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If $F(z)=L+O(1/z)$ uniformly on large circles, then $F(z)/z$ has residue $L$ at infinity in the large-contour sense and its counterclockwise large-circle integral tends to $2\pi iL$.

Two holomorphic functions on a connected domain that agree on a set with an interior accumulation point agree everywhere.

The Dirichlet beta function is $\beta (s)={\sum }_{n\ge 0}(-1{)}^n(2n+1{)}^{-s}$ and is the Dirichlet L-function for the nontrivial character modulo four.

The value of a holomorphic function at the center of a closed disc equals its average around every concentric circle lying in the domain.

The real and imaginary parts are harmonic functions, whose mean-value property explains this average identity.

A nonconstant holomorphic function on a connected domain cannot attain a local maximum of its modulus at an interior point.

A meromorphic function with finitely many specified simple zeros and poles is a rational product up to a nonzero holomorphic factor.

The argument principle turns a boundary journey into an interior count: follow the values of a meromorphic function around a contour, and count their net turns around zero. Every enclosed zero of a function contributes its multiplicity of a zero; every enclosed pole subtracts its order. There is no need to find each zero of a function separately.

Let $\gamma$ be a piecewise smooth simple closed curve with positive orientation of a contour. Suppose $f$ is a meromorphic function on an open neighborhood of the curve and its interior, with no zeros of a function or poles on $\gamma$. ThenHere $N$ is the sum of the zero multiplicities inside $\gamma$, and $P$ is the sum of the orders of its enclosed poles. The image contour $f\circ \gamma$ may cross itself or revisit a point. Its signed winding number, rather than the number of visually bounded regions, is the quantity that counts.^[ref]

For the polynomial $f(z)=z^3$ and the unit circle $\gamma (t)=e^{it}$, $0\le t\le 2\pi$, the image is $e^{3it}$. It goes around zero three times although its drawn outline is just one circle. The interior contains one distinct zero of a function, at zero, with multiplicity of a zero three: the count is three. Reversing the orientation of a contour reverses the winding number and the sign of the contour integral.

An argument must be followed continuously along the journey. Subtracting the principal arguments at the endpoints loses whole turns: a closed image contour has the same endpoint value even when its winding number is nonzero. Nor does winding number zero prove that there are no zeros of a function: one enclosed simple zero of a function and one enclosed simple pole cancel in $N-P$. Known poles must be added back to recover $N$.

The mechanism is short. Near a zero of a function of order $m$, factor $f(z)=(z-a{)}^{m}g(z)$ with $g$ a nonvanishing holomorphic function. Its logarithmic derivative is $f^{\prime }/f=m/(z-a)+g^{\prime }/g$, so its residue is $m$. A pole of order $m$ gives residue $-m$. The residue theorem adds those contributions. Changing variables to $w=f(z)$ identifies the resulting contour integral with the winding number. This is a local algebraic count joined to a global geometric count.^[ref]

The usefulness goes beyond explicit polynomials. Rouché's theorem keeps a count unchanged when a boundary perturbation cannot push the image contour through zero. The Nyquist stability criterion applies the same counting idea to poles governing a system under feedback control.^[ref]

An upper-half-plane self-map is holomorphic and has positive imaginary part in the upper half-plane.

A holomorphic function real on a boundary interval extends across it by conjugate reflection.

Locally uniform limits of holomorphic functions are holomorphic and their derivatives converge locally uniformly.

Suppose $f$ is a holomorphic function on an open set containing the closed disc $|z-a|\le R$, with $R>0$. For any nonnegative integer $n$ and $|w-a|<R$, the counterclockwise boundary contour satisfiesThe case $n=0$ is the Cauchy integral formula. On a smaller closed disc, the denominator stays uniformly away from zero, allowing repeated differentiation under the contour integral. This proves that a holomorphic function has complex derivatives of every order. ^[ref]

At the centre, if $M={\max }_{|z-a|=R}|f(z)|$, taking absolute values gives the Cauchy estimate $|f^{(n)}(a)|\le n!M/R^n$. Boundary size therefore limits interior variation. For $f(z)=z^2$ on the unit disc, $M=1$, and the $n=2$ estimate is exact: $|f^{\prime \prime }(0)|=2$. This is one quantitative reason that holomorphic functions are so rigid.

Every bounded entire function is constant.

A univalent function is a holomorphic injective function.

