# Info / Tips / Techniques

A great deal of content borrowed from https://web.stanford.edu/~chriseur/notes_pdf/Eur_ComplexAnalysis_Notes.pdf

Some useful notation:

• $${\mathbb{D}}_r(a) \coloneqq\left\{{z\in {\mathbb{C}}{~\mathrel{\Big\vert}~}0\leq {\left\lvert {z-a} \right\rvert}< a}\right\}$$ an open disc about $$a$$
• $$\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu_r(a) \coloneqq\left\{{z\in {\mathbb{C}}{~\mathrel{\Big\vert}~}0\leq {\left\lvert {z-a} \right\rvert} \leq a}\right\}$$ a closed disc about $$a$$.
• $${\mathbb{D}}^*(a) \coloneqq\left\{{z\in {\mathbb{C}}{~\mathrel{\Big\vert}~}0 < {\left\lvert {z-a} \right\rvert} < r}\right\}$$ a punctured disc about $$a$$.
• $$\Delta \coloneqq{\mathbb{D}}_1(0)$$ the standard unit disc
• $$\mkern 1.5mu\overline{\mkern-1.5mu\Delta \mkern-1.5mu}\mkern 1.5mu\coloneqq\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu_1(0)$$ the closed unit disc
• $$\Delta^* \coloneqq{\mathbb{D}}_1^*(0)$$ the punctured unit disc.
• $$\Omega$$ an open simply-connected subset of $${\mathbb{C}}$$.
• $${\mathcal{O}}(\Omega), \mathop{\mathrm{Hol}}(\Omega), \mathop{\mathrm{Hol}}(\Omega, {\mathbb{C}})$$ the holomorphic functions $$f:\Omega \to {\mathbb{C}}$$ equipped with the structure of a $${\mathbb{C}}{\hbox{-}}$$algebra..

## Greatest Hits

Things to know well:

For just the statements of most of these theorems: see this doc.

## Common tricks

• Virtually any time: consider $$1/f(z)$$ and $$f(1/z)$$.
• Set $$w=e^z$$.
• If $$f$$ has no zeros, put it in the denominator! No one can stop you!
• If $$f$$ is holomorphic in a neighborhood of $${\mathbb{D}}$$ and $${\left\lvert {f} \right\rvert} = 1$$ on $${{\partial}}{\mathbb{D}}$$, then $$f$$ is a finite Blaschke product.
• If $$\Omega$$ is connected, $$f$$ admits a log and exponential, so try setting $$f^{1\over n} = \exp\qty{{1\over n}\log(f)}$$.

## Holomorphic

• To show a function is holomorphic,
• Use Morera’s theorem
• Find a primitive (sufficient but not necessary)
• Express $$f$$ as a convergent power series
• Holomorphic functions have isolated zeros.

## Arithmetic

Some silly arithmetic tricks:

• Absolutely essential: $${\left\lvert {f} \right\rvert}^2 = f\mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu$$.
• $$z$$ is purely imaginary $$\iff \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu = -z$$.
• $$z\in {\mathbb{R}}\iff \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu = z$$.
• $$\log\qty{{\left\lvert {z} \right\rvert}} = {1\over 2}\log\qty{{\left\lvert {z} \right\rvert}^2} = {1\over 2}\log\qty{x^2 + y^2}$$, which is easier to differentiate.
• To prove $$a=b$$, try $$a/b = 1$$ or $$a-b=0$$.
• $$\int_0^{2\pi} e^{i(m-n)\theta}\,d\theta= \chi_{m=n}\cdot 2\pi$$.

## Showing a function is constant (or zero)

• Show $$f' = 0$$.
• Can write $$f=u+iv$$ and show $$u_x, u_y = 0$$ and apply CR.
• Show $${\left\lvert {f} \right\rvert}=0$$ on the boundary and apply the MMP.
• Show that $$f$$ attains a minimum or maximum on the interior of a domain where it is nonzero.
• Show that $$f$$ is entire and bounded.
• If you additionally want to show $$f$$ is zero, show $$\lim_{z\to\infty} f(z) = 0$$.
• Useful trick: show that either $${\left\lvert {f} \right\rvert} \geq M$$ or $${\left\lvert {f} \right\rvert} \leq M$$, then by Liouville on $$f$$ or $$1/f$$ respectively, $$f$$ must be constant.
• Similar trick: show either $$e^f$$ or $$e^{-f}$$ is bounded.
• If the function is periodic, just bound it on a fundamental domain.
• Show that $$f({\mathbb{C}})$$ is not an open set (e.g. $${\mathbb{R}}$$ or $${{\partial}}{\mathbb{D}}_r(0)$$, and apply the open mapping theorem.
• More generally, the image can be dimension 0 or 2, but never 1.
• E.g. if $$\operatorname{im}(f) \subseteq {\mathbb{R}}$$ or $${\left\lvert {f} \right\rvert} = R$$ is constant.
• A holomorphic function with a non-isolated zero is identically zero.
• How to use: show $$f-g$$ has uncountably many zeros
• Show that $$f$$ omits at least 2 values and apply little Picard.
• E.g. if $$f$$ misses an open set, or $${\left\lvert {f} \right\rvert} \geq M$$ or $${\left\lvert {f} \right\rvert} \leq M$$.
• Define $$g\coloneqq e^f$$, then $${\left\lvert {g} \right\rvert} = e^{\Re(f)}$$ and if $$g$$ is constant then $$f$$ is constant.
• Show any of the following are constant:
• $$u = \Re(f)$$
• $$v = \Im(f)$$
• $${\left\lvert {f} \right\rvert}$$
• $$\operatorname{Arg}(f)$$
• Show that $$f$$ preserves $${{\partial}}{\mathbb{D}}$$, so $${\left\lvert {f(z)} \right\rvert} = 1$$ when $${\left\lvert {z} \right\rvert} = 1$$, and has no zeros in $${\mathbb{D}}$$.
• To show $$f(z) = g(z)$$ infinitely often, show $$f(z)/g(z)$$ (or $$f(1/z)/g(1/z)$$) has an essential singularity and apply Picard or Casorati.

