1 Useful Techniques

1.1 Notation

Notation Definition
\({\mathbb{D}}\mathrel{\vcenter{:}}=\left\{{z {~\mathrel{\Big|}~}{\left\lvert {z} \right\rvert} \leq 1}\right\}\) The unit disc
\({\mathbb{H}}\mathrel{\vcenter{:}}=\left\{{x+iy {~\mathrel{\Big|}~}y > 0}\right\}\) The upper half-plane
\(X_{1\over 2}\) A “half version of \(X\),” see examples
\({\mathbb{H}}_{1\over 2}\) The first quadrant
\({\mathbb{D}}_{1\over 2}\) The portion of the first quadrant inside the unit disc
\(S \mathrel{\vcenter{:}}=\left\{{x + iy {~\mathrel{\Big|}~}x\in {\mathbb{R}},\, 0<y<\pi}\right\}\) The horizonta strip

If you want to show that a function \(f\) is constant, try one of the following:

If you additionally want to show \(f\) is zero, try one of these:

1.2 Greatest Hits

Things to know well:

1.3 Basic but Useful Facts

\begin{align*} \sum_{k=1}^{n} k &=\frac{n(n+1)}{2} \\ \sum_{k=1}^{n} k^{2} &=\frac{n(n+1)(2 n+1)}{6} \\ \sum_{k=1}^{n} k^{3} &=\frac{n^{2}(n+1)^{2}}{4} \\ \log(z) &= \sum_{n=0}^\infty { (-1)^n \over n} \qty{z-a}^n \\ {\frac{\partial }{\partial z}\,} \sum_{j=0}^\infty a_j z^j &= \sum_{j=0}^\infty a_{j+1}z^j \end{align*}

1.4 Advice

2 Definitions

A function \(f:\Omega \to {\mathbb{C}}\) is analytic at \(z_0\in \Omega\) iff there exists a power series \(g(z) = \sum a_n (z-z_0)^n\) with radius of convergence \(R>0\) and a neighborhood \(U\ni z_0\) such that \(f(z) = g(z)\) on \(U\).

\begin{align*} u_x = v_y \quad\text{and}\quad u_y = -v_x \\ \frac{\partial u}{\partial r}=\frac{1}{r} \frac{\partial v}{\partial \theta} \quad \text { and } \quad \frac{\partial v}{\partial r}=-\frac{1}{r} \frac{\partial u}{\partial \theta} \\ .\end{align*}

A function that is holomorphic on \({\mathbb{C}}\) is said to be entire.

A singularity \(z_0\) is essential iff it is neither removable nor a pole.

Equivalently, a Laurent series expansion about \(z_0\) has a principal part with infinitely many terms.

A function \(f:{\mathbb{C}}\to {\mathbb{C}}\) is holomorphic at \(z_0\) if the following limit converges: \begin{align*} \lim_{h\to 0} {1\over h} \qty{f(z_0 + h) - f(z_0)} \mathrel{\vcenter{:}}= f'(z_0) .\end{align*}

A real function of two variables \(u(x, y)\) is harmonic iff its Laplacian vanishes: \begin{align*} \Delta u \mathrel{\vcenter{:}}=\qty{{\frac{\partial ^2}{\partial x^2}\,} + {\frac{\partial ^2}{\partial y^2}\,}}u = 0 .\end{align*}

A function \(f:\Omega\to{\mathbb{C}}\) is meromorphic iff there exists a sequence \(\left\{{z_n}\right\}\) such that

If \(f\) is either holomorphic or has a pole at \(z=\infty\) is said to be meromorphic on \({\mathbb{CP}}^1\).

A pole \(z_0\) of a meromorphic function \(f(z)\) is a zero of \(g(z) \mathrel{\vcenter{:}}={1\over f(z)}\). If there exists an \(n\) such that \begin{align*} \lim_{z\to z_0}\qty{z-z_0}^nf(z) \end{align*} is holomorphic and nonzero in a neighborhood of \(z_0\), then the minimal such \(n\) is the order of the pole. A pole of order 1 is said to be a simple pole.

