A nonvanishing holomorphic function on a simply connected domain is \(e^g\) for a holomorphic \(g\). Weierstrass factorization theorem: with \(E_0(u)=1-u\) and \(E_p(u)=(1-u)e^{u+u^2/2+\cdots+u^p/p}\), every entire function \(f\) with a zero of order \(m\) at \(0\) and nonzero zeros \(a_1,a_2,\ldots\), listed with multiplicity, is \(f(z)=z^me^{g(z)}\prod_nE_n(z/a_n)\) for an entire function \(g\), the product converging uniformly on compact sets. Hadamard’s theorem: if \(f\) has finite order \(\rho\), the factors can be taken to be \(E_p(z/a_n)\) with \(p=\lfloor\rho\rfloor\) and \(g\) a polynomial of degree at most \(\rho\). Examples: \(\sin\pi z=\pi z\prod_{n\ge1}(1-z^2/n^2)\), whose logarithmic derivative is \(\pi\cot\pi z=\frac1z+\sum_{n\ge1}\frac{2z}{z^2-n^2}\), a partial fraction expansion also obtained from residues and contour integration; \(\Gamma(z)=\int_0^\infty t^{z-1}e^{-t}\,dt\), defined for \(\operatorname{Re}z>0\), which extends to a meromorphic function on \(\CC\) with simple poles at \(0,-1,-2,\ldots\); and \(\zeta(s)=\sum_{n\ge1}n^{-s}\), defined for \(\operatorname{Re}s>1\), which extends to a meromorphic function on \(\CC\) with a single simple pole, at \(s=1\).
Hadamard factorization (Tag T-2WJ4U)
For an integer \(p\ge0\), let \(E_p\) be the canonical factor \[ E_0(z)=1-z,\qquad E_p(z)=(1-z)\exp\Big(z+\frac{z^2}{2}+\cdots+\frac{z^p}{p}\Big)\quad(p\ge1). \] Let \(f\) be an entire function of order \(\rho<\infty\), where \[ \rho\coloneqq\inf\ts{\sigma\ge0\st\text{there is }R>0\text{ with }\abs{f(z)}\le e^{\abs z^\sigma}\text{ for all }\abs z>R}, \] and let \(p\coloneqq\lfloor\rho\rfloor\). Suppose \(f\) is not identically zero, has a zero of order \(m\ge0\) at \(0\), and let \((z_k)_{k\ge1}\) be its nonzero zeros, repeated with multiplicity. Then there is a polynomial \(g\) of degree at most \(p\) such that \[ f(z)=z^me^{g(z)}\prod_{k\ge1}E_p\Big(\frac{z}{z_k}\Big), \] where the product converges uniformly on compact subsets of \(\CC\).
Weierstrass factorization (Tag T-IEJFA)
Let \(D\subseteq\CC\) be a simply connected connected open set and let \(f\) be meromorphic on \(D\) and not identically zero. Then there are holomorphic functions \(g\), \(h_1\), \(h_2\) on \(D\) such that \[ f(z)=e^{g(z)}\frac{h_1(z)}{h_2(z)}, \] where the zeros of \(h_1\) are the zeros of \(f\), with the same multiplicities, and the zeros of \(h_2\) are the poles of \(f\), with multiplicities equal to their orders.
Gamma function (Tag D-Q3MYK)
For \(z\in\CC\) with \(\operatorname{Re}z>0\), the gamma function is \[ \Gamma(z)\coloneqq\int_0^\infty t^{z-1}e^{-t}\dt, \] where \(t^{z-1}\coloneqq e^{(z-1)\log t}\) for \(t>0\), with \(\log\) the real natural logarithm.
$\Gamma$ is holomorphic on the right half-plane (Tag PR-NLV6Q)
For \(\Re s>0\) the integral \[ \Gamma(s)=\int_0^\infty e^{-t}t^{s-1}\dt \] converges absolutely, and it defines a holomorphic function on the right half-plane \(\ts{s\in\CC\st\Re s>0}\).
For \(t>0\), \(\abs{t^{s-1}}=\abs{e^{(s-1)\ln t}}=t^{\Re s-1}\). For \(0<\varepsilon<1\) put \[ \Gamma_\varepsilon(s)\coloneqq\int_\varepsilon^{1/\varepsilon}e^{-t}t^{s-1}\dt. \] The integrand is continuous in \((t,s)\) and holomorphic in \(s\), and the interval is compact, so each \(\Gamma_\varepsilon\) is entire.
