0  0

f

is holomorphic on an open set

U

, its conjugate reflection

f^{*}

is holomorphic on

U^{*} = {z : \overline{z} \in U}

, with

(f^{*})^{'} (z) = \overline{f^{'} (\overline{z})}

. The identity follows by conjugating the difference quotient with increment

\overline{h}

; the two conjugations restore complex differentiability.

U

is invariant under complex conjugation and is connected, and

f

is real at a set of real points accumulating inside

U

, then

f = f^{*}

by the identity theorem. In particular, an entire function real at every

1/ n

has this symmetry. This is a symmetry statement on an already defined domain; Schwarz reflection also constructs extensions across a boundary.

Complex differentiability ( $f^{'} (z) = lim_{h \to 0} (f (z + h) - f (z)) / h$ )

 0  0

A complex-valued function is complex differentiable at

z_{0}

f^{'} (z_{0}) = lim_{h \to 0 h \in C ∖ {0}} \frac{f ( z _{0} + h ) - f ( z _{0} )}{h}

(1)

exists. The same limit must hold for every approach, including curved paths. Equivalently,

f (z_{0} + h) = f (z_{0}) + f^{'} (z_{0}) h + o (∣ h ∣) .

(2)

The error divided by

∣ h ∣

tends to zero. Thus the derivative is a single complex multiplier, not an arbitrary real linear map of the plane. If the real coordinate map is differentiable, complex differentiability is equivalent to its satisfying the Cauchy-Riemann equations at that point.

The simplest failure is complex conjugation,

f (z) = \overline{z}

. At every

z_{0}

its increment quotient is

\overline{h} / h

. For real

h = t \neq = 0

it is

1

, while for

h = i t

it is

- 1

. More generally,

h = t e^{i θ}

gives

e^{- 2 i θ}

. Although complex conjugation is a smooth real reflection and has every real directional derivative, these incompatible quotients rule out a complex derivative everywhere.

Agreement on two axes is necessary, but does not by itself establish a limit over the whole complex plane. The Cauchy-Riemann equations need the real differentiability condition above, or a sufficient regularity condition such as continuous partial derivatives near the point. There is a second distinction:

f (z) = ∣ z ∣^{2}

has complex derivative zero at zero because its quotient there is

\overline{h} \to 0

, but its Cauchy-Riemann equations fail at every nonzero point. It is complex differentiable at zero without being a holomorphic function on any neighbourhood of zero.

Complex differentiability at a point without holomorphy nearby ( $f (z) = z / (1 + ∣ z ∣)$ )

 0  0

At zero,

[f (h) - f (0)] / h \to 1

, so the complex derivative exists. At a nonzero point, write

r = ∣ z ∣

. The Cauchy-Riemann equations for

x / (1 + r)

and

y / (1 + r)

force

x^{2} = y^{2}

and

x y = 0

, impossible when

r > 0

. Hence the function is complex differentiable only at zero and is holomorphic on no nonempty open subset. A complex derivative at one point gives no neighborhood of complex differentiability.

Nevanlinna class ( $N (D) = {f : sup_{r < 1} \int lo g^{+} ∣ f (r ξ) ∣ d m (ξ) < \infty}$ )

 0  0

The analytic Nevanlinna class consists of holomorphic functions on the unit disc with uniformly bounded means of their positive logarithmic modulus. Here

m

is normalized angular measure. A bounded holomorphic function belongs to this class, but the class also permits unbounded functions. It is the analytic version of bounded characteristic.

Bounded characteristic

 0  0

A holomorphic function on the unit disc has bounded characteristic when

sup_{r < 1} \int lo g^{+} ∣ f (r ξ) ∣ d m (ξ) < \infty

, where

lo g^{+} t = max (0, lo g t)

. Thus it belongs to the Nevanlinna class. A general meromorphic function requires the additional pole-counting term in its characteristic; the displayed analytic definition has no such term.

Quotient characterization of bounded characteristic ( $f = u / v, u, v \in H^{\infty} (D), v \neq \equiv 0$ )

 0  0

A meromorphic function on the unit disc has bounded characteristic if and only if it has this representation. Its pole Blaschke product

B

makes

g = B f

analytic with bounded positive logarithmic means. The least harmonic majorant by expanding disk lifts gives

H \geq lo g^{+} ∣ g ∣

. A harmonic conjugate gives an analytic

A

with

Re A = H

, so

u = g e^{- A}

and

v = B e^{- A}

are bounded. Conversely the Nevanlinna first main theorem bounds the characteristic of

u / v

by that of

u

and

1/ v

, hence by a constant.

Multiplicity of a zero ( $ord_{z_{0}} f$ )

 0  0

A holomorphic function that is not identically zero has a zero of multiplicity

m

z_{0}

f (z) = (z - z_{0})^{m} h (z)

for a holomorphic function

h

with

h (z_{0}) \neq = 0

. Its Taylor series identifies

m

as the first nonzero coefficient index. Logarithmic differentiation gives

f^{'} / f = m / (z - z_{0}) + h^{'} / h

, so the residue of

f^{'} / f

m

. Counting zeros with zero multiplicity repeats that zero

m

times.

Periodic holomorphic descent through the exponential map

 0  0

If a holomorphic function of

w

is periodic with period one, then

F ((lo g z) / (2 πi))

is independent of the choice of a local logarithm: two choices differ by

2 πik

, hence their arguments differ by an integer. Consequently it defines a holomorphic function on the image of the original domain under

w \mapsto e^{2 πi w}

, provided that domain is invariant under integer shifts. Local logarithms prove holomorphy across an arbitrary global branch cut. Singularities on an integer lattice all map to

z = 1

Constant-modulus holomorphic function

 0  0

A holomorphic function with constant modulus on a connected domain is constant. If

h = u + i v

and

u^{2} + v^{2} = c^{2} > 0

, differentiation and the Cauchy-Riemann equations give

u u_{x} + v v_{x} = 0

and

- u v_{x} + v u_{x} = 0

. Their coefficient determinant is

- c^{2}

, so

u_{x} = v_{x} = 0

, and all first derivatives vanish. When

c = 0

, the conclusion is immediate. Consequently two nonvanishing holomorphic functions with equal moduli differ by a single constant phase on a connected domain.

Banach-space-valued holomorphic function ( $F^{'} (z) = lim_{h \to 0} \frac{F ( z + h ) - F ( z )}{h}$ )

 0  0

A map from an open subset of

C

to a complex Banach space is holomorphic when the displayed limit exists in the Banach-space norm at every point. A norm-convergent power series is holomorphic inside its radius of convergence: on every smaller disk the series of derivatives converges uniformly by the geometric coefficient bound, allowing differentiation term by term. Applying a continuous linear functional produces an ordinary scalar holomorphic function. Scalar Cauchy identities therefore imply the corresponding vector-integral identities when bounded linear functionals separate points, as they do by the Hahn-Banach theorem.

Spectral-radius maximum principle ( $r (f (z)) \leq sup_{w \in \partial K} r (f (w))$ )

 0  0

For a Banach algebra-valued holomorphic function, set

p_{n} (w) = ∥ f (w)^{2^{n}} ∥^{1/ 2^{n}}

. These continuous functions decrease to the spectral radius by the spectral radius formula. If

M = sup_{\partial K} r (f (w))

, the continuous functions

max {p_{n}, M}

\partial K

decrease to the continuous constant

M

. The Dini theorem gives uniform convergence. Apply the norm maximum principle for a Banach-space-valued holomorphic function to each power, and then let

n \to \infty

. This proof does not assume continuity of spectral radius.