The pole \(z_0\) is isolated iff there exists a neighborhood of \(z_0\) containing no other poles of \(f\).

In a Laurent series \(f(z) \mathrel{\vcenter{:}}=\sum_{n\in {\mathbb{Z}}} c_n (z-z_0)^n\), the principal part of \(f\) at \(z_0\) consists of terms with negative degree: \begin{align*} P_f(z) \mathrel{\vcenter{:}}=\sum_{n=1}^\infty c_{-n}(z-z_0)^{-n} .\end{align*}

The residue of \(f\) at \(z_0\) is the coefficient \(c_{-1}\).

If \(z_0\) is a singularity of \(f\) and there exists a \(g\) such that \(f(z) = g(z)\) for all \(z\) in some deleted neighborhood \(U\setminus\left\{{z_0}\right\}\), then \(z_0\) is a removable singularity of \(f\).

A map of the following form is a linear fractional transformation: \begin{align*} T(z) = {az + b \over cz + d} ,\end{align*} where the denominator is assumed to not be a multiple of the numerator.

These have inverses given by \begin{align*} T^{-1}(w) = {dw-b \over -cw + a} .\end{align*}

A bijective holomorphic map is a conformal (or angle-preserving) map, a.k.a. a biholomorphism.

Note that some authors just require the weaker condition that \(f'(z) \neq 0\) for any point.

3 Theorems

3.1 Basics

If \(\Omega \subseteq {\mathbb{C}}\) is bounded with \({{\partial}}\Omega\) piecewise smooth and \(f, g\in C^1(\mkern 1.5mu\overline{\mkern-1.5mu\Omega\mkern-1.5mu}\mkern 1.5mu)\), then \begin{align*}\int_{{{\partial}}\Omega} f\, dx + g\, dy = \iint_{\Omega} \qty{ {\frac{\partial g}{\partial x}\,} - {\frac{\partial f}{\partial y}\,} } \, dA.\end{align*}

Define the forward difference operator \(\Delta f_k = f_{k+1} - f_k\), then \begin{align*} \sum_{k=m}^n f_k \Delta g_k + \sum_{k=m}^{n-1} g_{k+1}\Delta f_k = f_n g_{n+1} - f_m g_m \end{align*}

Note: compare to \(\int_a^b f \, dg + \int_a^b g\, df = f(b) g(b) - f(a) g(a)\).

3.2 Holomorphic and Entire Functions

3.2.1 Key Theorems

If \(f\) is holomorphic on \(\Omega\), then \begin{align*} \int_{{{\partial}}\Omega} f(z) \, dz = 0 .\end{align*}

Slogan: closed path integrals of holomorphic functions vanish.

If \(f\) is continuous on a domain \(\Omega\) and \(\int_T f = 0\) for every triangle \(T\subset \Omega\), then \(f\) is holomorphic.

Slogan: if every integral along a triangle vanishes, implies holomorphic.

If \(f\) is holomorphic and nonconstant on an open connected region \(\Omega\), then \({\left\lvert {f} \right\rvert}\) can not attain a maximum on \(\Omega\). If \(\Omega\) is bounded and \(f\) is continuous on \(\mkern 1.5mu\overline{\mkern-1.5mu\Omega\mkern-1.5mu}\mkern 1.5mu\), then \(\max_{\mkern 1.5mu\overline{\mkern-1.5mu\Omega\mkern-1.5mu}\mkern 1.5mu} {\left\lvert {f} \right\rvert}\) occurs on \({{\partial}}\Omega\). Conversely, if \(f\) attains a local supremum at \(z_0 \in \Omega\), then \(f\) is constant on \(\Omega\).

If \(f\) is entire and bounded, \(f\) is constant.

