Complex Analysis

Complex analysis is the study of differentiable functions of a complex variable. The miracle of the subject is that complex differentiability, defined identically to the real case, is enormously stronger than real differentiability: a function differentiable once on an open set is automatically infinitely differentiable, equal to its Taylor series, and determined globally by its behaviour on any open subset or accumulation set. This rigidity makes complex analysis the most “geometric” of the analytic disciplines and the proper language for problems ranging from the evaluation of real integrals (residue calculus), through the location of zeros of polynomials (fundamental theorem of algebra) and the analytic continuation of the Riemann zeta function, to the conformal-mapping framework underlying two-dimensional fluid dynamics and the theory of Riemann surfaces that anchors algebraic geometry over $\mathbb{C}$. This note covers holomorphicity, the Cauchy theory, power series and the identity theorem, the maximum modulus principle, isolated singularities and Laurent series, residues and contour integration, conformal mapping and the Riemann mapping theorem, analytic continuation and Riemann surfaces, harmonic functions, special functions $(\Gamma ,\zeta )$, and the Picard theorems.

See also

• multivariate-calculus — real differentiation in ${\mathbb{R}}^2$; the Cauchy-Riemann equations are a rigidity condition on top of it.
• functional-analysis — Hardy spaces $H^p$ on the disk; bounded analytic functions $H^{\infty }$ as a Banach algebra.
• measure-theory-and-integration — Lebesgue framework underlying $L^p$ harmonic analysis on $\partial \mathbb{D}$.
• fft-spectral — Fourier series as boundary values of harmonic functions; Paley-Wiener.
• ode-numerical-methods — Frobenius method for ODE singular points.
• pde-methods — Laplace equation in 2D is the real part of holomorphic functions.
• algebraic-geometry-foundations — Riemann surfaces are 1-dimensional complex manifolds = smooth projective curves over $\mathbb{C}$.
• probability-fundamentals — characteristic functions are Fourier transforms; holomorphic moment generating functions.

1. The complex plane

1.1 Algebra of $\mathbb{C}$

$\mathbb{C}=\{z=x+iy:x,y\in \mathbb{R}\}$ with $i^2=-1$. Modulus $|z|=\sqrt{x^2+y^2}$, argument $\arg z\in (-\pi ,\pi ]$, polar form $z=r{e}^{i\theta }$. Conjugate $\bar{z}=x-iy$, satisfying $|z{|}^2=z\bar{z}$. Euler’s formula $e^{i\theta }=\cos \theta +i\sin \theta$ turns multiplication into a rotation: $r_1{e}^{i{\theta }_1}\cdot r_2{e}^{i{\theta }_2}=r_1{r}_2{e}^{i({\theta }_1+{\theta }_2)}$.

1.2 Topology

$\mathbb{C}$ inherits the Euclidean topology of ${\mathbb{R}}^2$. The Riemann sphere $\hat{\mathbb{C}}=\mathbb{C}\cup \{\infty \}$ is the one-point compactification, identified with $S^2$ via stereographic projection. The chordal metric on $\hat{\mathbb{C}}$ makes all Möbius transformations isometric up to scale.

1.3 Wirtinger derivatives

Introduce the formal operators $\frac{\partial }{\partial z}=\frac{1}{2}\left(\frac{\partial }{\partial x}-i\frac{\partial }{\partial y}\right),\frac{\partial }{\partial \bar{z}}=\frac{1}{2}\left(\frac{\partial }{\partial x}+i\frac{\partial }{\partial y}\right).$ These commute with each other and treat $z$ and $\bar{z}$ as independent. The Cauchy-Riemann equations become $\partial f/\partial \bar{z}=0$.

2. Holomorphic functions

2.1 Complex differentiability

$f:\Omega \to \mathbb{C}$ defined on open $\Omega \subseteq \mathbb{C}$ is holomorphic at $z_0$ if $f^{\prime }(z_0)={\lim }_{h\to 0}\frac{f(z_0+h)-f(z_0)}{h}$ exists, where $h\in \mathbb{C}$ approaches $0$ from any direction. Holomorphic on $\Omega$ means holomorphic at every point.

2.2 Cauchy-Riemann equations

Write $f(x+iy)=u(x,y)+iv(x,y)$. $f$ is holomorphic iff $u,v$ are real-differentiable and satisfy $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}.$ (Augustin-Louis Cauchy 1814; Bernhard Riemann 1851.)

Consequences when $f$ is $C^2$:

• $u$ and $v$ are both harmonic: $\Delta u=u_{xx}+u_{yy}=0$, $\Delta v=0$.
• $u$ and $v$ are conjugate harmonic functions.
• The Jacobian of $f$ as a map ${\mathbb{R}}^2\to {\mathbb{R}}^2$ equals $|f^{\prime }(z){|}^2$.

2.3 Holomorphic = analytic

A function is analytic at $z_0$ if it equals a convergent power series in a neighbourhood of $z_0$: $f(z)={\sum }_{n=0}^{\infty }{a}_n(z-z_0{)}^n.$ Theorem (Cauchy 1825 / Weierstrass 1841): every holomorphic function is analytic. This is the most consequential rigidity statement in the subject.