Norm maximum principle for a Banach-space-valued holomorphic function ( $∥ f (z) ∥ \leq sup_{w \in \partial K} ∥ f (w) ∥$ )

 0  0

For a compact set

K

in the domain, a norming bounded linear functional at

f (z)

reduces the assertion to the scalar maximum modulus principle. If

z

lies inside

K

, use its connected component of the interior; that component's boundary lies in

\partial K

. If

z \in \partial K

, the inequality is immediate. No smoothness or connectedness assumption on

K

is required.

Prescribed zeros in a plane domain via Runge approximation

 0  0

Every closed discrete subset of a plane domain is the zero set of a holomorphic function, with assigned finite positive multiplicities if desired. Exhaust the domain by relatively filled compact sets. For a prescribed point outside an earlier compact, choose a rational factor with that zero and its possible pole outside the domain in the same complementary component. A slit joining zero and pole allows a holomorphic logarithm near the earlier compact. Approximate that logarithm by Runge theorem and multiply the rational factor by the negative exponential of the approximant. The resulting factors tend to one with summable errors on each compact. Their infinite product has the prescribed zeros and no additional zeros.

// Destination: analysis.bigb

Antiholomorphic function ( $\partial_{z} f = 0$ )

 1  0

A complex-valued function is antiholomorphic when its complex conjugate is a holomorphic function. In one complex dimension it locally depends on

\overline{z}

and satisfies

\partial_{z} f = 0

. A nonconstant antiholomorphic map reverses the orientation induced by the complex coordinates. Thus complex conjugating a holomorphic sphere map reverses its topological degree while preserving its Dirichlet energy.

Holomorphic convex hull ( $K_{Ω}$ )

 0  0

For compact

K

in a plane domain

Ω

, its holomorphic convex hull is

K_{Ω} = {w \in Ω : ∣ f (w) ∣ \leq sup_{K} ∣ f ∣ for every f \in O (Ω)} .

(1)

It is

K

together with the components of

C ∖ K

lying entirely in

Ω

. Inclusion of those components follows from the maximum modulus principle. For exclusion, suppose

w

lies in a complementary component meeting

C ∖ Ω

. Choose an allowed pole there. The open-and-closed resolvent argument in the proof of Runge theorem approximates

(z - w)^{- 1}

uniformly on

K

by functions

h

holomorphic on

Ω

. Then

1 - (z - w) h (z)

has value one at

w

and arbitrarily small modulus on

K

, separating

w

from the hull. For

Ω = C

, this is the polynomial hull in one complex variable.

Blaschke product ( $B (z) = λ z^{m} \prod_{n} b_{a_{n}} (z)$ )

 1  0

For zeros

a_{n}

in the unit disc satisfying

\sum_{n} (1 - ∣ a_{n} ∣) < \infty

, with

m

occurrences at zero, a Blaschke product is

B (z) = λ z^{m} \prod_{a_{n} \neq = 0} \frac{∣ a _{n} ∣}{a _{n}} \frac{a _{n} - z}{1 - a _{n} z}, ∣ λ ∣ = 1.

(1)

For

∣ z ∣ \leq R < 1

and

a \neq = 0

, the normalized factor

b_{a}

satisfies

∣1 - b_{a} (z) ∣ \leq (1 - ∣ a ∣) (1 + R) / (1 - R)

. The product therefore converges locally uniformly; outside its prescribed zeros the logarithms of its tail factors converge absolutely, so it has no additional zeros. Every finite product has modulus at most one in the disc, and the limit does too. Since

∣ b_{a} (w) ∣

is pseudohyperbolic distance from

w

a

, its modulus can be estimated geometrically.

Interior zero count for a finite Blaschke product minus a constant ( $N (B - a) = de g B (∣ a ∣ < 1)$ )

 0  0

A finite Blaschke product is holomorphic across the unit circle and has modulus one there. For a constant

a

with

∣ a ∣ < 1

, Rouché's theorem makes

B - a

and

B

have the same number of interior zeros, counting multiplicity. Each factor contributes its prescribed zero; the poles are outside the closed disk.

Hyperbolic zero estimate for a Blaschke product ( $∣ B (w) ∣ \leq exp [- 2 \sum_{n} e^{- ρ (w, a_{n})}]$ )

 0  0

Use curvature-minus-one Hyperbolic distance in the Poincare disc. Each Blaschke factor has modulus

tanh (ρ (w, a) /2) = (1 - t) / (1 + t)

, with

t = e^{- ρ (w, a)}

. For

0 \leq t < 1

, differentiating shows

lo g [(1 + t) / (1 - t)] \geq 2 t

. Sum this logarithmic inequality over the zero factors and pass to the Blaschke product by locally uniform convergence. At a prescribed zero, its modulus vanishes and the bound is immediate. All sums count multiplicities.

Vanishing absolute logarithmic mean characterization of Blaschke products ( $\int ∣ lo g ∣ f (r ξ) ∣∣ d m \to 0 ⟺ f is a Blaschke product$ )

 0  0

For a holomorphic function on the unit disc, this boundary condition characterizes Blaschke products, including unimodular constants. The nondecreasing radial means of the nonnegative subharmonic function

lo g^{+} ∣ f ∣

must vanish, so

∣ f ∣ \leq 1

. In the Blaschke factorization of a bounded holomorphic function

f = B g

, both logarithmic means tend to zero. The mean value property for harmonic functions applied to

lo g ∣ g ∣

gives

∣ g (0) ∣ = 1

, and the maximum modulus principle makes

g

unimodular constant. The converse follows from the radial logarithmic mean of a Blaschke product. Omitting the outer absolute value changes the condition:

e^{z}

has zero mean logarithmic modulus, but is not a Blaschke product.

Blaschke factorization of a bounded holomorphic function ( $f = B g, ∥ g ∥_{\infty} \leq ∥ f ∥_{\infty}$ )

 0  0

Every nonzero bounded holomorphic function on the unit disc has this factorization, with

B

the Blaschke product of its zeros and

g

zero-free and holomorphic. The Blaschke condition follows from Jensen's formula. Dividing by any finite partial zero product gives a holomorphic function; the maximum modulus principle and the finite product's boundary modulus one bound that quotient by

∥ f ∥_{\infty}

. Locally uniform convergence of the partial products gives the same bound for

g

, with removable extensions at their zeros. This factorization alone does not make

g

constant or a full inner-outer factorization.

Radial logarithmic mean of a Blaschke product ( $\int lo g ∣ B (r ξ) ∣ d m = m lo g r + \sum_{n} lo g max (r, ∣ a_{n} ∣)$ )

 0  0

For

B (z) = λ z^{m} \prod_{n} b_{a_{n}} (z)

with nonzero zeros

a_{n}

counted with multiplicity, Jensen's formula gives the displayed identity. The negative logarithms of partial products increase, so the monotone convergence theorem permits termwise integration. The Blaschke condition implies

\sum_{n} - lo g ∣ a_{n} ∣ < \infty

, which dominates the absolute summands. The dominated convergence theorem therefore gives a limit of zero as

r ↑ 1

. Since

∣ B ∣ \leq 1

, the mean of

∣ lo g ∣ B ∣∣

tends to zero as well.