Suppose \(f\) is holomorphic on \(\Omega\), then \begin{align*} f(z) = {1 \over 2\pi i} \oint_{{{\partial}}\Omega} {f(\xi) \over \xi-z}\,d\xi \end{align*} and \begin{align*} {\frac{\partial ^nf }{\partial z^n}\,}(z) = {n! \over 2\pi i} \int_{{{\partial}}\Omega} {f(\xi) \over (\xi - z)^{n+1}} \,d\xi .\end{align*}

For \(z_0 \in D_R(z_0) \subset \Omega\), we have \begin{align*} {\left\lvert { f^{(n)} (z_0) } \right\rvert} \leq \frac{n !}{2 \pi} \int_{0}^{2 \pi} \frac{ {\left\lVert {f} \right\rVert}_{\infty} } {R^{n+1}} R \,d\theta = \frac{n !{\left\lVert {f} \right\rVert}_{\infty}}{R^n} ,\end{align*} where \({\left\lVert {f} \right\rVert}_{\infty}\mathrel{\vcenter{:}}=\sup_{z\in C_R} {\left\lvert {f(z)} \right\rvert}\).

The \(n\)th Taylor coefficient of an analytic function is at most \(\sup_{{\left\lvert {z} \right\rvert} = R} {\left\lvert {f} \right\rvert}/R^n\).

For \(f\) meromorphic in \(\gamma^\circ\), if \(f\) has no poles and is nonvanishing on \(\gamma\) then \begin{align*} \Delta_\gamma \arg f(z) = \int_\gamma {f'(z) \over f(z)} \,dz = 2\pi (Z_f - P_f) ,\end{align*} where \(Z_f\) and \(P_f\) are the number of zeros and poles respectively enclosed by \(\gamma\), counted with multiplicity.

If \(f, g\) are analytic on a domain \(\Omega\) with finitely many zeros in \(\Omega\) and \(\gamma \subset \Omega\) is a closed curve surrounding each point exactly once, where \({\left\lvert {g} \right\rvert} < {\left\lvert {f} \right\rvert}\) on \(\gamma\), then \(f\) and \(f+g\) have the same number of zeros.

Alternatively:

Suppose \(f = g + h\) with \(g \neq 0, \infty\) on \(\gamma\) with \({\left\lvert {g} \right\rvert} > {\left\lvert {h} \right\rvert}\) on \(\gamma\). Then \begin{align*}\Delta_\gamma \arg(f) = \Delta_\gamma \arg(h)\quad\text{ and } Z_f - P_f = Z_g - P_g.\end{align*}

If \(f\) is holomorphic on an open set \(\Omega\) containing a curve \(\gamma\) and its interior \(\gamma^\circ\), except for finitely many poles \(\left\{{z_k}\right\}_{k=1}^N \subset \gamma^\circ\). Then \begin{align*} \int_\gamma f(z) \,dz = 2\pi i \sum_{k=1}^N \mathop{\mathrm{Res}}_{z_k} f .\end{align*}

The fractional linear transformation given by \(F(z) = {i - z \over i + z}\) maps \({\mathbb{D}}\to {\mathbb{H}}\) with inverse \(G(w) = i {1-w \over 1 + w}\).

If \(f: {\mathbb{D}}\to {\mathbb{D}}\) is holomorphic with \(f(0) = 0\), then

  1. \({\left\lvert {f(z)} \right\rvert} \leq {\left\lvert {z} \right\rvert}\) for all \(z\in {\mathbb{D}}\)
  2. \({\left\lvert {f'(0)} \right\rvert} \leq 1\).

Moreover, if \({\left\lvert {f(z_0)} \right\rvert} = {\left\lvert {z_0} \right\rvert}\) for any \(z_0\in {\mathbb{D}}\) or \({\left\lvert {f'(0)} \right\rvert} = 1\), then \(f\) is a rotation

\begin{align*} f(z_0) = {1\over \pi r^2} \iint_{D_r(z_0)} f(z)\, dA .\end{align*}

If \(f\) is continuous and holomorphic on \({\mathbb{H}}^+\) and real-valued on \({\mathbb{R}}\), then the extension defined by \(F(z) = \mkern 1.5mu\overline{\mkern-1.5muf(\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu)\mkern-1.5mu}\mkern 1.5mu\) for \(z\in {\mathbb{H}}^-\) is a well-defined holomorphic function on \({\mathbb{C}}\).