2.4 Examples and counterexamples

Holomorphic on $\mathbb{C}$ (entire functions):

• Polynomials.
• $e^z=\sum z^n/n!$ — non-constant, never zero, $|e^z|=e^x$.
• $\sin z,\cos z$ — defined via series; unbounded on $\mathbb{C}$.
• $\sinh z,\cosh z$.

Holomorphic on subdomains:

• $1/z$ on $\mathbb{C}\setminus \{0\}$.
• $\log z$ on $\mathbb{C}\setminus (-\infty ,0]$ (a branch cut is required).
• $\sqrt{z}$ similar.

Not holomorphic (real-differentiable but fails Cauchy-Riemann):

• $\bar{z}$: $\partial \bar{z}/\partial \bar{z}=1\ne 0$.
• $|z{|}^2=z\bar{z}$: $\partial /\partial \bar{z}=z\ne 0$ except at $0$.
• $\text{Re}(z)=x$.

3. Power series

3.1 Radius of convergence

For a power series $\sum a_n(z-z_0{)}^n$, the radius of convergence is $R=\frac{1}{{\mathrm{limsup}}_{n\to \infty }|a_n{|}^{1/n}}$ (Cauchy-Hadamard 1821 / Hadamard 1888). The series converges absolutely on $|z-z_0|<R$, diverges on $|z-z_0|>R$, and may do either on the boundary circle.

3.2 Operations on power series

Within the radius of convergence, power series can be added, multiplied, differentiated, and integrated term by term. Composition holds when the inner series maps into the domain of convergence of the outer. The resulting series has its own radius of convergence determined by the singularities (if any) of the analytic continuation.

3.3 Taylor’s theorem

A holomorphic function on $|z-z_0|<R$ equals its Taylor series there: $f(z)={\sum }_{n=0}^{\infty }\frac{f^{(n)}(z_0)}{n!}(z-z_0{)}^n,$ with $R$ equal to the distance from $z_0$ to the nearest singularity in the analytic continuation of $f$.

3.4 Examples

• $e^z=\sum z^n/n!$, $R=\infty$.
• $\sin z=\sum (-1{)}^n{z}^{2n+1}/(2n+1)!$, $\cos z=\sum (-1{)}^n{z}^{2n}/(2n)!$, both $R=\infty$.
• $1/(1-z)=\sum z^n$, $R=1$.
• $\log (1+z)=\sum (-1{)}^{n+1}{z}^n/n$, $R=1$.
• $(1+z{)}^{\alpha }=\sum (\genfrac{}{}{0ex}{}{\alpha }{n})z^n$, $R=1$ for $\alpha \notin {\mathbb{Z}}_{\ge 0}$.

4. Cauchy’s integral theorem

4.1 Contour integration

For a piecewise-$C^1$ curve $\gamma :[a,b]\to \Omega$ and continuous $f:\Omega \to \mathbb{C}$, ${\int }_{\gamma }f(z)dz={\int }_a^{b}f(\gamma (t)){\gamma }^{\prime }(t)dt.$ ML-inequality: $\left|{\int }_{\gamma }fdz\right|\le L(\gamma )\cdot {\max }_{\gamma }|f|$ where $L(\gamma )$ is the arc length.

4.2 Cauchy-Goursat theorem

If $f$ is holomorphic on a simply connected open set $\Omega$ and $\gamma$ is a closed piecewise-$C^1$ curve in $\Omega$, then ${\oint }_{\gamma }f(z)dz=0.$

Cauchy 1825 with the additional hypothesis that $f^{\prime }$ is continuous; Édouard Goursat 1900 removed that hypothesis.

Proof sketch (Goursat): For a triangle $T$, subdivide into four sub-triangles. The contour integral over $T$ is the sum of the four sub-integrals. Inductively pick the sub-triangle with the largest absolute integral; we get a nested sequence of triangles whose intersection is a single point $z_0$. Holomorphicity at $z_0$ gives $f(z)=f(z_0)+f^{\prime }(z_0)(z-z_0)+o(z-z_0)$. The first two terms integrate to zero around any closed curve; the error term gives a bound vanishing as the triangles shrink.

For arbitrary closed curves in a simply connected domain: triangulate and apply additivity, plus a homotopy argument.

4.3 Cauchy’s integral formula

For $f$ holomorphic on an open set containing the closed disk $\overline{D(z_0,r)}$: $f(z_0)=\frac{1}{2\pi i}{\oint }_{|z-z_0|=r}\frac{f(z)}{z-z_0}dz.$

More generally, for $z$ inside the contour $\gamma$ (with winding number $n(\gamma ,z)$), $n(\gamma ,z)f(z)=\frac{1}{2\pi i}{\oint }_{\gamma }\frac{f(\zeta )}{\zeta -z}d\zeta .$

4.4 Cauchy’s integral formula for derivatives

Differentiating under the integral sign (justified by uniform convergence): $f^{(n)}(z)=\frac{n!}{2\pi i}{\oint }_{\gamma }\frac{f(\zeta )}{(\zeta -z{)}^{n+1}}d\zeta .$

This immediately shows: holomorphic $\Rightarrow$ infinitely differentiable $\Rightarrow$ analytic (expand $1/(\zeta -z)$ as a geometric series and integrate term by term).