Automorphy character of an orbit Blaschke product ( $B (T z) = χ (T) B (z)$ )

 0  0

Let a group

G

of Möbius transformations of the unit disk have an orbit satisfying the Blaschke condition, counted once per distinct orbit point. For its Blaschke product

B

, invariance of the pseudohyperbolic distance and permutation of the orbit give

∣ B (T z) ∣ = ∣ B (z) ∣

. The ratio

B (T z) / B (z)

extends across its zeros and has modulus one, so the open mapping theorem makes it a constant

χ (T)

. Composition gives a group homomorphism

χ : G \to {λ : ∣ λ ∣ = 1}

. It is essential to compare moduli of the products: two arbitrary bounded holomorphic functions with identical zeros need not have constant quotient.

Blaschke condition ( $\sum_{n} (1 - ∣ a_{n} ∣) < \infty$ )

 0  0

The Blaschke condition for a sequence in the unit disc is

\sum_{n} (1 - ∣ a_{n} ∣) < \infty

. It characterizes the zero sequences, counted with multiplicity, of nonzero bounded holomorphic functions. Sufficiency follows by constructing the Blaschke product. For necessity, Jensen's formula gives

\sum_{∣ a_{n} ∣ < r} lo g (r /∣ a_{n} ∣) \leq lo g ∥ f ∥_{\infty} - lo g ∣ f (0) ∣

when

f (0) \neq = 0

. Letting

r ↑ 1

and using

- lo g ∣ a ∣ \geq 1 - ∣ a ∣

proves the condition. A finite zero at zero is factored out first. Equivalently, for the curvature-minus-one hyperbolic metric,

\sum_{n} e^{- ρ (0, a_{n})} < \infty

, since

e^{- ρ (0, a)} = (1 - ∣ a ∣) / (1 + ∣ a ∣)

Blaschke sequence ( $\sum_{n} (1 - ∣ a_{n} ∣) < \infty$ )

 0  0

A sequence in the unit disc, counted with multiplicity, is a Blaschke sequence when it satisfies the Blaschke condition. The zeros of every nonzero bounded holomorphic function form such a sequence: Jensen's formula bounds

\sum_{∣ a_{n} ∣ < r} lo g (r /∣ a_{n} ∣)

uniformly as

r ↑ 1

, and

1 - ∣ a ∣ \leq - lo g ∣ a ∣

gives the assertion. There are only finitely many zeros at the origin, which can be factored out first.

Jensen's formula ( $lo g ∣ f (0) ∣ + \sum_{∣ a_{j} ∣ < R} lo g (R /∣ a_{j} ∣) = \frac{1}{2 π} \int_{0}^{2 π} lo g ∣ f (R e^{i θ}) ∣ d θ$ )

 0  0

Let

R > 0

and let

f

be a holomorphic function on an open neighbourhood of the closed disk

{z : ∣ z ∣ \leq R}

. Assume

f (0) \neq = 0

and that

f

has no zeros of a function on the boundary circle

∣ z ∣ = R

. List its interior zeros of a function as

a_{1}, \dots, a_{N}

, repeating each according to its multiplicity of a zero. With the natural logarithm, Jensen's formula states

\frac{1}{2 π} \int_{0}^{2 π} lo g ∣ f (R e^{i θ}) ∣ d θ = lo g ∣ f (0) ∣ + \sum_{j = 1}^{N} lo g \frac{R}{∣ a _{j} ∣} .

(1)

The empty sum is zero. This boundary-zero-free version is the holomorphic special case of the general identity in the source.^[ref]

Every interior zero of a function makes a positive contribution; a zero nearer the centre contributes more. Consequently boundary growth relative to

∣ f (0) ∣

limits how many zeros of a function can lie well inside the disk. The Jensen zero-count bound makes this quantitative, while the integrated zero count retains the individual radial weights. The boundary statistic here is the circle log-mean of a holomorphic function; exponentiating gives a circle geometric mean of a holomorphic function.

Circle log-mean of a holomorphic function ( $L_{f} (R) = \frac{1}{2 π} \int_{0}^{2 π} lo g ∣ f (R e^{i θ}) ∣ d θ$ )

 0  0

Under the hypotheses of Jensen's formula, the circle log-mean is the angular average of the natural logarithm of the complex modulus. The factor

1/ (2 π)

normalizes the integral to a mean. It averages values at equally weighted angles on the circle, and Jensen's formula computes it from

∣ f (0) ∣

and the interior zeros of a function.

Circle geometric mean of a holomorphic function ( $G_{f} (R) = exp L_{f} (R)$ )

 0  0

The continuous angular geometric mean of the positive boundary complex modulus is

G_{f} (R) = exp (\frac{1}{2 π} \int_{0}^{2 π} lo g ∣ f (R e^{i θ}) ∣ d θ) = ∣ f (0) ∣ \prod_{j = 1}^{N} \frac{R}{∣ a _{j} ∣} .

(1)

Here the empty product is one. This differs from the angular arithmetic mean

A_{f} (R) = (2 π)^{- 1} \int_{0}^{2 π} ∣ f (R e^{i θ}) ∣ d θ

. In general

G_{f} (R) \leq A_{f} (R)

by the concavity of the logarithm; equality requires constant boundary complex modulus. For a holomorphic function with no zeros of a function in the closed disk, Jensen's formula gives

G_{f} (R) = ∣ f (0) ∣

, even when the boundary complex modulus varies with angle. Thus replacing the mean of a logarithm by the logarithm of an arithmetic mean loses the exact zero identity.

Polynomial circle log-mean ( $L_{p} (R) = lo g ∣ c ∣ + \sum_{j} lo g max (R, ∣ a_{j} ∣)$ )

 0  0

For a polynomial

p (z) = c \prod_{j} (z - a_{j})

with

c \neq = 0

, no polynomial root at zero, and

R > 0

different from every

∣ a_{j} ∣

, Jensen's formula gives the displayed expression. The polynomial roots are repeated according to their multiplicity of a zero: an exterior root contributes

lo g ∣ a_{j} ∣

, whereas an interior root contributes

lo g R

For an explicit example, take the polynomial

f (z) = 3 (z - \frac{1}{4}) (z + \frac{1}{2}), f (0) = - \frac{3}{8} .

(1)

Its two simple zeros are

1/4

and

- 1/2

. The circle log-mean of a holomorphic function is therefore

L_{f} (R) = ⎩ ⎨ ⎧ lo g (3/8), lo g (3 R /2), lo g (3 R^{2}), 0 < R < 1/4, 1/4 < R < 1/2, R > 1/2.

(2)

R = 0.2

the open disc contains no zeros of a function, at

R = 0.4

it contains

1/4

only, and at

R = 0.75

it contains both. These three circles avoid the polynomial roots; the radii

1/4

and

1/2

are excluded from the displayed version.

Figure 1.
The zeros of $3 (z - 1/4) (z + 1/2)$ inside three concentric disks
. The three boundary circles enclose respectively zero, one and two simple zeros. Each zero contributes to Jensen's formula only after the radius exceeds its complex modulus. **Details**.