There are 8 major types of conformal maps:

Type/Domains Formula
Translation/Dilation/Rotation \(z\mapsto e^{i\theta}(cz + h)\)
Sectors to sectors \(z\mapsto z^n\)
\({\mathbb{D}}_{1\over 2} \to {\mathbb{H}}_{1\over 2}\), the first quadrant \(z\mapsto {1+z \over 1-z}\)
\({\mathbb{H}}\to S\) \(z\mapsto \log(z)\)
\({\mathbb{D}}_{1\over 2} \to S_{1\over 2}\) \(z\mapsto \log(z)\)
\(S_{1\over 2} \to {\mathbb{D}}_{1\over 2}\) \(z\mapsto e^{iz}\)
\({\mathbb{D}}_{1\over 2} \to {\mathbb{H}}\) \(z\mapsto {1\over 2}\qty{z + {1\over z}}\)
\(S_{1\over 2} \to {\mathbb{H}}\) \(z\mapsto \sin(z)\)

Conformal maps \({\mathbb{D}}\to{\mathbb{D}}\) have the form \begin{align*} g(z) = \lambda {1-a \over 1 - \mkern 1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5mu z}, \quad {\left\lvert {a} \right\rvert} < 1, \quad {\left\lvert {\lambda} \right\rvert} = 1 .\end{align*}

3.2.2 Others

If \(\Omega\) is simply connected, nonempty, and not \({\mathbb{C}}\), then for every \(z_{0}\in \Omega\) there exists a unique conformal map \(F:\Omega \to {\mathbb{D}}\) such that \(F(z_{0}) = 0\) and \(F'(z_{0}) > 0\).

Thus any two such sets \(\Omega_{1}, \Omega_{2}\) are conformally equivalent.

If \(f\) is holomorphic on \(\Omega\) except possibly at \(z_0\) and \(f\) is bounded on \(\Omega\setminus\left\{{z_0}\right\}\), then \(z_0\) is a removable singularity.

If \(f(z) = u(x, y) + iv(x, y)\) is holomorphic, then \(u, v\) are harmonic.

\(f\) is holomorphic at \(z_0\) iff there exists an \(a\in {\mathbb{C}}\) such that \begin{align*} f(z_0 + h) - f(z_0) - ah = h \psi(h), \quad \psi(h) \overset{h\to 0}\to 0 .\end{align*} In this case, \(a = f'(z_0)\).

If \(f = u+iv\) with \(u, v\in C^1({\mathbb{R}})\) satisfying the Cauchy-Riemann equations on \(\Omega\), then \(f\) is holomorphic on \(\Omega\) and \(f'(z) = {\frac{\partial f}{\partial z}\,} = {1 \over 2} \qty{{\frac{\partial }{\partial x}\,} + {1\over i} {\frac{\partial }{\partial y}\,}}f\).

\begin{align*} \frac{\partial u}{\partial r}=\frac{1}{r} \frac{\partial v}{\partial \theta} \quad \text { and } \quad \frac{1}{r} \frac{\partial u}{\partial \theta}=-\frac{\partial v}{\partial r} .\end{align*}

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Any holomorphic non-constant map is an open map.

3.3 Series and Analytic Functions

Any power series is smooth and its derivatives can be obtained using term-by-term differentiation.

A series of functions \(\sum_{n=1}^\infty f_n(x)\) converges uniformly iff \begin{align*} \lim_{n\to \infty} {\left\lVert { \sum_{k\geq n} f_k } \right\rVert}_\infty = 0 .\end{align*}

If \(\left\{{f_n}\right\}\) with \(f_n: \Omega \to {\mathbb{C}}\) and there exists a sequence \(\left\{{M_n}\right\}\) with \({\left\lVert {f_n} \right\rVert}_\infty \leq M_n\) and \(\sum_{n\in {\mathbb{N}}} M_n < \infty\), then \(f(x) \mathrel{\vcenter{:}}=\sum_{n\in {\mathbb{N}}} f_n(x)\) converges absolutely and uniformly on \(\Omega\).

Moreover, if the \(f_n\) are continuous, by the uniform limit theorem, \(f\) is again continuous.