4.5 Cauchy’s estimate

If $|f|\le M$ on the circle $|z-z_0|=r$, then $|f^{(n)}(z_0)|\le n!M/r^n$. Tight: take $f(z)=M(z/r{)}^n$.

5. Consequences of Cauchy’s theory

5.1 Morera’s theorem

Giacinto Morera 1886: a continuous function $f$ on $\Omega$ with ${\oint }_{T}fdz=0$ for every triangle $T\subseteq \Omega$ is holomorphic on $\Omega$. The converse of Cauchy-Goursat. Useful for proving holomorphicity of functions defined as integrals.

5.2 Liouville’s theorem

Joseph Liouville (announced by Cauchy 1844): a bounded entire function is constant.

Proof: For entire bounded $f$ with $|f|\le M$, apply Cauchy’s estimate at any $z_0$ with arbitrarily large $r$: $|f^{\prime }(z_0)|\le M/r\to 0$. So $f^{\prime }\equiv 0$, hence $f$ constant.

Refinements:

• Entire $f$ with $|f(z)|\le A+B|z{|}^N$ is a polynomial of degree $\le N$.
• Phragmén-Lindelöf principle: appropriate growth bounds in sectors force polynomial degree.

5.3 Fundamental theorem of algebra

Carl Friedrich Gauss 1799 (first proof, with a gap fixed by Argand 1814; rigorous proof 1849). Every non-constant polynomial $p\in \mathbb{C}[z]$ has a root in $\mathbb{C}$.

Proof via Liouville: If $p$ has no root, $1/p$ is entire. Since $p$ has degree $\ge 1$, $|p(z)|\to \infty$ as $|z|\to \infty$, so $1/p$ is bounded. Liouville gives $1/p$ constant, contradiction.

By induction $p$ has exactly $\mathrm{deg}p$ roots in $\mathbb{C}$ counted with multiplicity.

5.4 Identity theorem

Let $f,g$ holomorphic on a connected open $\Omega$. If $\{z:f(z)=g(z)\}$ has an accumulation point in $\Omega$, then $f\equiv g$ on $\Omega$.

Proof sketch: Let $E$ = set of accumulation points of $\{f=g\}$ in $\Omega$. $E$ is closed (limit of accumulation points is an accumulation point). $E$ is open: if $z_0\in E$, the Taylor coefficients of $f-g$ at $z_0$ must all vanish (the values at the accumulation points and the continuity force all derivatives to vanish), so $f\equiv g$ near $z_0$, putting an open ball in $E$. Connectedness gives $E=\Omega$ or $E=\varnothing$.

This rigidity has no real-analysis analogue. The function $e^{-1/x^2}$ on $\mathbb{R}$ has all derivatives zero at $0$ yet is nonzero elsewhere — impossible for a complex-analytic function.

5.5 Maximum modulus principle

If $f$ is holomorphic on a connected open $\Omega$ and $|f|$ attains a local maximum, then $f$ is constant. Equivalently: on a bounded connected $\Omega$ with continuous $f$ on $\bar{\Omega }$, $|f|$ attains its max on $\partial \Omega$.

Proof via mean value: Cauchy’s integral formula gives $f(z_0)=\frac{1}{2\pi }{\int }_0^{2\pi }f(z_0+r{e}^{i\theta })d\theta$ — the mean value property. If $|f(z_0)|$ is a maximum, the inequality $|f(z_0)|\le \frac{1}{2\pi }\int |f|d\theta$ forces $|f|$ constant on the circle, then on a neighbourhood by varying $r$, then the identity theorem gives $f$ constant.

5.6 Schwarz lemma

Hermann Schwarz 1870: let $f$ holomorphic on $\mathbb{D}=\{|z|<1\}$ with $f(0)=0$ and $|f|\le 1$. Then $|f(z)|\le |z|$ for all $z\in \mathbb{D}$ and $|f^{\prime }(0)|\le 1$. Equality at any $z\ne 0$ or at $0$ forces $f(z)=e^{i\alpha }z$ for some real $\alpha$.

The Schwarz-Pick refinement gives the hyperbolic distance on $\mathbb{D}$ is contracted by holomorphic self-maps.

6. Isolated singularities

6.1 Classification

Let $f$ be holomorphic on a punctured disk $0<|z-z_0|<r$. The singularity at $z_0$ is:

• Removable if $f$ extends to a holomorphic function on the full disk. Equivalent: $f$ is bounded near $z_0$ (Riemann’s removable singularity theorem).
• Pole (of order $k$) if $(z-z_0{)}^{k}f(z)$ has a removable singularity at $z_0$ but $(z-z_0{)}^{k-1}f(z)$ does not. Equivalent: $|f(z)|\to \infty$ as $z\to z_0$.
• Essential if neither removable nor a pole. Equivalent: no power of $(z-z_0)$ removes the singularity.