The corresponding values of the circle log-mean of a holomorphic function and the circle geometric mean of a holomorphic function are

R 0.2 0.4 0.75 n_{f} (R) 012 L_{f} (R) lo g (3/8) \approx - 0.980829 lo g (3/5) \approx - 0.510826 lo g (27/16) \approx 0.523248 G_{f} (R) 3/8 3/5 27/16

(3)

For example, at

R = 0.4

the contribution of the sole interior zero of a function is

lo g (0.4/0.25) = lo g (8/5)

; adding it to

lo g (3/8)

gives

lo g (3/5)

. The second polynomial root still affects

f (0)

, but contributes no additional interior-zero term at this radius.

Figure 2.
Circle log-mean versus log-radius for $3 (z - 1/4) (z + 1/2)$
. The log-radius slope of the circle log-mean is zero, one and two in the three intervals, matching the radial zero-counting function. Open markers exclude the boundary-zero radii $1/4$ and $1/2$ ; filled markers show the three computed examples. **Details**.

Log-radius slope of the circle log-mean ( $\frac{d}{d l o g R} L_{f} (R) = n_{f} (R)$ )

 0  0

Fix a holomorphic function on a neighbourhood of a closed disk, with

f (0) \neq = 0

. On an interval of positive radii whose circles contain no zeros of a function, differentiating Jensen's formula with respect to

x = lo g R

gives

\frac{d}{d x} L_{f} (e^{x}) = n_{f} (e^{x}) .

(1)

Each interior zero of a function contributes slope one for each unit of its multiplicity of a zero. Between consecutive zero moduli,

L_{f} (e^{x})

is an affine function; passing the modulus of a zero of multiplicity of a zero

m

increases the adjacent slopes by

m

. The finite sum

lo g ∣ f (0) ∣ + \sum_{j} max (0, x - lo g ∣ a_{j} ∣)

defines a continuous convex function of

x

whose values agree with the circle log-mean of a holomorphic function at every boundary-zero-free radius. This describes the adjacent line segments without asserting a boundary-singularity extension of the integral.

Radial zero-counting function ( $n_{f} (r) = \sum_{∣ a ∣ < r} ord_{a} f$ )

 0  0

For a nonzero holomorphic function,

n_{f} (r)

counts its zeros of a function in the centred open disc of radius

r

, with multiplicity of a zero. Thus a double zero counts twice, even though it is a single point. On every smaller closed disk in the domain the count is finite. The strict inequality

∣ a ∣ < r

specifies the convention at a zero modulus; elsewhere it agrees with counting in the closed disk. In the polynomial circle log-mean example the count is zero, one and two on the three displayed intervals.

Integrated zero count ( $N_{f} (R) = \int_{0}^{R} \frac{n _{f} ( t )}{t} d t$ )

 0  0

Under the hypotheses of Jensen's formula,

f (0) \neq = 0

ensures that the radial zero-counting function vanishes for sufficiently small positive

t

. Integrating each zero's contribution separately gives

N_{f} (R) = \int_{0}^{R} \frac{n _{f} ( t )}{t} d t = \sum_{∣ a ∣ < R} ord_{a} f \int_{∣ a ∣}^{R} \frac{d t}{t} = \sum_{∣ a ∣ < R} ord_{a} f lo g \frac{R}{∣ a ∣} = L_{f} (R) - lo g ∣ f (0) ∣.

(1)

The values of

n_{f}

at its finitely many jump points do not change the integral. For

0 < r < R

, every zero in the smaller open disc supplies at least

lo g (R / r)

per unit of multiplicity of a zero, so

n_{f} (r) lo g (R / r) \leq N_{f} (R) \leq lo g M_{f} (R) - lo g ∣ f (0) ∣, M_{f} (R) = max_{∣ z ∣ = R} ∣ f (z) ∣.

(2)

The second inequality bounds the circle log-mean of a holomorphic function by the logarithm of the largest boundary complex modulus, recovering the Jensen zero-count bound. An upper bound on boundary growth and a fixed nonzero centre value therefore constrain interior zeros. The factor

lo g (R / r)

also explains why the same estimate becomes weaker as the smaller radius approaches the boundary.

Jensen formula proof outline

 0  0

Divide

f

by its finitely many interior zero factors, with each factor repeated according to its multiplicity of a zero. The resulting holomorphic function

g

has no zeros of a function in the closed disk, so

lo g ∣ g ∣

is a harmonic function and its circle average is

lo g ∣ g (0) ∣

by the mean value property for harmonic functions. For each removed factor

z - a

, with

0 < ∣ a ∣ < R

, the circle average of

lo g ∣ R e^{i θ} - a ∣

lo g R

: factor out

R e^{i θ}

and apply the mean value property for harmonic functions to

lo g ∣1 - (a / R) w ∣

on a neighbourhood of the closed unit disk. Reassembling the factors gives

lo g (R /∣ a ∣)

for each zero, proving Jensen's formula.

Poisson-Jensen formula ( $lo g ∣ f (z) ∣ = Poisson (lo g ∣ f ∣) (z) - \sum m_{a} G_{R} (z, a) + \sum m_{p} G_{R} (z, p)$ )

 0  0

For a meromorphic function with no zero or pole on

∣ w ∣ = R

, set

G_{R} (z, a) = lo g ∣ (R^{2} - \overline{a} z) / (R (z - a)) ∣

. At source points that are neither zeros nor poles, the displayed identity expresses its logarithmic modulus as the Poisson integral of its boundary values, minus its zero Green potentials plus its pole Green potentials. The Poisson kernel is

(R^{2} - ∣ z ∣^{2}) /∣ R e^{i θ} - z ∣^{2}

with normalized angular measure. Subtracting the corresponding logarithmic singularities leaves a harmonic function, so the harmonic Poisson formula proves the identity. At zero it reduces to Jensen's formula.

Annular spherical Jensen formula ( $T (r) = I (a) lo g (1/ r) + N (r; a) + \overline{m} (r; a)$ )

 0  0

For a meromorphic function near

0 < ∣ z ∣ \leq 1

, and a target

a

off its image of the unit circle, let

u_{a} = - lo g k (f, a)

for the chordal metric,

M_{a} (s)

its normalized angular mean, and

I (a) = M_{a}^{'} (1)

. The weighted spherical area

T (r) = (4 π)^{- 1} \int_{r < ∣ z ∣ < 1} lo g (∣ z ∣/ r) (f_{round}^{#})^{2} d A

satisfies the displayed formula, where

N

counts

a

-points with multiplicity and weight

lo g (∣ z ∣/ r)

and

\overline{m} = M_{a} (r) - M_{a} (1)

. Indeed

Δ u_{a} = (f_{round}^{#})^{2} /2 - 2 π \sum de g_{z} (f) δ_{z}

, and Green's second identity with the radial logarithmic weight proves the formula. The angular mean must include

1/ (2 π)

Jensen zero-count bound ( $n_{f} (r) lo g (R / r) \leq lo g M_{f} (R) - lo g ∣ f (0) ∣$ )

 0  0

Every zero in the smaller disk contributes at least

lo g (R / r)

in Jensen's formula. Bounding the outer circle average by its maximum proves the estimate. With

R = 2 r

, an exponential

O (r lo g r)

maximum bound gives an

O (r lo g r)

zero count.

