Cauchy’s theorem: if \(f\) is holomorphic on a simply connected domain \(\Omega\), then \(\oint_\gamma f\,dz=0\) for every closed piecewise smooth curve \(\gamma\) in \(\Omega\). Cauchy’s integral formula: if \(f\) is holomorphic on an open set containing the closed disc \(\abs{w-a}\le r\), then \(f^{(n)}(z)=\frac{n!}{2\pi i}\oint_{\abs{w-a}=r}\frac{f(w)}{(w-z)^{n+1}}\,dw\) for \(\abs{z-a}<r\) and \(n\ge0\). Expanding \(\frac{1}{w-z}\) as a geometric series in \(\frac{z-a}{w-a}\) shows that \(f\) equals its Taylor series on \(\abs{z-a}<r\), so holomorphic functions are analytic. If \(\abs{f}\le M\) on \(\abs{w-a}=r\), Cauchy’s estimate gives \(\abs{f^{(n)}(a)}\le n!\,M/r^n\), and Liouville’s theorem follows: a bounded entire function is constant. The case \(n=0\) gives the mean value property \(f(a)=\frac{1}{2\pi}\int_0^{2\pi}f(a+re^{i\theta})\,d\theta\), from which the maximum modulus principle follows: if \(\abs{f}\) attains a maximum at a point of a domain, then \(f\) is constant. Identity theorem: two holomorphic functions on a domain that agree on a set with a limit point in the domain are equal. Morera’s theorem: a continuous function on \(\Omega\) whose integral over the boundary of every closed triangle in \(\Omega\) vanishes is holomorphic.
Contour integral (Tag D-6DAXB)
Let \(\gamma\colon[a,b]\to\CC\) be a piecewise \(C^1\) curve and let \(f\) be a continuous complex-valued function on \(\gamma([a,b])\). The contour integral of \(f\) along \(\gamma\) is \[ \int_{\gamma}f\dz\coloneqq\int_a^b f(\gamma(t))\,\gamma'(t)\dt. \]
Write \(f=u+iv\) with \(u,v\) real-valued and \(z=x+iy\), so that \(dz=dx+i\,dy\). Then \(f\dz=(u+iv)\dx+(-v+iu)\dy\), and the contour integral is the line integral of this complex-valued \(1\)-form: \[ \int_{\gamma}f\dz=\int_\gamma (u+iv)\dx+(-v+iu)\dy. \]
Cauchy's theorem for simply connected regions (Tag T-2OVCI)
Let \(\Omega\subseteq\CC\) be a simply connected region and let \(f\) be holomorphic on \(\Omega\). Then for every closed piecewise \(C^1\) curve \(\gamma\) in \(\Omega\), \[ \int_\gamma f(z)\dz=0. \]
Cauchy integral formula on a disc (Tag T-6DEUH)
Let \(\Omega\subseteq\CC\) be open and let \(f\) be holomorphic on \(\Omega\). Let \(z_0\in\Omega\) and \(R>0\) satisfy \(\overline{D_R(z_0)}\subseteq\Omega\), and let \(\gamma=\bd D_R(z_0)\) be the positively oriented circle. Then \[ f(z_0)=\frac{1}{2\pi i}\int_\gamma\frac{f(\xi)}{\xi-z_0}\dxi, \] and for every \(n\ge0\), \[ f^{(n)}(z_0)=\frac{n!}{2\pi i}\int_\gamma\frac{f(\xi)}{(\xi-z_0)^{n+1}}\dxi. \]
Power series expansion of a holomorphic function on a disc (Tag T-GROTS)
Let \(\Omega\subseteq\CC\) be open, let \(f\) be holomorphic on \(\Omega\), and let \(z_0\in\Omega\) and \(R>0\) satisfy \(\overline{D_R(z_0)}\subseteq\Omega\). Then for every \(z\in D_R(z_0)\), \[ f(z)=\sum_{n=0}^{\infty}a_n(z-z_0)^n, \] where for every \(n\geq0\) and every \(r\in(0,R]\), \[ a_n=\frac{f^{(n)}(z_0)}{n!}=\frac{1}{2\pi r^n}\int_0^{2\pi}f(z_0+re^{i\theta})e^{-in\theta}\dtheta. \]