6.2 Laurent series

For $f$ holomorphic in the annulus $r_1<|z-z_0|<r_2$: $f(z)={\sum }_{n=-\infty }^{\infty }{a}_n(z-z_0{)}^n,a_n=\frac{1}{2\pi i}{\oint }_{|z-z_0|=r}\frac{f(\zeta )}{(\zeta -z_0{)}^{n+1}}d\zeta$ for any $r$ with $r_1<r<r_2$. The principal part is ${\sum }_{n<0}{a}_n(z-z_0{)}^n$.

At an isolated singularity:

• Removable iff all $a_n=0$ for $n<0$.
• Pole of order $k$ iff $a_{-k}\ne 0$ and $a_n=0$ for $n<-k$.
• Essential iff infinitely many $a_n\ne 0$ for $n<0$.

6.3 Casorati-Weierstrass theorem

Felice Casorati 1868 / Karl Weierstrass 1876: near an essential singularity, $f$ takes values arbitrarily close to every complex number. That is, for every $c\in \mathbb{C}$ and every neighbourhood $U$ of $z_0$, $f(U\setminus \{z_0\})$ is dense in $\mathbb{C}$.

Strengthened to Picard’s great theorem (Émile Picard 1879): $f$ assumes every complex value, with at most one exception, infinitely often in every neighbourhood of an essential singularity. Example: $e^{1/z}$ near $0$ misses the value $0$ but takes every other value infinitely often.

6.4 Meromorphic functions

A function is meromorphic on $\Omega$ if it is holomorphic except for a set of isolated poles. Meromorphic functions on a connected $\Omega$ form a field (the field of meromorphic functions). On $\hat{\mathbb{C}}$, meromorphic = rational ($\hat{\mathbb{C}}$ has only rational functions as global meromorphic functions).

7. Residue calculus

7.1 Residue

For $f$ with isolated singularity at $z_0$, the residue is the coefficient $a_{-1}$ in the Laurent expansion: $\text{Res}(f,z_0)=a_{-1}=\frac{1}{2\pi i}{\oint }_{|z-z_0|=r}f(z)dz.$

For a simple pole at $z_0$ ($k=1$): $\text{Res}(f,z_0)={\lim }_{z\to z_0}(z-z_0)f(z).$

For a pole of order $k$: $\text{Res}(f,z_0)=\frac{1}{(k-1)!}{\lim }_{z\to z_0}\frac{d^{k-1}}{d{z}^{k-1}}\left[(z-z_0{)}^{k}f(z)\right].$

For $f=g/h$ with $g(z_0)\ne 0$ and $h$ having a simple zero at $z_0$: $\text{Res}(f,z_0)=g(z_0)/h^{\prime }(z_0)$.

7.2 Residue theorem

For $f$ meromorphic on a simply connected $\Omega$ with finitely many poles $z_1,\dots ,z_N$ inside a closed curve $\gamma$ avoiding the poles: ${\oint }_{\gamma }f(z)dz=2\pi i{\sum }_{k=1}^{N}n(\gamma ,z_k)\cdot \text{Res}(f,z_k).$

For a simple positively oriented loop encircling each pole once, this reduces to $2\pi i{\sum }_k\text{Res}(f,z_k)$.

7.3 Applications to real integrals

Type I: rational functions of $\cos ,\sin$ on $[0,2\pi ]$

Substitute $z=e^{i\theta }$, $d\theta =dz/(iz)$, $\cos \theta =(z+1/z)/2$, $\sin \theta =(z-1/z)/(2i)$. The integral becomes a contour integral on $|z|=1$.

Example: ${\int }_0^{2\pi }\frac{d\theta }{a+b\cos \theta }$ for $a>|b|>0$. Converts to $\oint \frac{dz/(iz)}{a+b(z+1/z)/2}=\oint \frac{2dz}{i(b{z}^2+2az+b)}$ with poles at $z=(-a\pm \sqrt{a^2-b^2})/b$. Only one pole lies inside $|z|=1$; the residue gives the answer $2\pi /\sqrt{a^2-b^2}$.

Type II: rational functions on $\mathbb{R}$

For $f=P/Q$ rational with $\mathrm{deg}Q\ge \mathrm{deg}P+2$ and no real poles: close the contour with a semicircle in the upper half-plane. ${\int }_{-\infty }^{\infty }f(x)dx=2\pi i{\sum }_{\text{poles in UHP}}\text{Res}(f,z_k).$

Example: ${\int }_{-\infty }^{\infty }dx/(1+x^2{)}^n$. Pole at $z=i$ of order $n$. Residue computation gives $\pi \left(\genfrac{}{}{0ex}{}{2n-2}{n-1}\right)/4^{n-1}$.