Lattice zeros force quadratic exponential growth ( $lim sup_{R \to \infty} R^{- 2} lo g M_{F} (R) > 0$ )

 0  0

A nonzero entire function vanishing at every nonzero point of the integer lattice has at least a constant times

R^{2}

zeros in the disc of radius

R /2

. Each contributes at least

lo g 2

to Jensen's formula at radius

R

. Therefore the circle mean of

lo g ∣ F ∣

, and hence the logarithm of its maximum modulus, is bounded below by

c R^{2}

for all sufficiently large boundary-zero-free radii. Choosing maximizing points yields

∣ F (z_{j}) ∣ > e^{c^{'} ∣ z_{j} ∣^{2}}

for some

c^{'} > 0

and

∣ z_{j} ∣ \to \infty

. If zero is also a zero, first factor out its finite multiplicity; the additional

m lo g R

term does not weaken the quadratic lower bound.

Holomorphic primitive ( $F^{'} = f$ )

 0  0

A holomorphic primitive of

f

is a holomorphic function

F

with

F^{'} = f

. A holomorphic

f

on a domain has such a primitive exactly when its integrals around all closed piecewise smooth curves vanish. Integrating from a fixed starting point then constructs a path-independent primitive.

Complex differentiability at a point

 0  0

A function is complex differentiable at

p

when

f (p + h) = f (p) + f^{'} (p) h + o (∣ h ∣) .

(1)

f^{'} (p) \neq = 0

, the real derivative is multiplication by a nonzero complex number, hence a rotation and scaling; this is the local source of angle preservation by holomorphic maps.

Complex derivative ( $f^{'} (z) = lim_{h \to 0} (f (z + h) - f (z)) / h$ )

 0  0

The complex derivative exists when this limit has the same finite value for increments

h

approaching zero from every complex direction. For

f = u + i v

, real and imaginary increments give the necessary Cauchy-Riemann equations. Existence of those partial derivatives at one point alone is not sufficient for complex differentiability at a point; continuous partial derivatives in a neighbourhood, together with the Cauchy-Riemann equations there, give a standard sufficient condition for

f

to be a holomorphic function.

Space of holomorphic functions ( $O (U)$ )

 0  0

For an open set

U \subseteq C

, the space

O (U)

consists of all holomorphic functions on

U

. Its standard topology is locally uniform convergence, equivalently the compact-open topology defined by the seminorms

f \mapsto sup_{z \in K} ∣ f (z) ∣

for compact

K \subset U

Weighted holomorphic norm on a shrinking time domain ( $∥ u ∥_{α} = 0 < s < 1, 0 < ∣ t ∣ < α (1 - s), ∣ x ∣ \leq s R sup ∣ u (x, t) ∣ \frac{α ( 1 - s ) - ∣ t ∣}{∣ t ∣}$ )

 0  0

Let

R, α > 0

and

C_{α} = {(x, t) \in C^{2} : ∣ x ∣ < R, ∣ t ∣ < α (1 - ∣ x ∣/ R)}

. The displayed norm makes the holomorphic functions with finite norm a Banach space. Finiteness forces

u (x, 0) = 0

: for fixed

x

, the weight grows like

1/∣ t ∣

near zero. A Cauchy sequence in this norm converges uniformly on every compact subset of

C_{α}

, including at

t = 0

. Its limit is holomorphic by locally uniform convergence of holomorphic functions, and the pointwise norm bounds show convergence in the original norm. The shrinking domain compensates for loss of an

x

-derivative in a Cauchy-Kovalevskaya theorem argument.

Contraction estimate on a shrinking holomorphic domain ( $\int_{0}^{t} \partial_{x} u (x, z) d z_{α} \leq \frac{4 α}{R} ∥ u ∥_{α}$ )

 0  0

Write

N = ∥ u ∥_{α}

r = ∣ t ∣

and

A = α (1 - s) > r

. Along the straight integration segment, choose

σ (τ) = s + (A - τ) / (2 α)

. A Cauchy estimate on a disc of radius

R (σ - s)

bounds the integrand by

4 α N τ / [R (A - τ)^{2}]

. Therefore

\frac{A - r}{r} \int_{0}^{t} \partial_{x} u (x, z) d z \leq \frac{4 α N}{R} [1 + \frac{A - r}{r} lo g (1 - r / A)] \leq \frac{4 α N}{R} .

(1)

The integral is taken at fixed

x

, and the disc and segment lie inside the shrinking domain. Consequently

u \mapsto \int_{0}^{t} (i u_{x} + f) d z

is a contraction mapping when

0 < α < R /4

and the holomorphic source

f

is bounded on the relevant closed polydisc. The source term has weighted norm at most

α sup ∣ f ∣

, so the Banach fixed-point theorem gives a holomorphic solution with zero initial data.

Order of a zero of a holomorphic function

 0  0

If a nonzero holomorphic function has expansion

f (z) = (z - z_{0})^{m} g (z), g (z_{0}) \neq = 0,

(1)

then

m

is the order or multiplicity of its zero at

z_{0}

Zero sets in the unit disc

 0  0

A nonzero holomorphic function on the unit disc can have exactly a prescribed sequence of zeros, counted with multiplicity, if and only if that sequence is locally finite in the disc. Necessity is the identity theorem for holomorphic functions. For sufficiency, remove a finite number

m

of zeros at zero and enumerate the others as

a_{n}

, so

∣ a_{n} ∣ \to 1

. Use the Weierstrass elementary factors

E_{n} (t) = (1 - t) exp (\sum_{j = 1}^{n} t^{j} / j)

. On every compact subdisc, eventually

∣ z / a_{n} ∣ \leq q < 1

and

∣ lo g E_{n} (z / a_{n}) ∣ \leq q^{n + 1} / ((n + 1) (1 - q))

. Hence

z^{m} \prod_{n} E_{n} (z / a_{n})

converges locally uniformly and has exactly the prescribed zeros. Finite sequences simply give finite products.

Simple zero

 0  0

A simple zero has order one, equivalently

f (z_{0}) = 0

and

f^{'} (z_{0}) \neq = 0

Analytic continuation

 1  0

An analytic continuation extends a holomorphic function beyond its original domain without changing its existing values. Formally, if

U \subset V

are domains and

f

is holomorphic on

U

, a holomorphic function

F

V

with

F ∣_{U} = f

is an analytic continuation. More locally, holomorphic functions on overlapping open sets continue one another when they agree on the overlap, allowing a chain of patches along a path. The identity theorem for holomorphic functions forces agreement throughout a connected overlap once they agree on a nonempty open set in it. It also makes an extension to a fixed connected domain unique. ^[ref]

The geometric series illustrates why a formula can reach farther than the power series that first represents it. For

z \neq = 1

, its partial sum is

S_{N} (z) = \sum_{n = 0}^{N} z^{n} = \frac{1 - z ^{N + 1}}{1 - z} .

(1)

When

∣ z ∣ < 1

z^{N + 1} \to 0

, so the series converges to

f (z) = 1/ (1 - z)

. The rational function

F (z) = 1/ (1 - z)

is holomorphic on

C ∖ {1}

and agrees there with the sum on the unit disc. It therefore continues

f

to every point except its simple pole at

1

. In particular,

f (1/2) = 2, F (2) = - 1, S_{N} (2) = 2^{N + 1} - 1 ⟶ + \infty.

(2)

Thus analytic continuation gives a value at

2

without making the original geometric series converge there. Its radius of convergence remains

1

: on

∣ z ∣ \geq 1

the terms

z^{n}

do not even tend to zero.

Figure 1.
The original geometric series converges only inside the unit circle, while its rational continuation exists everywhere except the pole at 1
. **Details**.

