Complex Analysis for Analytic Number Theory

I wrote this as a quick reference for the essential facts of complex analysis one is expected to know before any serious course in analytic number theory.

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 $D$.
A function $f$ is meromorphic at a point $a$ if there is some neighbourhood of $a$ on which either $f$ or $1/f$ is holomorphic. In this case there is some $n$ such that $(s-a)^nf(s)$ is holomorphic and non-zero in a neighbourhood of $a$. If $n>0$ then $a$ is a pole of $f$ of order $n$. If $n<0$ then $a$ is a zero of $f$ of order $-n$. If $f$ is meromorphic at $a$ then there is some neighbourhood of $a$ in which $f$ can be expressed as a Laurent series, \[\sum_{n=-k}^\infty c_n(z-a)^n,\] for some finite integer $k$. The coefficient $c_{-1}$ is the residue of $f$ at $a$, and is denoted by $\mathrm{Res}(f,a)$.
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 $D$.
If $U$ is a simply connected open set and $f$ is holomorphic and non-zero on $U$ then we define $\log f(z)$ on $U$ as \[\log f(z)=a+\int_b^z \frac{f'(s)}{f(s)}\td s,\] where $b\in U$ and $a$ is such that $\exp(a)=f(b)$. The integral can be taken over any path between $b$ and $z$. This function is well-defined up to a constant (depending on the choice of $a$ and $b$) which is always an integral multiple of $2\pi i$.