Type III: ${\int }_{-\infty }^{\infty }f(x)e^{iax}dx$, $a>0$

For $f$ rational with $\mathrm{deg}Q\ge \mathrm{deg}P+1$ and no real poles: by Jordan’s lemma the upper-half-plane semicircle contributes nothing in the limit, so ${\int }_{-\infty }^{\infty }f(x)e^{iax}dx=2\pi i{\sum }_{\text{UHP}}\text{Res}(f(z)e^{iaz},z_k).$

Standard: ${\int }_0^{\infty }\frac{\cos (ax)}{1+x^2}dx=\frac{\pi }{2}{e}^{-a}$.

Type IV: with branch cuts

Keyhole contour for ${\int }_0^{\infty }{x}^{s-1}f(x)dx$, etc. Example: ${\int }_0^{\infty }\frac{x^{s-1}}{1+x}dx=\pi /\sin (\pi s)$ for $0<s<1$.

7.4 Argument principle and Rouché’s theorem

Argument principle: for meromorphic $f$ on a region containing a closed curve $\gamma$ avoiding zeros and poles of $f$, $\frac{1}{2\pi i}{\oint }_{\gamma }\frac{f^{\prime }(z)}{f(z)}dz=N_0-N_p$ where $N_0$ = number of zeros and $N_p$ = number of poles inside $\gamma$, both counted with multiplicity. Equivalently, the left side counts the winding number of $f\circ \gamma$ around $0$.

Rouché’s theorem (Eugène Rouché 1862): if $f,g$ holomorphic on $\Omega$ containing $\gamma$ with $|g|<|f|$ on $\gamma$, then $f$ and $f+g$ have the same number of zeros inside $\gamma$. Use for locating roots of polynomials.

7.5 Hurwitz’s theorem

Adolf Hurwitz 1889: a uniform limit on compacts of holomorphic functions $f_n$ with no zeros is either identically zero or has no zeros. Refinement: zeros of $f_n$ accumulate only at zeros of $f$.

8. Conformal mappings

8.1 Conformal = angle-preserving

A map $f:\Omega \to \mathbb{C}$ is conformal at $z_0$ if it preserves oriented angles between intersecting curves at $z_0$. For holomorphic $f$ with $f^{\prime }(z_0)\ne 0$, $f$ is conformal at $z_0$ — the local linear approximation is multiplication by $f^{\prime }(z_0)$, a rotation by $\arg f^{\prime }(z_0)$ followed by scaling by $|f^{\prime }(z_0)|$.

A conformal map of two domains is a biholomorphism: bijective, holomorphic, with holomorphic inverse.

8.2 Möbius transformations

The maps $z\mapsto (az+b)/(cz+d)$ with $ad-bc\ne 0$ form the group $\text{PSL}(2,\mathbb{C})$ acting on $\hat{\mathbb{C}}$. Properties:

• Bijective conformal self-maps of $\hat{\mathbb{C}}$.
• Send circles and lines to circles and lines (with the convention that lines pass through $\infty$).
• 3-transitive: any three distinct points map to any three.
• Preserve cross-ratio $(z_1,z_2;z_3,z_4)=\frac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)}$.

The automorphism group of the open unit disk $\mathbb{D}$ is the subgroup of Möbius transformations preserving $\mathbb{D}$: $f(z)=e^{i\alpha }\frac{z-a}{1-\bar{a}z},|a|<1,\alpha \in \mathbb{R}.$

8.3 Riemann mapping theorem

Bernhard Riemann 1851 (statement, with a gap fixed by Koebe and others). For any simply connected proper open subset $\Omega ⊊\mathbb{C}$, there is a biholomorphism $\phi :\Omega \to \mathbb{D}$. The map is unique up to post-composition with an automorphism of $\mathbb{D}$ (three real parameters); normalisation $\phi (z_0)=0$, ${\phi }^{\prime }(z_0)>0$ at a fixed $z_0\in \Omega$ gives uniqueness.

Proof outline: Consider the family $\mathcal{F}$ of holomorphic injections $\Omega \to \mathbb{D}$ sending $z_0\mapsto 0$. Show $\mathcal{F}$ is nonempty (using simple connectedness to define a square root). The family is bounded, hence normal (Montel). Take the maximiser of $|{\phi }^{\prime }(z_0)|$ (achieved by uniform convergence on compacts) and prove it is surjective via a Schwarz-lemma argument.

8.4 Carathéodory’s boundary extension

Constantin Carathéodory 1913: for $\Omega ⊊\mathbb{C}$ a Jordan domain (bounded by a Jordan curve), the Riemann map extends to a homeomorphism $\bar{\Omega }\to \overline{\mathbb{D}}$. Extends the conformal map to the boundary.

8.5 Schwarz-Christoffel formula

Elwin Schwarz and Elwin Christoffel 1869: the conformal map from the upper half-plane $\mathbb{H}$ to a polygon with vertices $w_1,\dots ,w_n$ and interior angles ${\alpha }_k\pi$ is $f(z)=A+C{\int }_0^z{\prod }_{k=1}^n(\zeta -x_k{)}^{{\alpha }_k-1}d\zeta$ for some constants $A,C$ and pre-images $x_k\in \mathbb{R}$. The pre-image locations are determined (up to a Möbius transformation) by the polygon shape — a transcendental system.

