Info / Tips / Techniques

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

Some useful notation:

  • \({\mathbb{D}}_r(a) \coloneqq\left\{{z\in {\mathbf{C}}{~\mathrel{\Big\vert}~}0\leq {\left\lvert {z-a} \right\rvert}< a}\right\}\) an open disc about \(a\)
  • \(\overline{{\mathbb{D}}}_r(a) \coloneqq\left\{{z\in {\mathbf{C}}{~\mathrel{\Big\vert}~}0\leq {\left\lvert {z-a} \right\rvert} \leq a}\right\}\) a closed disc about \(a\).
  • \({\mathbb{D}}^*(a) \coloneqq\left\{{z\in {\mathbf{C}}{~\mathrel{\Big\vert}~}0 < {\left\lvert {z-a} \right\rvert} < r}\right\}\) a punctured disc about \(a\).
  • \(\Delta \coloneqq{\mathbb{D}}_1(0)\) the standard unit disc
  • \(\overline{\Delta }\coloneqq\overline{{\mathbb{D}}}_1(0)\) the closed unit disc
  • \(\Delta^* \coloneqq{\mathbb{D}}_1^*(0)\) the punctured unit disc.
  • \(\Omega\) an open simply-connected subset of \({\mathbf{C}}\).
  • \({\mathcal{O}}(\Omega), \mathop{\mathrm{Hol}}(\Omega), \mathop{\mathrm{Hol}}(\Omega, {\mathbf{C}})\) the holomorphic functions \(f:\Omega \to {\mathbf{C}}\) equipped with the structure of a \({\mathbf{C}}{\hbox{-}}\)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: \({\left\lvert {f} \right\rvert}^2 = f\overline{f}\).
  • \(z\) is purely imaginary \(\iff \overline{z} = -z\).
  • \(z\in {\mathbf{R}}\iff \overline{z} = z\).
  • \(\log\qty{{\left\lvert {z} \right\rvert}} = {1\over 2}\log\qty{{\left\lvert {z} \right\rvert}^2} = {1\over 2}\log\qty{x^2 + y^2}\), 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 \(f({\mathbf{C}})\) is not an open set (e.g. \({\mathbf{R}}\) 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 \(\operatorname{im}(f) \subseteq {\mathbf{R}}\) 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 \(\overline{\Omega}\) 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*}