Fix \(n\ge1\) and the strip \(A_n\coloneqq\ts{s\st \frac1n<\Re s<n}\). For \(s\in A_n\), \[ \abs{\Gamma(s)-\Gamma_\varepsilon(s)}\le\int_0^\varepsilon t^{1/n-1}\dt+\int_{1/\varepsilon}^\infty e^{-t}t^{n-1}\dt, \] using \(t^{\Re s-1}\le t^{1/n-1}\) for \(t\le1\) and \(t^{\Re s-1}\le t^{n-1}\) for \(t\ge1\). Both integrals are finite (so the integral defining \(\Gamma\) converges absolutely) and tend to \(0\) as \(\varepsilon\to0\), independently of \(s\). Hence \(\Gamma_\varepsilon\to\Gamma\) uniformly on each \(A_n\). A uniform limit of holomorphic functions on an open set is holomorphic, so \(\Gamma\) is holomorphic on each \(A_n\), and the strips \(A_n\) cover the right half-plane.
Meromorphic continuation of $\Gamma$ (Tag PR-YQZI3)
The Gamma function, defined by \(\Gamma(s)=\int_0^\infty t^{s-1}e^{-t}\dt\) for \(\Re s>0\), extends to a meromorphic function on \(\CC\) whose only singularities are simple poles at \(s=0,-1,-2,\ldots\).
For \(\Re s>0\), integration by parts gives \(\Gamma(s+1)=s\Gamma(s)\), and \(\Gamma\) is holomorphic there by PR-NLV6Q. For \(n\ge1\) define, on \(\ts{s\st\Re s>-n}\), \[ \Gamma_n(s)\coloneqq\frac{\Gamma(s+n)}{s(s+1)\cdots(s+n-1)}. \] Since \(\Re(s+n)>0\), the numerator is holomorphic on this half-plane, so \(\Gamma_n\) is meromorphic there with at most simple poles at \(0,-1,\ldots,-(n-1)\). Iterating \(\Gamma(s+1)=s\Gamma(s)\) shows \(\Gamma_n=\Gamma\) on \(\Re s>0\), so by the identity theorem \(\Gamma_n\) and \(\Gamma_{n+1}\) agree on \(\ts{\Re s>-n}\) minus the poles. The functions \(\Gamma_n\) therefore define one meromorphic function on \(\CC\). At \(s=-k\) with \(0\le k\le n-1\), the numerator is \(\Gamma(n-k)=(n-k-1)!\neq0\) and exactly one factor of the denominator vanishes, simply, so \(-k\) is a simple pole.
Beta function (Tag D-2WZLB)
For \(z,w\in\CC\) with \(\operatorname{Re}z>0\) and \(\operatorname{Re}w>0\), the beta function is \[ B(z,w)\coloneqq\int_0^1 t^{z-1}(1-t)^{w-1}\dt, \] where \(t^{s}\coloneqq e^{s\log t}\) for \(t>0\), with \(\log\) the real natural logarithm.
Riemann zeta function (Tag D-HJYH3)
For \(s\in\CC\) with \(\operatorname{Re}s>1\), the Riemann zeta function is \[ \zeta(s)\coloneqq\sum_{n=1}^{\infty}\frac{1}{n^{s}}, \] where \(n^{s}\coloneqq e^{s\log n}\) with the real logarithm. The series converges absolutely because \(\abs{n^{-s}}=n^{-\operatorname{Re}s}\).
(Euler product)
For \(\operatorname{Re}s>1\), \[ \zeta(s)=\prod_{p\text{ prime}}\frac{1}{1-p^{-s}}, \] where the product is over all prime numbers \(p\) and converges as the limit of its partial products over \(p\le N\).