8.6 Riemann uniformization theorem

Henri Poincaré 1907 / Paul Koebe 1907: every simply connected Riemann surface is biholomorphic to exactly one of:

• The Riemann sphere $\hat{\mathbb{C}}$ (compact).
• The complex plane $\mathbb{C}$ (parabolic).
• The unit disk $\mathbb{D}$ (hyperbolic, equivalent to the upper half-plane).

This generalises the Riemann mapping theorem and is the foundation of the geometric trichotomy in 1D complex geometry (spherical / Euclidean / hyperbolic).

9. Analytic continuation

9.1 Direct continuation

Two power series $f_1,f_2$ centred at $z_1,z_2$ with overlapping disks of convergence are direct analytic continuations of each other if they agree on the overlap. By the identity theorem the continuation is unique.

A complete analytic function is the maximal connected set of all direct analytic continuations of a given germ.

9.2 The monodromy theorem

If $f$ has an analytic continuation along every path in a simply connected domain $\Omega$, the continuations along homotopic paths agree, and the function is single-valued on $\Omega$. Failure of simple connectedness can produce multi-valued functions: $\log z$ on $\mathbb{C}\setminus \{0\}$ acquires $2\pi i$ on each loop around the origin.

9.3 Riemann surfaces

The natural domain of a multi-valued holomorphic function is a Riemann surface — a one-dimensional complex manifold on which the function becomes single-valued. Examples:

• $\sqrt{z}$: double cover of $\mathbb{C}\setminus \{0\}$ glued at $0$; surface = $\mathbb{C}$ (parametrise as $z=w^2$).
• $\log z$: universal cover of $\mathbb{C}\setminus \{0\}$; surface = $\mathbb{C}$.
• $\sqrt{(z-a_1)(z-a_2)(z-a_3)(z-a_4)}$: branched double cover; surface = torus (an elliptic curve!).

A compact Riemann surface is the analytic / topological analogue of a smooth projective algebraic curve over $\mathbb{C}$, by GAGA / Riemann’s existence theorem.

9.4 Genus

Compact connected Riemann surfaces are classified topologically by genus $g$:

• $g=0$: $\hat{\mathbb{C}}$.
• $g=1$: elliptic curves $\mathbb{C}/\Lambda$ for a lattice $\Lambda$.
• $g\ge 2$: hyperbolic surfaces $\mathbb{D}/\Gamma$ for Fuchsian groups $\Gamma$.

Riemann-Roch on a Riemann surface (genus $g$): for a divisor $D$ on $X$, $\ell (D)-\ell (K-D)=\mathrm{deg}D+1-g$ where $K$ is a canonical divisor. See algebraic-geometry-foundations.

10. Harmonic functions

10.1 Definition and properties

$u:\Omega \to \mathbb{R}$ is harmonic if $\Delta u=u_{xx}+u_{yy}=0$. On simply connected $\Omega$, every harmonic $u$ is the real part of a holomorphic function (the harmonic conjugate $v$ is determined up to a constant by the Cauchy-Riemann equations).

Harmonic functions on ${\mathbb{R}}^2$ inherit from holomorphic functions:

• Mean value property: $u(z_0)=\frac{1}{2\pi }{\int }_0^{2\pi }u(z_0+r{e}^{i\theta })d\theta$.
• Maximum principle.
• Smoothness: $C^{\infty }$ if $C^2$ (and even real-analytic).
• Identity principle: zero on an open set $\Rightarrow$ zero on the connected component.

10.2 Poisson integral formula

The Dirichlet problem on the unit disk: given $g:\partial \mathbb{D}\to \mathbb{R}$ continuous, find harmonic $u$ on $\mathbb{D}$ with boundary values $g$. Solution: $u(r{e}^{i\theta })=\frac{1}{2\pi }{\int }_0^{2\pi }\frac{1-r^2}{1-2r\cos (\theta -\phi )+r^2}g(e^{i\phi })d\phi .$

The Poisson kernel $P_r(\theta -\phi )=(1-r^2)/(1-2r\cos (\theta -\phi )+r^2)$ is an approximate identity as $r\to 1^{-}$.

10.3 Subharmonic functions

$u$ is subharmonic if $u\le$ mean value over circles. Equivalently, $\Delta u\ge 0$ in the distribution sense. The maximum principle still applies: subharmonic $u$ attains max on the boundary. Plays the role in 2D potential theory that convex functions play on the real line.

11. Special functions

11.1 Gamma function

Leonhard Euler 1729-1730. Defined for $\text{Re}(s)>0$: $\Gamma (s)={\int }_0^{\infty }{t}^{s-1}{e}^{-t}dt.$ Functional equation: $\Gamma (s+1)=s\Gamma (s)$, with $\Gamma (n+1)=n!$ for $n\in {\mathbb{Z}}_{\ge 0}$.

Analytic continuation to all of $\mathbb{C}$ with simple poles at $0,-1,-2,\dots$ with residues $(-1{)}^n/n!$. Never zero.