\(f(z) = e^z\) is uniformly convergent in any disc in \({\mathbb{C}}\).

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Apply the estimate \begin{align*} {\left\lvert {e^z} \right\rvert} \leq \sum {{\left\lvert {z} \right\rvert}^n \over n!} = e^{{\left\lvert {z} \right\rvert}} .\end{align*} Now by the \(M{\hbox{-}}\)test, \begin{align*} {\left\lvert {z} \right\rvert} \leq R < \infty \implies {\left\lvert {\sum {z^n \over n!}} \right\rvert} \leq e^R < \infty .\end{align*}

For a power series \(f(z) = \sum a_n z^n\), define \(R\) by \begin{align*} {1\over R}\mathrel{\vcenter{:}}=\limsup {\left\lvert {a_n} \right\rvert}^{1\over n} .\end{align*}

Then \(f\) converges absolutely on \({\left\lvert {z} \right\rvert} < R\) and diverges on \({\left\lvert {z} \right\rvert} > R\).

3.4 Others

If \(f\) is holomorphic on \(\Omega\setminus\left\{{z_0}\right\}\) where \(z_0\) is an essential singularity, then for every \(V\subset \Omega\setminus\left\{{z_0}\right\}\), \(f(V)\) is dense in \({\mathbb{C}}\).

The image of a disc punctured at an essential singularity is dense in \({\mathbb{C}}\).

If \(f:{\mathbb{C}}\to {\mathbb{C}}\) is entire and nonconstant, then \(\operatorname{im}(f)\) is either \({\mathbb{C}}\) or \({\mathbb{C}}\setminus\left\{{z_0}\right\}\) for some point \(z_0\).

If \(f\) is holomorphic on a bounded connected domain \(\Omega\) and there exists a sequence \(\left\{{z_i}\right\}\) with a limit point in \(\Omega\) such that \(f(z_i) = 0\), then \(f\equiv 0\) on \(\Omega\).

Two functions agreeing on a set with a limit point are equal on a domain.

The ring of holomorphic functions on a domain in \({\mathbb{C}}\) has no zero divisors.

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???

If \(z_0\) is a zero of \(f'\) of order \(n\), then \(f\) is \((n+1)\)-to-one in a neighborhood of \(z_0\).

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For \(\Omega\subseteq{\mathbb{C}}\), show that \(A({\mathbb{C}})\mathrel{\vcenter{:}}=\left\{{f: \Omega \to {\mathbb{C}}{~\mathrel{\Big|}~}f\text{ is bounded}}\right\}\) is a Banach space.

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Apply Morera’s Theorem and Cauchy’s Theorem

4 Residues

Check: do you need residues? You may be able to just compute an integral

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If \(z_0\) is a simple pole of \(f\), then \begin{align*} \mathop{\mathrm{Res}}_{z_0}f = \lim_{z\to z_0} (z-z_0) f(z) .\end{align*}

Let \(f(z) = \frac{1}{1+z^2}\), then \(\mathop{\mathrm{Res}}(i, f) = \frac{1}{2i}\).

If \(f\) has a pole \(z_0\) of order \(n\), then \begin{align*} \mathop{\mathrm{Res}}_{z=z_0} f = \lim_{z\to z_0} {1 \over (n-1)!} \qty{{\frac{\partial }{\partial z}\,}}^{n-1} (z-z_0)^n f(z) .\end{align*}

5 Conformal Maps

A bijective holomorphic map automatically has a holomorphic inverse. This can be weakened: an injective holomorphic map satisfies \(f'(z) \neq 0\) and \(f ^{-1}\) is well-defined on its range and holomorphic.