The pole at

1

limits the original Taylor series, although most points of its convergence circle are regular for

F

. Recentring at

- 1

gives another genuinely convergent power series,

F (z) = \frac{1}{2 - ( z + 1 )} = \frac{1}{2} \sum_{n = 0}^{\infty} (\frac{z + 1}{2})^{n}, ∣ z + 1∣ < 2.

(3)

z = - 3/2

this yields

\frac{1}{2} \sum_{n \geq 0} (- 1/4)^{n} = 2/5

, even though the original powers of

- 3/2

grow. This is the useful idea: overlapping local descriptions can represent one holomorphic function on a larger region.

Uniqueness on a connected overlap does not imply that every path leads to the same terminal germ of a holomorphic function. Continuing a complex logarithm around

0

changes its value by

2 πi

, giving monodromy; the return patch can carry a different branch of a multivalued function. If two patches have a disconnected overlap, agreement on one component imposes no agreement on the others. ^[ref] Continuation using the rational function

1/ (1 - z)

is single-valued: any path avoiding

1

gives the same terminal germ of a holomorphic function, even if it winds around the pole.

Natural boundary of a holomorphic function

 0  0

A boundary of a domain of a holomorphic function is natural if the function has no analytic continuation through any of its points. For

f (z) = \sum_{n \geq 1} z^{2^{n}}

, each root of unity of order a power of two is a singular boundary point: along

z = r ζ

all sufficiently late terms equal

r^{2^{n}}

and their sum tends to infinity as

r ↑ 1

. These roots are dense on the unit circle, so no boundary arc admits analytic continuation.

Zeta function regularization

 1  0

Zeta function regularization assigns a finite value to a divergent spectral sum by first forming a convergent complex-power series and then using its analytic continuation. For positive frequencies

ω_{n}

, continue

\sum_{n} ω_{n}^{- s}

s = - 1

to define a regulated frequency sum, when that continuation is regular there. This does not turn the divergent positive-term sum into an ordinary convergent sum. It is a useful subtraction convention for the normal-ordering constant of a string and for vacuum energies.

Half-integer zeta-regularized mode sum ( $\sum_{n \geq 0}^{reg} (n + \frac{1}{2}) = \frac{1}{24}$ )

 0  0

For

r = n + 1/2

n \geq 0

, the convergent frequency zeta function, expressed through the Riemann zeta function, is

\sum_{r} r^{- s} = (2^{s} - 1) ζ_{R} (s)

when

Re s > 1

. Its analytic continuation at

s = - 1

gives

(1/2 - 1) (- 1/12) = 1/24

. A common exponential frequency cutoff gives

\sum_{r} r e^{- ε r} = ε^{- 2} + 1/24 + O (ε^{2})

, so the same finite part follows by subtracting the leading divergence. In contrast, integer frequencies have regulated sum

- 1/12

Meromorphic continuation

 0  0

A meromorphic continuation extends a holomorphic function from a connected open set to a larger connected domain as a meromorphic function, allowing isolated poles. Equality on the initial open set determines the extension uniquely by the identity theorem. Integral formulas whose integrands depend holomorphically on a parameter often supply such an extension after contour deformation or subtraction of singular terms.

Sokhotski–Plemelj theorem

 0  0

As distributions on the real line,

\frac{1}{x \pm i 0} = PV \frac{1}{x} \mp iπ δ (x) .

(1)

The Sokhotski--Plemelj formula relates the two boundary values of a Cauchy transform across its contour and fixes the spectral delta functions generated by retarded and advanced prescriptions.

Closed-contour Cauchy singular integral is an involution ( $C^{2} = I$ )

 0  0

On a positively oriented smooth simple closed contour, set

C f = (πi)^{- 1} PV \int_{Γ} f (t) / (t - z) d t

. The Sokhotski–Plemelj theorem gives the sum and difference of the two Cauchy boundary values. Changing the sign of the exterior branch converts the equation

C f = g

into a normalized additive Riemann-Hilbert problem with jump

g

. Applying the boundary formula again yields

f = C g

, proving the involution on a suitable smooth or Hölder function class.

Cauchy principal value for an endpoint Jacobi weight ( $PV \int_{- 1}^{1} \frac{( 1 - t ) ^{α - 1}}{( 1 + t ) ^{α} ( t - x )} d t = π cot (π α) \frac{( 1 - x ) ^{α - 1}}{( 1 + x ) ^{α}}$ )

 0  0

For

0 < α < 1

and

- 1 < x < 1

, the endpoint powers are integrable and the interior pole is taken as a Cauchy principal value. The analytic function

w (z) = (z - 1)^{α - 1} / (z + 1)^{α}

, positive for

z > 1

and decaying like

1/ z

, has boundary values

- e^{\pm iπ α} (1 - t)^{α - 1} / (1 + t)^{α}

. Its jump gives a Cauchy reconstruction, while the mean of its boundary values and the Sokhotski–Plemelj formula give the displayed identity. At

α = 1/2

the principal value is zero.

Cauchy reconstruction from a jump across a finite interval ( $w (z) = \frac{1}{2 πi} \int_{0}^{1} \frac{w _{+} ( t ) - w _{-} ( t )}{t - z} d t$ )

 0  0

Suppose

w

is holomorphic off

[0, 1]

, tends to zero at infinity, has sufficiently regular boundary values in the interval interior, and has endpoint growth

O (∣ z - a ∣^{- α})

for some

α < 1

at each endpoint

a

. If its jump is integrable, the displayed reconstruction follows by deforming a Cauchy integral formula contour onto the two sides of the interval. An endpoint circle of radius

ε

contributes

O (ε^{1 - α})

, which vanishes. The two oppositely oriented banks leave precisely the jump. Without decay at infinity, an entire remainder can survive; a nonzero constant is the simplest example.

Signed Cauchy boundary operators ( $P^{\pm} = \pm I /2 + i H /2$ )

 0  0

With

H v (ρ) = π^{- 1} PV \int v (s) / (ρ - s) d s

, the upper and lower limiting Cauchy integrals are

P^{\pm} v = \pm v /2 + i H v /2

. They obey

P^{+} - P^{-} = I

. The actual complementary idempotent projections are

P^{+}

and

- P^{-}

; confusing this signed convention with two positive projectors changes reconstruction signs. These boundary operators naturally enter the spectral reconstruction of an attenuated Radon transform.

Monodromy theorem

 0  0

The monodromy theorem says that, when analytic continuation exists throughout a domain, its ambiguity depends on homotopy. Let

D \subset C

be a domain,

a \in D

, and

g_{a}

a germ of a holomorphic function at

a

. Assume

g_{a}

admits analytic continuation along every continuous path in

D

starting at

a

. Two such paths ending at

b

give the same terminal germ of a holomorphic function if they are homotopic with endpoints fixed in

D

. ^[ref]

Here a homotopy with endpoints fixed is a continuous map

H : [0, 1]^{2} \to D

satisfying

H (0, t) = γ_{0} (t), H (1, t) = γ_{1} (t), H (s, 0) = a, H (s, 1) = b .

(1)

Every intermediate path must stay in

D

. Equality means agreement on a neighbourhood of

b

, rather than merely equal values at

b

. ^[ref]

The identity theorem provides local rigidity; the monodromy theorem shows precisely how topology controls the remaining global freedom. Deforming a route without crossing a forbidden point cannot change the resulting germ of a holomorphic function. Thus a complicated analytic continuation calculation can be replaced by one on a simpler homotopic route. Distinct homotopy classes can still give the same continuation: a rational function on its domain already gives a single-valued answer along every route, even around a pole.