Identities:

• Reflection: $\Gamma (s)\Gamma (1-s)=\pi /\sin (\pi s)$.
• Duplication (Legendre): $\Gamma (s)\Gamma (s+1/2)=2^{1-2s}\sqrt{\pi }\Gamma (2s)$.
• Stirling: $\Gamma (s)\sim \sqrt{2\pi /s}(s/e{)}^s$ as $|s|\to \infty$, $|\arg s|<\pi -\delta$.
• Weierstrass product: $1/\Gamma (s)=s{e}^{\gamma s}{\prod }_{n=1}^{\infty }(1+s/n)e^{-s/n}$, with $\gamma$ the Euler-Mascheroni constant.

11.2 Riemann zeta function

Bernhard Riemann 1859. Defined for $\text{Re}(s)>1$: $\zeta (s)={\sum }_{n=1}^{\infty }\frac{1}{n^s}={\prod }_p\frac{1}{1-p^{-s}}$ (Euler product over primes $p$).

Analytic continuation to $\mathbb{C}\setminus \{1\}$ with a simple pole at $s=1$ of residue $1$. Functional equation: $\zeta (s)=2^s{\pi }^{s-1}\sin (\pi s/2)\Gamma (1-s)\zeta (1-s).$ Equivalently $\xi (s)=\xi (1-s)$ where $\xi (s)=\frac{1}{2}s(s-1){\pi }^{-s/2}\Gamma (s/2)\zeta (s)$.

Trivial zeros: $s=-2,-4,-6,\dots$ (from the $\sin (\pi s/2)$ factor in the functional equation).

Critical strip: $0<\text{Re}(s)<1$. Riemann hypothesis (Riemann 1859, Hilbert problem 8, Clay Millennium): all non-trivial zeros have $\text{Re}(s)=1/2$. Verified computationally for the first $10^{13}$ zeros; widely believed.

Zeros control the distribution of primes. The Prime Number Theorem ($\pi (x)\sim x/\log x$) is equivalent to $\zeta (s)\ne 0$ on $\text{Re}(s)=1$.

11.3 Other classical functions

• Beta function: $B(p,q)={\int }_0^1{t}^{p-1}(1-t{)}^{q-1}dt=\Gamma (p)\Gamma (q)/\Gamma (p+q)$.
• Hypergeometric ${}_2{F}_1(a,b;c;z)={\sum }_{n=0}^{\infty }\frac{(a{)}_n(b{)}_n}{(c{)}_n}\frac{z^n}{n!}$ — solves the hypergeometric ODE; analytic continuation to $\mathbb{C}\setminus [1,\infty )$.
• Bessel functions $J_{\nu },Y_{\nu },H_{\nu }^{(1)},H_{\nu }^{(2)}$ — solutions to Bessel’s equation, central in cylindrical-coordinate PDE.
• Weierstrass $\wp$ function — doubly periodic meromorphic function on $\mathbb{C}/\Lambda$; the basic example of an elliptic function.
• Theta functions — quasi-periodic functions on $\mathbb{C}$; generate modular forms.

12. Applications to ODEs

12.1 Frobenius method

Ferdinand Frobenius 1873. For a linear ODE $y^{\prime \prime }+p(z)y^{\prime }+q(z)y=0$ with regular singular point at $z_0$ (meaning $(z-z_0)p$ and $(z-z_0{)}^{2}q$ are holomorphic at $z_0$), solutions exist in the form $y(z)=(z-z_0{)}^r{\sum }_{n=0}^{\infty }{a}_n(z-z_0{)}^n$ where $r$ satisfies the indicial equation $r(r-1)+p_{0}r+q_0=0$ ($p_0,q_0$ leading coefficients). When the roots differ by a non-integer, both give independent series solutions; when they coincide or differ by an integer, the second solution involves a logarithm.

12.2 Irregular singular points and Stokes phenomenon

At irregular singularities (where the regular-singular condition fails) formal power series may diverge. Asymptotic expansions are valid in sectors and may differ in different sectors — the Stokes phenomenon (George Gabriel Stokes 1857 in the context of the Airy function).

12.3 Linear ODE on Riemann surfaces

Linear ODEs on the punctured Riemann sphere lead to Fuchsian equations and the Riemann-Hilbert correspondence: classifying flat holomorphic connections on a punctured surface up to gauge equivalence is equivalent to classifying their monodromy representations. Modern incarnation: the geometric Langlands program.

13. Picard’s theorems

13.1 Little Picard

Émile Picard 1879: a non-constant entire function $f:\mathbb{C}\to \mathbb{C}$ omits at most one value of $\mathbb{C}$.

Sharp: $e^z$ omits only $0$. Polynomials omit nothing.

13.2 Great Picard

Near an essential singularity, an analytic function takes every complex value with at most one exception, infinitely often.

Equivalent form: a non-constant meromorphic function on $\mathbb{C}$ (i.e., a transcendental meromorphic function) omits at most two values of $\hat{\mathbb{C}}$.