## Singularities

• Let $$z_0$$ be a singularity of $$f$$. To show $$z_0$$ is…
• Removable: show that $$\lim_{z\to z_0} f(z)$$ is bounded.
• A pole of some order: show $$\lim_{z\to z_o}f(z) = \infty$$.
• A pole of order $$m$$: write $$f(z) = (z-z_0)^mg(z)$$ where $$g(z_0)\neq 0$$ (or check the Laurent expansion directly).
• Can also check that $${\partial}_z^k f(z_0) \neq 0$$ for $$k<m$$ but $${\partial}_z^m f(z_0) = 0$$.
• Essential: show that $$\lim_{z\to z_0} f(z)$$ doesn’t exist (e.g. if it’s oscillatory).
• Alternatively, show $$z_0$$ is neither removable nor a pole, or that $$f$$ has a Laurent expansion about $$z_0$$ with infinitely many negative terms.
• It can be useful to take a specific sequence $$\left\{{z_k}\right\}\to z_0$$.
• $$f$$ and $$f'$$ have the same poles.

## Zeros

• To show that a zero $$z_0$$ is order $$n$$, show that $$f^{(<n)}(z_0) = 0$$ but $$f^{(n)}(z_0) \neq 0$$.
• Getting rid of zeros: divide by a Blaschke product.
• To count zeros:
• Rouche’s theorem
• The argument principle
• If $$f(z_0)\neq 0$$, by continuity there is some neighborhood where $$f$$ is nonzero.
• Conversely, if $$f$$ is holomorphic at $$z_0$$ is a zero, there is punctured neighborhood of $$z_0$$ where $$f$$ is nonzero.

## Estimating

• To prove $$a\leq b$$, try showing $${a\over b} \leq 1$$ and reason about $${\mathbb{D}}$$, or show $$b-a\geq 0$$,
• To bound a rational function, use the reverse triangle inequality: \begin{align*} {\left\lvert {a\pm b} \right\rvert} \geq {\left\lvert { {\left\lvert {a} \right\rvert} - {\left\lvert {b} \right\rvert}} \right\rvert} \implies {1\over {\left\lvert {a\pm b} \right\rvert}} \leq {1\over {\left\lvert {{\left\lvert {a} \right\rvert} - {\left\lvert {b} \right\rvert} } \right\rvert} } .\end{align*}
• Bounding a derivative using the original function: Cauchy’s formula.
• Also works to bound a function in terms of its integral, e.g. over a compact set like a curve.
• If $${\left\lvert {f} \right\rvert} = M$$ on $${{\partial}}\Omega$$, then if (importantly) $$f\neq 0$$ in $$\Omega$$ then $${\left\lvert {f} \right\rvert} = M$$ on all of $$\mkern 1.5mu\overline{\mkern-1.5mu\Omega\mkern-1.5mu}\mkern 1.5mu$$ by apply the MMP to $$f$$ and $$1/f$$.
• Why $$f\neq 0$$ is necessary: take $$f(z) = z$$.
• To show that a sequence of harmonic functions converge on e.g. a disc or rectangle, find good estimates on the boundary and apply the MMP.
• For real analysis: if $$f' < M$$, apply the mean value theorem to show $$f$$ is Lipschitz: $${\left\lvert {f(x) - f(y)} \right\rvert} = {\left\lvert {f'(\xi)} \right\rvert} {\left\lvert {x-y} \right\rvert} < M{\left\lvert {x-y} \right\rvert}$$.
• To show $${\left\lvert {f} \right\rvert} \leq {\left\lvert {g} \right\rvert}$$: if you have a factor of $$z$$ to play with, try to apply Schwarz to $$f/g$$ to get $${\left\lvert {f/g} \right\rvert}\leq {\left\lvert {z} \right\rvert}$$.

## Polynomials

• $$f$$ is polynomial when:
• $$f^{(n)} =0$$ for every $$n$$ large enough (e.g. using Cauchy’s inequality)
• $$f$$ is entire and only has poles at $$\infty$$.

## Series

• A common trick: \begin{align*} \frac{1}{z-w}=\frac{1}{(z-a)\left(1-\frac{w-a}{z-a}\right)}=\sum_{n=0}^{n} \frac{(w-a)^{n}}{(z-a)^{n+1}} . \end{align*}