5.1 Plane to Disc

\begin{align*} \phi: {\mathbb{H}}&\to {\mathbb{D}}\\ \phi(z) &= {z - i \over z + i} \qquad f^{-1}(z) = i\qty{1 + w \over 1 - w} .\end{align*}

5.2 Sector to Disc

For \(S_\alpha \mathrel{\vcenter{:}}=\left\{{z\in{\mathbb{C}}{~\mathrel{\Big|}~}0 < \arg(z) < \alpha }\right\}\) an open sector for \(\alpha\) some angle, first map the sector to the half-plane: \begin{align*} g: S_\alpha &\to {\mathbb{H}}\\ g(z) &= z^{\pi \over \alpha} .\end{align*}

Then compose with a map \({\mathbb{H}}\to{\mathbb{D}}\): \begin{align*} f: S_\alpha &\to {\mathbb{D}}\\ f(z) &= (\phi \circ g)(z) = {z^{\pi\over \alpha} - i \over z^{\pi\over\alpha} + i} .\end{align*}

5.3 Strip to Disc

6 Schwarz Reflection

\({\mathbb{H}}^+, {\mathbb{H}}^-\) can be replaced with any region symmetric about a line segment \(L\subseteq {\mathbb{R}}\).

7 Zeros and Poles

7.1 Counting Zeros

8 Linear Fractional Transformations

9 Appendix: Proofs of the Fundamental Theorem of Algebra

9.0.1 Fundamental Theorem of Algebra: Argument Principle

9.0.2 Fundamental Theorem of Algebra: Rouche’s Theorem

\[\begin{align*} |g(z)| &\coloneqq|a_{n-1}z^{n-1} + \cdots + a_1 z + a_0 | \\ &\leq |a_{n-1}z^{n-1}| + \cdots + |a_1 z| + |a_0 | \quad\text{by the triangle inequality} \\ &= |a_{n-1}|\cdot |z^{n-1}| + \cdots + |a_1|\cdot| z| + |a_0 | \\ &= |a_{n-1}|\cdot R^{n-1} + \cdots + |a_1| R + |a_0 | \\ &\leq |a_{n-1}|\cdot R^{n-1}+|a_{n-2}|\cdot R^{n-1} + \cdots + |a_1| \cdot R^{n-1} + |a_0 |\cdot R^{n-1} \quad\text{since } R>1 \implies R^{a+b} \geq R^a \\ &= R^{n-1} \left( |a_{n-1}| + |a_{n-2}| + \cdots + |a_1| + |a_0| \right) \\ &\leq R^{n-1} \left( |a_n|\cdot R \right) \quad\text{by choice of } R \\ &= R^{n} |a_n| \\ &= |a_n z^n| \\ &\coloneqq|f(z)| \end{align*}\]

9.0.3 Fundamental Theorem of Algebra: Liouville’s Theorem

9.0.4 Fundamental Theorem of Algebra: Open Mapping Theorem

10 Appendix

10.1 Misc Basic Algebra

Mnemonic: Write \(f(x, y) = Ax^2 + Bxy + Cy^2 + \cdots\), then consider the discriminant \(\Delta = B^2 - 4AC\):

\begin{align*} x^2 - bx = (x - s)^2 - s^2 \quad\text{where} s = \frac{b}{2} \\ x^2 + bx = (x + s)^2 - s^2 \quad\text{where} s = \frac{b}{2} .\end{align*}

The sum of the interior angles of an \(n{\hbox{-}}\)gon is \((n-2)\pi\), where each angle is \(\frac{n-2}{n}\pi\).

Basics

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11 Draft of Problem Book

What is a pair of conformal equivalences between \({\mathbb{H}}\) and \({\mathbb{D}}\)?

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\begin{align*} F: HH &\to {\mathbb{D}}\\ z & \mapsto {i-z \over i+z} \\ \\ G: {\mathbb{D}}&\to {\mathbb{H}}\\ w &\mapsto i{1-w \over 1 + w} .\end{align*}

Mnemonic: any point in \({\mathbb{H}}\) is closer to \(i\) than \(-i\), so \({\left\lvert {F(z)} \right\rvert} < 1\).

  • Maps \({\mathbb{R}}\to S^1\setminus\left\{{-1}\right\}\).

What is conformal equivalence \({\mathbb{H}}\rightleftharpoons S \mathrel{\vcenter{:}}=\left\{{w\in {\mathbb{C}}{~\mathrel{\Big|}~}0 < \arg(w) < \alpha \pi}\right\}\)?

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\begin{align*} f(z) = z^ \alpha .\end{align*}

12 Bibliography