13.3 Bloch’s theorem and the proof strategy

Modern proofs of Picard’s theorems flow through Bloch’s theorem (André Bloch 1925): the image of a holomorphic $f:\mathbb{D}\to \mathbb{C}$ with $f(0)=0$ and $f^{\prime }(0)=1$ contains a disk of radius $B\ge \sqrt{3}/4$ — a universal constant. Iterating to entire functions and noting that omitting two values puts $f$ into the universal cover of $\mathbb{C}\setminus \{a,b\}$ (a hyperbolic surface) gives Picard’s theorems via the Schwarz-Pick lemma.

The sharp Bloch constant is conjectured to be $\frac{\Gamma (1/3)\Gamma (11/12)}{(1+\sqrt{3}{)}^{1/2}\Gamma (1/4)}\approx 0.4719$; only the lower bound $\sqrt{3}/4\approx 0.4330$ is rigorously established (Ahlfors, Heins; recent improvements push the lower bound slightly higher).

14. Hardy spaces and boundary behaviour

14.1 Definition

For $0<p\le \infty$, the Hardy space $H^p(\mathbb{D})$ consists of holomorphic functions on $\mathbb{D}$ with $\Vert f{\Vert }_{H^p}={\sup }_{0<r<1}{\left(\frac{1}{2\pi }{\int }_0^{2\pi }|f(r{e}^{i\theta }){|}^{p}d\theta \right)}^{1/p}<\infty$ (with the obvious modification for $p=\infty$).

14.2 Fatou’s theorem

Pierre Fatou 1906: every $f\in H^p$ ($1\le p\le \infty$) has non-tangential boundary values $f^{\ast }(e^{i\theta })$ almost everywhere on $\partial \mathbb{D}$, and $\Vert f{\Vert }_{H^p}=\Vert f^{\ast }{\Vert }_{L^p(\partial \mathbb{D})}$.

14.3 Inner-outer factorization

Beurling 1949: every $f\in H^p$ factors uniquely as $f=B\cdot S\cdot O$ with

• $B$ a Blaschke product (encoding zeros),
• $S$ a singular inner function (encoding boundary singularities),
• $O$ an outer function (with $|O|$ determined by $|f^{\ast }|$).

Blaschke product for zeros $\{a_n\}\subset \mathbb{D}$ with $\sum (1-|a_n|)<\infty$: $B(z)={\prod }_n\frac{{\bar{a}}_n}{|a_n|}\cdot \frac{a_n-z}{1-{\bar{a}}_{n}z}.$

15. Connections and open problems

• Number theory: the zeta function, $L$-functions, modular forms; see algebraic-geometry-foundations.
• Algebraic geometry: compact Riemann surfaces = smooth projective curves over $\mathbb{C}$.
• Differential geometry: Kähler manifolds = complex manifolds with compatible Riemannian metric.
• Mathematical physics: 2D conformal field theory, partition functions, quantum gravity in 2D.
• PDE: complex methods solve the Laplace equation in 2D and the Stokes / Navier-Stokes equations in stream-function formulation.
• Inverse problems: Calderón problem in 2D solved via complex-geometric optics (Astala-Päivärinta 2006 Annals of Math 163).
• Random matrices: zeros of the zeta function on the critical line are conjecturally distributed like eigenvalues of random Hermitian matrices (Montgomery 1973, Odlyzko 1989).
• Open: Riemann hypothesis. Lindelöf hypothesis on $\zeta$ growth in the critical strip. Generalised Riemann hypothesis for $L$-functions. Sarnak’s conjecture on Möbius randomness.

Further reading

• Ahlfors, L. V. 1979. Complex Analysis (3rd ed.). McGraw-Hill.
• Conway, J. B. 1978. Functions of One Complex Variable (2nd ed.). Springer GTM 11.
• Stein, E. M. and R. Shakarchi. 2003. Complex Analysis. Princeton Lectures II.
• Rudin, W. 1987. Real and Complex Analysis (3rd ed.). McGraw-Hill.
• Lang, S. 1999. Complex Analysis (4th ed.). Springer.
• Marsden, J. E. and M. J. Hoffman. 1999. Basic Complex Analysis (3rd ed.). Freeman.
• Needham, T. 1997. Visual Complex Analysis. Oxford.
• Remmert, R. 1991, 1998. Theory of Complex Functions and Classical Topics in Complex Function Theory. Springer.
• Hörmander, L. 1990. An Introduction to Complex Analysis in Several Variables (3rd ed.).
• Farkas, H. M. and I. Kra. 1992. Riemann Surfaces (2nd ed.). Springer GTM 71.
• Edwards, H. M. 2001. Riemann’s Zeta Function. Dover.
• Titchmarsh, E. C. (Heath-Brown rev.) 1986. The Theory of the Riemann Zeta-function (2nd ed.). Oxford.
• Whittaker, E. T. and G. N. Watson. 1927. A Course of Modern Analysis (4th ed.). Cambridge. Classic reference for special functions.
• Olver, F. W. J. et al. (eds.) 2010. NIST Handbook of Mathematical Functions. Cambridge.

Adjacent

• multivariate-calculus
• functional-analysis
• measure-theory-and-integration
• fft-spectral
• ode-numerical-methods
• pde-methods
• algebraic-geometry-foundations
• probability-fundamentals