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An analytic continuation extends a holomorphic function through overlapping connected open sets while preserving its values on their overlap. Continuation along different paths can produce different germs when the domain contains branch points.

If $f=u+iv$ is a holomorphic function on a connected domain andthen the Cauchy-Riemann equations force both components of $f^{\prime }$ to vanish. Hence $f$ is constant.

Near an isolated essential singularity, a holomorphic function has dense image in the complex plane. If it avoided a disk about $w$, then $1/(f-w)$ would be bounded and extend across the singularity by the Riemann removable singularity theorem. If its extended value is nonzero, $f$ extends holomorphically; if zero, its finite-order zero makes $f$ a pole. Either contradicts an essential singularity.

A holomorphic function on an open neighbourhood of a closed triangle has zero contour integral around its boundary. Goursat's subdivision proof uses complex differentiability alone: repeatedly choose a quarter-size triangle carrying at least one quarter of the integral, and subtract the affine expansion at the limiting point. No continuity of the derivative is assumed.

For a finite-index subgroup of the modular group, the quotient of the complex upper half-plane becomes a compact Riemann surface after adjoining its cusp classes. At an elliptic fixed point use the quotient coordinate for the finite stabilizer; at a cusp use its exponential local parameter. Weight-zero modular forms descend to holomorphic functions on this compact surface, and hence are constant by the maximum modulus principle.

If $B$ is planar Brownian motion in a domain $D$ and $f:D\to D^{\prime }$ is a nonconstant holomorphic function, then $f(B)$, run with the clock ${\int }_0^t|f^{\prime }(B_s){|}^{2}ds$, is planar Brownian motion in $D^{\prime }$ until the relevant exit time. In particular, conformal maps transport harmonic measure between planar domains.

Equip holomorphic functions on an open spectral neighborhood with the compact-open topology. A fixed admissible contour gives a bound on $\Vert {\Theta }_x(f)\Vert$ by a constant times the supremum of $|f|$ on the contour, proving continuity. The Runge theorem makes rational functions with poles outside the open set dense. A unital homomorphism sending the coordinate function to $x$ must send each inverse coordinate difference to the corresponding resolvent, so its rational values and then all its holomorphic values are forced.

A contour deformation changes the path of a contour integral through a region where its integrand is a holomorphic function. The Cauchy integral theorem preserves the integral when the endpoints are fixed and the connecting boundary terms vanish. Crossing a pole instead contributes the appropriately oriented residue; a branch cut limits which deformations are permitted on one analytic sheet.

For a piecewise differentiable parametrized contour $z(t)$, the contour integral is $\int f(z(t))z^{\prime }(t)dt$. It depends on the orientation of the contour. For a holomorphic function, the Cauchy integral theorem allows a contour deformation through regions without singularities.

A prescribed inhomogeneous Wirtinger derivative equation. The distributional identity ${\partial }_{\bar{z}}(1/(\pi z))={\delta }_0$ gives its whole-plane right inverse through the Cauchy-Green operator. Boundary or decay conditions select a solution from the freedom to add a holomorphic function.

For $\Re s>0$, the alternating Dirichlet seriesconverges locally uniformly and defines a holomorphic function. For $\Re s>1$,

For a bounded operator on a complex Banach space, partition a disconnected spectrum into two nonempty compact pieces. The holomorphic function equal to one near the first piece and zero near the second defines a Riesz projection. The holomorphic spectral mapping theorem makes it neither zero nor identity. Its closed range is a proper nonzero invariant vector subspace. This need not be an orthogonal projection, and the argument does not require a normal operator.

An entire function is a holomorphic function whose domain is the whole complex plane.

For $\mathrm{Re}z,\mathrm{Re}{z}^{\prime }>0$, the series ${\sum }_{d,e}\mu (d)\mu (e)d^{-z}{e}^{-z^{\prime }}/[d,e]$ has local factor $1-p^{-1-z}-p^{-1-z^{\prime }}+p^{-1-z-z^{\prime }}$. Dividing by the displayed zeta function ratio leaves a holomorphic function $H$ near zero with $H(0,0)=1$. The local factor of $H$ is $(1-a-b+c)(1-c)/((1-a)(1-b))$, where $a=p^{-1-z}$, $b=p^{-1-z^{\prime }}$, $c=p^{-1-z-z^{\prime }}$, and equals one exactly at zero. With $z=(1+it)/\log X$, the zeta function poles produce the kernel $(1+it)(1+i{t}^{\prime })/((2+i(t+t^{\prime }))\log X)$ that determines the smooth divisor-square sieve asymptotic.

A real-valued function $u$ on a simply connected domain is harmonic exactly whenfor some holomorphic function $f$. One construction observes that $u_x-i{u}_y$ is holomorphic and takes its holomorphic primitive.

The Kähler Laplacian identity ${\Delta }_d^{\mathrm{H}}=2{\Delta }_{\bar{\partial }}$ implies that every holomorphic function on a Kähler manifold is a harmonic function. On functions, $\bar{\partial }u=0$ and ${\bar{\partial }}^{\ast }u=0$ by degree, so both terms of the Dolbeault Laplacian vanish. Compactness is unnecessary. Changing the sign convention for the scalar Laplacian does not change this conclusion.

A nonsingular holomorphic function of chiral superfields remains chiral because ${\bar{D}}_{\dot{\alpha }}f(\Phi )={\sum }_i{f}_i(\Phi ){\bar{D}}_{\dot{\alpha }}{\Phi }_i=0$. Dependence on conjugate superfields generally destroys the constraint. This explains the holomorphic form of a superpotential.

For the bounded Dirichlet Poisson integral in a quadrant, the Wirtinger derivative isThis follows by differentiating the Schwarz integral formula composed with $z^2$, remembering that for real boundary data $u_z$ is half the derivative of a holomorphic function whose real part is $u$. For complex boundary data, apply the construction to the real and imaginary parts separately and use linearity. If the data and their derivatives decay sufficiently, integration by parts givesThe endpoint terms cancel exactly when $g_1(0)=g_2(0)$. Interior denominators do not vanish because $\mathrm{Im}{z}^2>0$.

A map between Riemann surfaces is holomorphic when its expression in every pair of local complex coordinates is a holomorphic function.

A renormalizable MSSM superpotential is a holomorphic function of chiral superfields and cannot use their conjugates. Gauge-invariant up-type Yukawa couplings require $H_u$ of hypercharge $+1/2$, while down-type and charged-lepton Yukawa couplings require $H_d$ of hypercharge $-1/2$. A single doublet and its conjugate cannot play both roles in the superpotential. This is distinct from higgsino anomaly cancellation.