Techniques

\[ \begin{aligned} dz &= dx + i\,dy, \\ d\bar z &= dx - i\,dy, \\ f_z &= f_x = f_y / i \quad \text{for } f \text{ holomorphic}, \end{aligned} \] and for \(\ell\in\ZZ\), \[ \int_{0}^{2 \pi} e^{i \ell x} \dx = \begin{cases} 2 \pi & \ell=0, \\ 0 & \ell\neq 0. \end{cases} \]

• \(\abs{f}^2 = f\bar{f}\).
• \(z\) is purely imaginary if and only if \(\bar{z} = -z\), and \(z\in \RR\) if and only if \(\bar z = z\).
• \(\log\abs{z} = {1\over 2}\log\qty{\abs{z}^2} = {1\over 2}\log\qty{x^2 + y^2}\) for \(z = x+iy\neq 0\).
• For \(w, z, a\in\CC\) with \(\abs{w-a}<\abs{z-a}\), \[ \frac{1}{z-w} =\frac{1}{(z-a)\qty{1-\frac{w-a}{z-a}}} =\sum_{n=0}^{\infty}\frac{(w-a)^n}{(z-a)^{n+1}}. \] The standard expansions, generalized binomial coefficients, Cauchy products, and inverses of power series are on Series reference.

(Sufficient conditions for holomorphy)

A function \(f\) on an open set \(U\) is holomorphic if any of the following holds:

• \(f\) is continuous and \(\int_{\bd T} f = 0\) for every closed triangle \(T\subseteq U\) (Morera's theorem);
• \(f\) has a holomorphic primitive on \(U\);
• \(f\) is locally the sum of a convergent power series.

• If \(f\) is holomorphic with no zeros, then \(1/f\) is holomorphic; applying the maximum modulus principle or Liouville’s theorem to \(1/f\) gives the minimum modulus principle and constancy of entire functions bounded below by a positive constant.
• If \(f\) is holomorphic and nonvanishing on a simply connected domain \(\Omega\), then \(f = e^g\) for some holomorphic \(g\) on \(\Omega\), and \(f^{1/n} \coloneqq \exp\qty{{1\over n}g}\) is a holomorphic \(n\)th root.
• If \(f\) is holomorphic on a neighborhood of \(\bar\DD\) and \(\abs{f} = 1\) on \(\bd \DD\), then \(f\) is a unimodular constant times a finite Blaschke product.
• The substitutions \(z\mapsto 1/z\), \(f\mapsto 1/f\), and \(w = e^z\) transfer statements between \(0\) and \(\infty\), between zeros and poles, and between strips and annuli.

(Criteria for a holomorphic function to be constant)

Let \(f\) be holomorphic on a domain \(U\). Then \(f\) is constant if any of the following holds:

• \(f' = 0\) on \(U\); by the Cauchy–Riemann equations it suffices that \(u_x = u_y = 0\) for \(u = \Re f\).
• One of \(\Re f\), \(\Im f\), \(\abs{f}\), or \(\arg f\) (for a continuous branch) is constant.
• \(\abs f\) attains a maximum in \(U\), or \(f\) is nonvanishing and \(\abs f\) attains a minimum in \(U\).
• \(f(U)\) is not open, for instance \(f(U)\subseteq\RR\) or \(f(U)\subseteq \bd\DD_r(0)\) (open mapping theorem).
• \(U\) is bounded, \(f\) extends continuously to \(\bar U\), \(f\) has no zeros in \(U\), and \(\abs f\) is constant on \(\bd U\): the maximum modulus principle applied to \(f\) and to \(1/f\) shows \(\abs f\) is constant.

If moreover \(U = \CC\), then \(f\) is constant if any of the following holds:

• \(f\) is bounded (Liouville's theorem); if also \(f(z)\to 0\) as \(z\to\infty\), then \(f\equiv 0\).
• \(\abs f\geq M\) for some \(M>0\), by Liouville’s theorem applied to \(1/f\).
• \(e^f\) or \(e^{-f}\) is bounded, since \(\abs{e^{f}} = e^{\Re f}\).
• \(f\) is periodic with two \(\RR\)-linearly independent periods, since then \(f\) is bounded by its bound on a closed fundamental parallelogram.
• \(f\) omits two values of \(\CC\), for instance when \(f(\CC)\) misses an open set (little Picard theorem).

