As is traditional in this area, we will often write $s=\sigma+it\in\bbc$ for an arbitrary complex variable, in which case $\sigma=\Re s$ is the real part of $s$ and $t=\Im s$ is the imaginary part. The derivative of a function $f:\bbc\to\bbc$ at a point $s$ is defined to be \[f'(s)=\lim_{z\to s}\frac{f(z)-f(s)}{z-s}.\] Implicit in this definition is the fact the limit exists and remains the same for any sequence of $(z)$ which has $s$ as a limit. A neighbourhood of $s$ is a bounded open set which contains $s$. We say that $f$ is holomorphic on an open set $U$ if $f'(s)$ exists for every $s\in U$, and that $f$ is holomorphic at $s$ if $f$ is holomorphic on some neighbourhood of $s$. A function is entire if it is holomorphic on $\bbc$.

A smooth curve is a continuous function $\gamma:[a,b]\to\bbc$ with a non-vanishing continuous derivative which is injective (except possibly at the endpoints). More generally, a contour is a finite sequence of smooth curves joined at the endpoints. The contour integral of $f$ along a smooth curve $\gamma$ is \[\int_\gamma f(s)\td s=\int_a^b f(\gamma(t))\gamma'(t)\td t,\] which is extended in the obvious fashion for general contours.

Cauchy's theorem

If $U$ is an open simply connected set, $f$ is holomorphic on $U$, and $\gamma$ is a closed contour in $U$, then
\[\int_\gamma f(s)\td s=0.\]
Cauchy integral formula

If $D$ is a closed disc with boundary circle $C$ and $f$ is holomorphic on a neighbourhood of $D$ then for every $a$ in the interior of $D$
\[f(a)=\frac{1}{2\pi i}\int_C \frac{f(s)}{s-a}\td s.\]
Every holomorphic function is analytic. That is, if $f$ is holomorphic on some neighbourhood of $a$ then there is some open disc centred at $a$ in which $f$ can be expanded as a convergent power series
\[f(s)=\sum_{n=0}^\infty c_n(s-a)^n.\]
The coefficients $c_n$ are
\[c_n=\frac{f^{(n)}(a)}{n!}=\frac{1}{2\pi i}\int_C\frac{f(w)}{(w-a)^{n+1}}\td w,\]
where $C$ is any circle centred at $a$ on and within which $f$ is holomorphic.

Identity Theorem

If $f$ and $g$ are both holomorphic on an open and connected set $D$ and $f=g$ for all $s\in S\subset D$, where $S$ is such that there is some $x\in D$ such that every neighbourhood of $x$ in $D$ contains some point in $S$, then $f\equiv g$ on $D$.
Residue theorem

If $U$ is a simply connected open set which contains a simple closed curve $\gamma$, $f$ is holomorphic on $\gamma$, and is holomorphic inside $\gamma$ except for a finite sequence $a_1,\ldots,a_k$, then
\[\int_\gamma f(s)\td s=2\pi i\sum_{i=1}^k\mathrm{Res}(f,a_k).\]
Maximum modulus principle

If $U$ is a connected open set and $f$ is holomorphic on $U$, and if there exists some $a\in U$ such that $\abs{f(a)}\geq \abs{f(s)}$ for all $s$ in a neighbourhood of $a$, then $f$ is constant on $D$.