Cauchy's inequalities (Tag T-22RQZ)
Let \(\Omega\subseteq\CC\) be open, let \(f\) be holomorphic on \(\Omega\), and let \(z_0\in\Omega\) and \(R>0\) satisfy \(\overline{D_R(z_0)}\subseteq\Omega\). Let \(\gamma\) be the circle \(\abs{z-z_0}=R\) and \(M\coloneqq\max_{z\in\gamma}\abs{f(z)}\). Then for every \(n\ge0\), \[ \abs{f^{(n)}(z_0)}\le\frac{n!\,M}{R^n}. \]
By the Cauchy integral formula for derivatives, with \(\gamma\) parametrized by \(z=z_0+Re^{i\theta}\), \[ \abs{f^{(n)}(z_0)}=\abs{\frac{n!}{2\pi i}\int_\gamma\frac{f(z)}{(z-z_0)^{n+1}}\dz}\le\frac{n!}{2\pi}\int_0^{2\pi}\frac{M}{R^{n+1}}R\dtheta=\frac{n!\,M}{R^n}. \]
Liouville's theorem (Tag T-QHIHJ)
Let \(f\) be an entire function. If there is \(M>0\) with \(\abs{f(z)}\leq M\) for all \(z\in\CC\), then \(f\) is constant.
Mean value property for holomorphic functions (Tag T-5BLYU)
Let \(f\) be holomorphic on an open set containing the closed disc \(\overline{D_r(z_0)}\). Then \[ f(z_0)=\frac{1}{2\pi}\int_0^{2\pi}f(z_0+re^{i\theta})\dtheta=\frac{1}{\pi r^2}\iint_{D_r(z_0)}f(z)\dA. \] The same identities hold for \(u=\Re f\) in place of \(f\), by taking real parts.
The Cauchy integral formula on the circle \(z=z_0+\rho e^{i\theta}\), \(0<\rho\le r\), gives \[ f(z_0)=\frac{1}{2\pi i}\int_0^{2\pi}\frac{f(z_0+\rho e^{i\theta})}{\rho e^{i\theta}}\,i\rho e^{i\theta}\dtheta=\frac{1}{2\pi}\int_0^{2\pi}f(z_0+\rho e^{i\theta})\dtheta. \] Multiplying by \(\rho\) and integrating over \(0\le\rho\le r\) in polar coordinates gives \(\frac{r^2}{2}f(z_0)=\frac{1}{2\pi}\iint_{D_r(z_0)}f\dA\).
Maximum modulus principle (Tag T-BYNL5)
Let \(\Omega\subseteq\CC\) be open and let \(f\) be holomorphic on \(\Omega\).
If \(\abs f\) has a local maximum at \(z_0\in\Omega\), then \(f\) is constant on a neighborhood of \(z_0\).
If \(\Omega\) is bounded and connected and \(f\) extends continuously to \(\overline\Omega\), then either \(f\) is constant or \(\abs f\) attains its maximum \(M\coloneqq\max_{z\in\overline\Omega}\abs{f(z)}\) only at points of \(\bd\Omega\).
Equivalent conditions for a holomorphic function to vanish identically (Tag T-SVF2W)
Let \(\Omega\subseteq\CC\) be a nonempty connected open set and let \(f\) be holomorphic on \(\Omega\). The following are equivalent.
\(f(z)=0\) for all \(z\in\Omega\).
There are \(z_0\in\Omega\) and a sequence \((z_k)_{k\geq1}\) of distinct points of \(\Omega\) with \(z_k\to z_0\) and \(f(z_k)=0\) for all \(k\).
There is \(z_0\in\Omega\) with \(f^{(n)}(z_0)=0\) for all \(n\geq0\).
Morera's theorem (Tag T-LHSMY)
Let \(\Omega\subseteq\CC\) be open and let \(f\colon\Omega\to\CC\) be continuous. If \(\int_{\partial T}f(z)\dz=0\) for every closed triangle \(T\subseteq\Omega\), then \(f\) is holomorphic on \(\Omega\).