If \(f\) and \(g\) are holomorphic on a domain \(U\) and \(f-g\) has a zero set with a limit point in \(U\), then \(f = g\) (identity principle).

(Classifying an isolated singularity)

Let \(f\) be holomorphic on a punctured disc about \(z_0\).

• \(z_0\) is removable if and only if \(f\) is bounded near \(z_0\).
• \(z_0\) is a pole if and only if \(\abs{f(z)}\to\infty\) as \(z\to z_0\).
• \(z_0\) is a pole of order \(m\) if and only if \(f(z) = (z-z_0)^{-m}g(z)\) with \(g\) holomorphic near \(z_0\) and \(g(z_0)\neq 0\), equivalently if \(1/f\) has a zero of order \(m\) at \(z_0\), equivalently if the Laurent expansion at \(z_0\) has lowest term of degree \(-m\).
• \(z_0\) is essential if and only if \(\lim_{z\to z_0} f(z)\) exists neither in \(\CC\) nor as \(\infty\), equivalently if the Laurent expansion at \(z_0\) has infinitely many terms of negative degree. Two sequences \(z_k\to z_0\) along which \(f\) has different limits show this.

A meromorphic function \(f\) and its derivative \(f'\) have the same poles.

If \(h\) has an essential singularity at \(z_0\), then by the great Picard theorem, in every punctured neighborhood of \(z_0\), \(h\) takes every value of \(\CC\) infinitely often with at most one exception. With \(h \coloneqq f/g\), this shows \(f(z) = cg(z)\) for infinitely many \(z\) near \(z_0\), for all but at most one \(c\in\CC\).

Let \(f\) be holomorphic near \(z_0\).

• \(f\) has a zero of order \(n\) at \(z_0\) if and only if \(f^{(k)}(z_0) = 0\) for \(0\leq k<n\) and \(f^{(n)}(z_0) \neq 0\).
• If \(f(z_0)\neq 0\), then \(f\) is nonzero on a neighborhood of \(z_0\) by continuity.
• If \(f(z_0) = 0\) and \(f\) is not identically zero near \(z_0\), then \(f\) is nonzero on a punctured neighborhood of \(z_0\).
• Dividing by a finite Blaschke product with the same zeros in \(\DD\) gives a function without zeros in \(\DD\) with the same modulus on \(\bd\DD\).
• Zeros in a region are counted by Rouché's theorem and the argument principle.

• For \(\abs a\neq \abs b\), the reverse triangle inequality gives \[ {1\over \abs{a\pm b}} \leq {1\over \abs{\abs{a} - \abs{b} } }. \]
• The Cauchy integral formula bounds derivatives of \(f\) by the maximum of \(\abs f\) on a circle, and bounds \(\abs{f}\) at interior points by an integral of \(\abs f\) over a curve.
• If \(u_n\) are harmonic on a bounded domain \(U\), continuous on \(\bar U\), and \(u_n\) converges uniformly on \(\bd U\), then \(u_n\) converges uniformly on \(\bar U\), by the maximum principle applied to \(u_n - u_m\).
• For \(f\colon[a,b]\to\RR\) differentiable with \(\abs{f'} \leq M\), the mean value theorem gives \(\abs{f(x) - f(y)} \leq M\abs{x-y}\).
• If \(h\) is holomorphic on \(\DD\) with \(\abs h\leq 1\) and \(h(0) = 0\), then \(\abs{h(z)}\leq\abs z\) by the Schwarz lemma; applied to \(h \coloneqq f/g\) this gives \(\abs{f(z)}\leq\abs{z}\abs{g(z)}\).

An entire function \(f\) is a polynomial if and only if \(f^{(n)} = 0\) for all sufficiently large \(n\), if and only if \(f\) has a pole or a removable singularity at \(\infty\). Cauchy’s inequality shows \(f^{(n)} = 0\) for \(n>d\) when \(\abs{f(z)}\leq C\abs z^d\) for large \(\abs z\).