Simply connected corollary of the monodromy theorem

 0  0

On a simply connected domain

D

, a germ of a holomorphic function admitting analytic continuation along every path from its base point determines a unique holomorphic function

F : D \to C

with that initial germ of a holomorphic function. ^[ref]

Indeed, any two paths with the same endpoints are homotopic with endpoints fixed. Define

F (b)

using the terminal germ of a holomorphic function. The monodromy theorem makes this independent of the path. In a small open disc about

b

, extend the chosen route by a short segment; the same local representative supplies

F

throughout the open disc, proving holomorphy. The identity theorem proves uniqueness. The existence hypothesis of the monodromy theorem must still be checked.

Logarithm and square-root continuation along different paths

 0  0

Start at

a = 1

with complex logarithm germ of a holomorphic function

L (1) = 0

and holomorphic square root germ of a holomorphic function

S (1) = 1

. On the punctured plane

D = C ∖ {0}

, both admit analytic continuation along every path. To construct it locally, for any

c \neq = 0

choose a small open disc avoiding

0

, select a holomorphic logarithm there, and select its additive multiple of

2 πi

to agree with the arriving germ of a holomorphic function; exponentiating half this logarithm supplies the arriving holomorphic square root. A finite chain of these open discs follows any given path.

Compare two paths from

1

i

, parametrized by

0 \leq t \leq 1

γ_{A} (t) = e^{iπ t /2}, γ_{B} (t) = e^{- 3 iπ t /2} .

(1)

The first follows a counterclockwise quarter-circle; the second follows a clockwise three-quarter-circle. Following

γ_{A}

and then path reversal of

γ_{B}

makes one positive turn about

0

. Its winding number is

1

, so the two paths cannot be homotopic with endpoints fixed in the punctured plane.

Figure 1.
Two paths from 1 to i enclose the puncture, while a negative-real-axis cut excludes the clockwise route and permits a deformation between the remaining routes
. **Details**.

The continued values along the chosen paths are

L_{A} (t) = \frac{πi t}{2}, L_{B} (t) = - \frac{3 πi t}{2}, S_{A} (t) = e^{iπ t /4}, S_{B} (t) = e^{- 3 iπ t /4} .

(2)

At their common endpoint,

L_{A} (1) = \frac{πi}{2}, L_{B} (1) = - \frac{3 πi}{2}, L_{A} (1) - L_{B} (1) = 2 πi, S_{A} (1) = \frac{1 + i}{2}, S_{B} (1) = - \frac{1 + i}{2} .

(3)

Both complex logarithm values exponentiate to

i

, and both holomorphic square root values square to

i

. These give different terminal germ of a holomorphic function representatives, consistently obtained from the same starting germ of a holomorphic function. The monodromy theorem applies, but does not equate these two non-homotopic paths.

Figure 2.
Tracking continuation from 1 to i along a quarter-circle and a clockwise three-quarter-circle gives logarithms separated by 2 pi i and opposite square roots
. **Details**.

The closed path

λ (t) = e^{2 πi t}

instead returns to

1

, with continued values

L_{λ} (t) = 2 πi t

and

S_{λ} (t) = e^{πi t}

. One counterclockwise turn changes the complex logarithm by

2 πi

and flips the holomorphic square root from

1

- 1

. Hence neither initial germ of a holomorphic function can come from a global holomorphic logarithm or a global holomorphic square root of

z

on the punctured plane. Repeating a loop of winding number

n

gives the changes

2 πin

and

(- 1)^{n}

. ^[ref]

Now restrict to the slit domain

D_{slit} = C ∖ (- \infty, 0]

. The principal logarithm and

exp (Log z /2)

realize the same initial germ of a holomorphic function for each expression globally. To see that this is a simply connected domain, polar form identifies this domain with

(0, \infty) \times (- π, π)

, a convex set; every loop contracts there. The simply connected corollary of the monodromy theorem now forces equality along every pair of paths with matching endpoints. In particular,

γ_{A}

and the segment

σ (t) = 1 - t + i t

both give

L (i) = πi /2

and

S (i) = (1 + i) / 2

. The explicit homotopy

H (s, t) = (1 - s) γ_{A} (t) + s σ (t)

stays in the closed first quadrant away from

0

, hence in

D_{slit}

The other route

γ_{B}

crosses the removed ray at

t = 2/3

and is no longer admissible. In the punctured plane, that crossing is locally regular: analytic continuation passes through

- 1

smoothly, although evaluating the principal logarithm would reset its argument. The branch cut changes the allowed domain; it does not create additional branch points or poles. That distinction is the beauty of the obstruction: perfectly compatible local holomorphic functions can fail to make a global choice solely because of the allowed based loops. The branches of a multivalued function article develops how moving the branch cut changes the selected values.

Existence hypothesis of the monodromy theorem

 0  0

The monodromy theorem gives equality of continuations that exist under its hypothesis. It does not assert that every holomorphic function admits analytic continuation everywhere, even within a simply connected domain.

For an explicit obstruction, take

f (z) = 1/ (1 - z)

near

a = 0

and attempt to apply the theorem on

D = C

. The path

η (t) = t

ends at

1

, where no terminal germ of a holomorphic function exists. If a holomorphic representative

G

near

1

continued

f

, then the identity theorem would extend

(1 - z) G (z) = 1

to that neighbourhood. Substituting

z = 1

would give

0 = 1

. Thus the pole prevents the existence hypothesis, although

C

is a simply connected domain. The analytic continuation article explains why the same rational function does continue throughout

C ∖ {1}

Existence at an endpoint by some route is also weaker than existence along every path from the base point. Along the real segment from

1

towards

0

, a holomorphic square root of

z

tends to

0

, yet cannot continue holomorphically through

0

: a local representative

G

would satisfy

G^{2} = z

by the identity theorem, and differentiating at

0

would give

2 G (0) G^{'} (0) = 1

with

G (0) = 0

. A finite limiting value therefore does not remove a branch point. More severe obstructions appear at a natural boundary of a holomorphic function.

Local continuation proof of the monodromy theorem

 0  0

Along one path, analytic continuation is a chain of function elements on overlapping open discs, each agreeing with the preceding germ of a holomorphic function near the crossing point. The identity theorem makes this chain unique along that fixed path. Equality at a crossing means equality on a neighbourhood, not just equality of a single value.

For the deformation step, fix one intermediate path

t \mapsto H (s_{0}, t)

and its continuation. By compactness of

[0, 1]

, choose a finite chain of representative open discs and subdivide the parameter so that each subarc fits in its assigned open disc, with consecutive junctions in overlap neighbourhoods where the representatives agree. Joint continuity of

H

ensures that every sufficiently nearby intermediate path uses that same chain and those same overlaps. The endpoints stay fixed, so the terminal germ of a holomorphic function stays unchanged. Existence along every intermediate path supplies this argument at each

s_{0}

; local constancy and connectedness of

[0, 1]

give equality at

s = 0

and

s = 1

. This proves the monodromy theorem without assuming that every analytic germ projection is a covering map.