Contour integration integrates complex functions along oriented curves and evaluates many real integrals through residues.

An isolated singularity at $a$ is a point at which a function is not holomorphic although it is holomorphic throughout some punctured neighbourhood of $a$.

A complex number is $z=x+iy$, where $x,y$ are real numbers and the imaginary unit satisfies $i^2=-1$. The real part is $x$ and the imaginary part is $y$: for $3+4i$ they are $3$ and $4$, respectively. Equality requires both coordinates to agree. Thus the complex plane gives each complex number a concrete position, while ordinary addition and multiplication become geometric operations.

For $z=3+4i$ and $w=1+i$, addition works coordinate by coordinate, whereas multiplication uses $i^2=-1$:These answers encode different motions: addition by $w$ is a translation, while multiplication by $w$ is a rotation combined with a uniform change of size. Keeping these pictures distinct prevents a common beginner's confusion.

The complex numbers form a field: every nonzero $z$ has a multiplicative inverse, computable using its complex conjugate and modulus. They also supply the missing solutions of a quadratic equation such as $u^2+1=0$, namely $u=\pm i$. The surprising payoff is that extending arithmetic also makes Euclidean geometry easier to calculate. For a systematic introduction to these definitions, see Jeremy Orloff, MIT 18.04, topic 1, sections 1.3–1.5.

A branch point is a point around which a sufficiently small loop can change a germ of a holomorphic function under analytic continuation. This is a local condition: a large loop around a regular point might also enclose a distant branch point, so that large loop alone cannot classify the regular point. For the complex logarithm and square root, every nonzero point has a local holomorphic branch, but a loop once around $0$ changes the chosen branch. ^[ref]

For $z=r{e}^{it}$ with fixed $r>0$, start at $t=0$ with values $\sqrt{r}$ and $\log r$. Their continuations areOne counterclockwise turn gives $w(2\pi )=-\sqrt{r}$ and $L(2\pi )=\log r+2\pi i$. At $r=4$, the square root therefore changes from $2$ to $-2$, while the logarithm changes from $\log 4$ to $\log 4+2\pi i$. Two turns restore the square root to $2$; no nonzero number of turns restores the logarithm. This distinguishes a two-sheeted algebraic branch point from a logarithmic singularity.

A pole need not be a branch point: $1/z$ is unbounded near $0$, yet $1/(r{e}^{it})=r^{-1}{e}^{-it}$ returns to its starting value after one turn. Conversely, the square-root values tend to $0$ at their branch point. The issue is whether a single holomorphic choice survives going around the point, rather than whether the values become large. On the Riemann sphere, infinity is also a branch point of both $\sqrt{z}$ and $\log z$: in the local coordinate $\zeta =1/z$, these are ${\zeta }^{-1/2}$ and $-\log \zeta$, whose continuations around $\zeta =0$ change sign and subtract $2\pi i$, respectively. ^[ref]

A Bromwich contour is a vertical line in the complex frequency plane lying to the right of the singularities in the inverse Laplace integral.

A homotopy through closed paths avoiding the base point cannot change the integer winding number.

The winding number records how many net turns a closed piecewise smooth curve makes around a chosen point $w$. Counterclockwise turns contribute positively, clockwise turns negatively. It measures a traversal, including repeated visits; a drawing of the curve's outline alone can lose that information. For a closed curve $\gamma$ avoiding $w$,Equivalently, continuously follow an argument of $\gamma (t)-w$ along one journey and divide its total change by $2\pi$. The start and endpoint agree as complex numbers, so the change is an integer number of full turns. The principal argument at the endpoints cannot retain those turns; a continuous choice along the parametrized path can.^[ref]

For $\gamma (t)=e^{it}$, $0\le t\le 2\pi$, the winding number about zero is one. Traversing the same unit circle twice makes it two; reversing its orientation of a contour makes it negative. The winding number about $w=2$ is zero: the same journey now stays to one side of the point being measured. Joining one counterclockwise loop and one clockwise loop around zero gives net winding number zero even though both loops encircle zero.

The winding number does not change during a homotopy of closed curves avoiding $w$. This is why it is useful: complicated curves can often be deformed to simple ones whose count is obvious. The argument principle transfers that robust geometric integer to a count of zeros of a function and poles. The example in boundary sampling can miss winding shows why finite samples of a curve need control between samples before they can certify the same winding number.