Let \(\sigma\coloneqq\operatorname{Re}s>1\). For each prime \(p\), \(\abs{p^{-s}}=p^{-\sigma}<1\), so \(\frac{1}{1-p^{-s}}=\sum_{k\ge0}p^{-ks}\), an absolutely convergent series. Multiplying these finitely many series for the primes \(p\le N\) and using unique factorization in \(\ZZ\), \[ \prod_{p\le N}\frac{1}{1-p^{-s}}=\sum_{n\in S_N}n^{-s}, \] where \(S_N\) is the set of positive integers all of whose prime factors are at most \(N\). Every \(n\le N\) lies in \(S_N\), so \[ \abs{\zeta(s)-\prod_{p\le N}\frac{1}{1-p^{-s}}}\le\sum_{n>N}n^{-\sigma}\to0\qquad(N\to\infty). \]
Meromorphic continuation of $\zeta$ (Tag PR-K4KTF)
The Riemann zeta function \(\zeta(s)=\sum_{n\ge1}n^{-s}\), defined for \(\Re(s)>1\), extends to a meromorphic function on \(\CC\) whose only singularity is a simple pole at \(s=1\). The extension satisfies the functional equation \[ \zeta(s) = \pi^{s-1/2}\,\frac{\Gamma\qty{\frac{1-s}{2}}}{\Gamma\qty{\frac{s}{2}}}\,\zeta(1-s). \]
For \(t>0\) let \(\theta(t)\coloneqq\sum_{n\in\ZZ}e^{-\pi n^2t}\). By Poisson summation \(\theta(1/t)=t^{1/2}\theta(t)\), and \(\abs{\theta(t)-1}\le ce^{-\pi t}\) for \(t\ge1\). For \(\Re(s)>1\) put \[ \xi(s)\coloneqq\frac12\int_0^\infty r^{s/2-1}\qty{\theta(r)-1}\,dr. \] Integrating term by term, \(\int_0^\infty r^{s/2-1}e^{-\pi n^2r}\,dr=\pi^{-s/2}n^{-s}\Gamma(s/2)\), so \(\xi(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)\) for \(\Re(s)>1\). Splitting the integral at \(r=1\) and applying \(\theta(1/r)=r^{1/2}\theta(r)\) on \((0,1)\) gives \[ \xi(s)=\frac{1}{s-1}-\frac1s+\frac12\int_1^\infty\qty{r^{s/2-1}+r^{-s/2-1/2}}\qty{\theta(r)-1}\,dr, \] where the integral is entire in \(s\) by the exponential decay of \(\theta-1\). So \(\xi\) extends meromorphically to \(\CC\) with simple poles only at \(0\) and \(1\), and the right side is unchanged under \(s\mapsto1-s\), so \(\xi(s)=\xi(1-s)\). Since \(1/\Gamma\) is entire with a simple zero at \(s=0\), \(\zeta(s)=\pi^{s/2}\xi(s)/\Gamma(s/2)\) is meromorphic on \(\CC\); the pole of \(\xi\) at \(0\) cancels against the zero of \(1/\Gamma(s/2)\), leaving only the simple pole at \(s=1\). Writing \(\xi(s)=\xi(1-s)\) in terms of \(\zeta\) gives the functional equation.
Holomorphic functions on a compact connected complex manifold are constant (Tag T-BBQLQ)
Let \(X\) be a compact connected complex manifold. Then every holomorphic function \(f\colon X\to\CC\) is constant.
Since \(X\) is compact, \(\abs f\) attains its maximum \(M\) at some point. The set \(S\coloneqq\ts{x\in X\st f(x)=c}\), for \(c\) a value with \(\abs c=M\) attained by \(f\), is nonempty and closed. At a point of \(S\), \(f\) read in a holomorphic chart is holomorphic on a connected open subset of \(\CC^n\) with an interior maximum of \(\abs f\); by the maximum modulus principle, applied on complex lines through the point, \(f\) is constant near that point, so \(S\) is open. As \(X\) is connected, \(S=X\).
Uniformization theorem (Tag T-ZAEWG)
Let \(S\) be a connected Riemann surface with universal cover \(\widetilde S\to S\). Then \(\widetilde S\) is a simply connected Riemann surface, and its group \(G\) of deck transformations acts on \(\widetilde S\) freely and properly discontinuously by biholomorphisms, with \(S\) biholomorphic to \(\widetilde S/G\). Moreover, \(\widetilde S\) is biholomorphic to exactly one of \(\CP^1\), \(\CC\), and \(\DD\).
Uniformization of simply connected domains in $\CC$ (Tag C-BKYF7)
Every nonempty simply connected open set \(\Omega\subseteq\CC\) is conformally equivalent to exactly one of \(\CC\) and the unit disc \(\DD\): to \(\CC\) if \(\Omega=\CC\), and to \(\DD\) otherwise.
By the uniformization theorem, every simply connected Riemann surface is conformally equivalent to exactly one of \(\CP^1\), \(\CC\), and \(\DD\). For open subsets of \(\CC\) the case \(\CP^1\) does not occur because \(\CP^1\) is compact, and \(\CC\) is not conformally equivalent to \(\DD\) by Liouville’s theorem.