Monodromy group of a covering

 0  0

For a covering map

p : X \to Y

and base point

y

, the path lifting theorem lifts each based loop from each point of the fibre

p^{- 1} (y)

. Its endpoint defines a permutation of that fibre; path reversal gives the inverse. The homotopy lifting theorem for a covering map makes the permutation depend only on the based homotopy class. These permutations form the monodromy group, the image of the fundamental group action

π_{1} (Y, y) ⟶ Sym (p^{- 1} (y)) .

(1)

The image is independent of the convention for composing routes; one may reverse the loop classes to obtain a left group action when route multiplication means following the first route and then the second.

For

p (w) = w^{2}

on the punctured plane, the fibre over

1

{1, - 1}

, and a once-around based loop exchanges its two points. The monodromy group has two elements: only the parity of the winding number matters. For the covering map

p (w) = e^{w}

from the complex plane to the punctured plane, the fibre over

1

2 πi Z

, and a positive turn acts by

2 πik \mapsto 2 πi (k + 1)

. Its monodromy group is an infinite cyclic group. These are the geometric versions of logarithm and square-root continuation along different paths: the base point returns, while its lifted endpoint can move to another sheet.

Monodromy transposition in a three-sheeted cover

 0  0

A connected three-sheeted cover has transitive monodromy. If it also has a simple branch point, its monodromy contains a transposition and must be the full symmetric group on three letters. A local triple branch supplies a three-cycle explicitly.

Monodromy group of z squared plus z to the minus two ( $z \mapsto z^{2} + z^{- 2}$ )

 0  0

The degree-four rational map

F (z) = z^{2} + z^{- 2}

has branch values

- 2, 2, \infty

. Away from their inverse images, the transformations

z \mapsto - z, z \mapsto z^{- 1}

(1)

are commuting deck involutions. They generate a Klein four-group acting transitively on each fibre

{z, - z, z^{- 1}, - z^{- 1}} .

(2)

Consequently the monodromy group is

V_{4}

, acting by the identity and the three double transpositions.

Analytic continuation by contour deformation

 0  0

If a parameter-dependent contour integral has a moving pole, deforming the integration contour continuously so that the pole never crosses it preserves a holomorphic branch. Returning the contour to its original path after crossing a pole adds or subtracts the corresponding residue contribution.

Branch phase in a figure-eight analytic continuation integral ( $J (z) = - 2 i sin (π z) \int_{- 1}^{1} e^{t} (1 - t^{2})^{z} d t$ )

 0  0

For a figure-eight contour around

1

anticlockwise and

- 1

clockwise, start

(t^{2} - 1)^{z}

t = 0

with argument

- π

. Shrinking the two loops gives the displayed identity. If the interval integral instead uses

(t^{2} - 1)^{z} = e^{- iπ z} (1 - t^{2})^{z}

, the corresponding identity includes the extra factor

e^{iπ z}

. Omitting this phase changes the value, rather than merely the locations of the meromorphic continuation's poles.

Multivalued inverse hyperbolic sine ( $arcsinh z$ )

 0  0

The solutions of

sinh w = z

are

w = Arcsinh z + 2 πik or w = - Arcsinh z + (2 k + 1) πi, k \in Z,

(1)

for any chosen local branch

Arcsinh

It is the complex multivalued sine member of the inverse hyperbolic functions.

Entire function ( $f : C \to C$ holomorphic everywhere)

 1  0

An entire function is a holomorphic function whose domain of a function is the whole complex plane. Its derivative exists at every

z \in C

. Every polynomial, the complex exponential function

e^{z}

, the product

z e^{z}

, and the complex sine function

sin z

are entire functions. The Taylor series of an entire function about any finite centre converges throughout the complex plane.

The complex exponential function is a particularly revealing entire function. Writing

z = x + i y

, Euler's formula gives

e^{x + i y} = e^{x} (cos y + i sin y), ∣ e^{x + i y} ∣ = e^{x} > 0.

(1)

Thus the complex exponential function never takes the value zero. Conversely, for any

w \neq = 0

, choose an argument

θ

w

. All its preimages under the complex exponential function are

z_{k} = ln ∣ w ∣ + i (θ + 2 πk), k \in Z .

(2)

Here

ln ∣ w ∣

is the real natural logarithm, and

k

ranges over the integers. Substitution gives

e^{z_{k}} = ∣ w ∣ e^{i θ} = w

; equality of moduli and arguments shows these are all the solutions. The complex exponential function therefore has image exactly

C ∖ {0}

, with infinitely many preimages for each nonzero value.

Figure 1.
Under the complex exponential, vertical lines become circles, horizontal lines become rays, and inputs separated by 2 pi i have the same image
. **Details**.

In the pictured vertical strip

- 1 \leq x \leq 1

, the real part sets the modulus between

e^{- 1}

and

e

, while the imaginary part sets the argument modulo

2 π

. The two drawn bands each make one full turn around the same annulus; their corresponding points differ by

2 πi

and have identical images. Allowing all real

x

gives every positive modulus, so the whole complex plane covers every nonzero output.

An entire function has exceptionally strong global constraints. The Liouville theorem says that a bounded function which is also an entire function must be a constant function. Accordingly, the nonconstant entire functions above are unbounded somewhere in the complex plane, even when their restrictions to the real axis are bounded: the complex sine function satisfies

sin (i y) = i sinh y

, growing through the hyperbolic sine. The Little Picard theorem adds a different constraint: an unbounded nonconstant entire function still cannot leave two distinct complex values out of its image. These results make entire functions useful when global bounds or restrictions on the image can establish rigidity without solving for the function explicitly.

In the opening of his 1880 memoir, Émile Picard credits Karl Weierstrass with the terminology of entire functions and describes them by everywhere-convergent power series. ^[ref]

Injective entire functions are affine ( $g (z) = a z + b, a \neq = 0$ )

 0  0

An injective holomorphic function cannot have an essential singularity: Casorati-Weierstrass theorem would give values from a small punctured neighbourhood inside the open image of a disjoint neighbourhood, contradicting injectivity. Apply this at infinity to an injective entire function. A removable singularity there would make the function constant by Liouville's theorem. A pole makes it a polynomial by the Cauchy coefficient formula; its derivative has no zero, so the fundamental theorem of algebra forces degree one.

Exponential polynomial ( $F (z) = \sum_{j} P_{j} (z) e^{λ_{j} z}$ )

 1  0

An exponential polynomial is a finite sum of polynomial multiples of exponentials. For distinct exponents, its terms are linearly independent over the polynomial ring: to isolate one term, apply the product of operators

(D - λ_{j})^{d e g P_{j} + 1}

for all the other terms. On the remaining polynomial each operator becomes

D + λ_{0} - λ_{j}

, which is injective because its nonzero constant preserves the leading coefficient. A nonzero exponential polynomial is an entire function of exponential type. Its maximum-modulus logarithm is

O (R + lo g R)

, and Jensen's formula gives at most

O (R)

zeros, counted with multiplicity, in a disk of radius

R

Order of an entire function ( $ρ_{f} = lim sup_{R \to \infty} lo g lo g M_{f} (R) / lo g R$ )

 0  0

For a nonconstant entire function, put

M_{f} (R) = max_{∣ s ∣ \leq R} ∣ f (s) ∣

. Its order measures the exponent in its exponential growth. Bounds

lo g M_{f} (R) = O (R lo g R)

imply order at most one; a matching lower bound along the positive real axis proves order exactly one. This distinction is stronger than mere entire analyticity and controls the genus in Hadamard factorization.

// Target: analytic-number-theory.bigb