Transforms

remark:

$\begin{align*} \widehat{f}(\xi) \coloneqq\int_{\mathbb{R}}e^{-i\xi x} f(x)\,dx\\ f(x) = {1\over 2\pi}\int_{\mathbb{R}}e^{i\xi x} \widehat{f}(\xi) \,d\xi .\end{align*}$

figures/2021-12-20_07-55-38.png

The Gamma Function

definition (Gamma function):

$\begin{align*} \Gamma(z) = \int_0^\infty t^{z-1}e^{-t} \,dt .\end{align*}$

remark:

Some interesting properties of $\Gamma$ : $\Gamma(z+1) = z\Gamma(z)$ and has simple poles at $z=0,-1,-2,\cdots$ with residues $\mathop{\mathrm{Res}}_{z=-m} \Gamma(z) = (-1)^m/m!$ . There is also a factorization $\begin{align*} \Gamma(z) = {1 \over ze^{\gamma z} \prod_{n=1}^\infty \qty{1 + {z\over n}}e^{-z\over n} } \end{align*}$ where $\gamma \coloneqq\lim_{N\to\infty } \sum_{n=1^N} {1\over n} - \log(N)$

$\begin{align*} \Gamma(z) \Gamma(1-z) = {\pi \over \sin(\pi z)} ,\end{align*}$ which yields a product factorization for $\sin(\pi z)$ .

${\mathcal{L}}(t^{z-1}, s=1) = \Gamma(z)$ and ${\mathcal{L}}(t^n, s=1) = \Gamma(n+1)$ .

The residues:

figures/2021-12-19_19-59-45.png

proposition (

$\Gamma$ is holomorphic on the right half-plane):

figures/2021-12-19_19-58-16.png

proposition (Functional equation for

$\Gamma$ ):

figures/2021-12-19_19-58-44.png

proposition (Meromorphic continuation of

$\Gamma$ ):

figures/2021-12-19_19-59-05.png

The Beta Function

definition (Beta function):

$\begin{align*} B(z, w)=\int_{0}^{1} t^{z-1}(1-t)^{w-1} \mathrm{~d} t .\end{align*}$

exercise (?):

Show that $\begin{align*} B(z, w) = {\Gamma(z) \Gamma(w) \over \Gamma(z+w)} .\end{align*}$

Hint: find ${\mathcal{L}}(t^{z-1})$ and ${\mathcal{L}}(t^{z-1}\ast t^{w-1})$ .

Riemann Zeta

definition (The Riemann Zeta Function):

figures/2021-12-19_20-00-12.png

There is a product expansion $\begin{align*} \zeta(s) = \prod_{p\in \operatorname{Spec}{\mathbb{Z}}} {1\over 1-p^{-s}} ,\end{align*}$ where $\operatorname{Spec}{\mathbb{Z}}$ are the primes of ${\mathbb{Z}}$ .

proposition (Meromorphic continuation of

$\zeta$ ):

figures/2021-12-19_20-00-54.png

There is a functional equation:

figures/2021-12-19_20-03-08.png

proposition (Zeros of

$\zeta$ ):

figures/2021-12-19_20-02-43.png

Weierstrass $\wp$

definition (The Weierstrass

$\wp$ function):

figures/2021-12-19_22-33-34.png

remark:

figures/2021-12-19_22-34-18.png

Elliptic Functions

#todo

Infinite Series and Products

fact (Infinite products):

figures/2021-12-14_17-36-04.png

theorem (Hadamard factorization):

Write $\begin{align*} E_{p}(z)= \begin{cases} 1-z & n=0 \\ (1-z) \exp \left(z+\frac{z^{2}}{2}+\cdots+\frac{z^{n}}{n}\right) & \text { otherwise } \end{cases} ,\end{align*}$ and define the order of an entire function $f$ to be the infimum over $p$ where there exists some $R$ such that ${\left\lvert {f(z)} \right\rvert} \leq e^{{\left\lvert {z} \right\rvert}^p}$ for ${\left\lvert {z} \right\rvert} > R$ . Suppose $f$ is entire of order $p$ , write $\left\{{z_k}\right\}_{k\leq n}$ for its set of nonzero zeros repeated with multiplicity, and suppose $z=0$ is a zero of $f$ of order $m$ . Then there is a decomposition $\begin{align*} f(z) = z^m e^{g(z)}\prod_{k\geq 1} E_p\qty{z\over z_k} ,\end{align*}$ where $\deg(g) \leq p$ .

exercise (Hadamard expansion of

$\sin$ ):

Find a Hadamard expansion of $\sin(\pi z)$ .

solution:

$\sin(\pi z)$ has order 1, and its zero set is $z_k = k$ for $k\in {\mathbb{Z}}$ . So one can write $\begin{align*} \sin(\pi z) = ze^{az+b} \prod_{k\in {\mathbb{Z}}\setminus\left\{{0}\right\}} \qty{1 - {z\over k}}e^{z\over k} = ze^{az+b} \prod_{k\geq 1}\qty{1 - {z^2\over k^2}} .\end{align*}$ Determine $e^b = \pi$ by considering $\sin(\pi z)/z$ as $z\to 0$ , and use that $\sin(\pi z)$ is odd and the product factor is even to conclude $e^{az}$ is even and thus equal to 1. This yields $\begin{align*} \sin(\pi z) = \pi z \prod_{k\geq 1}{1- {z^2\over k^2}} .\end{align*}$

theorem (Weierstrass factorization):

figures/2021-12-14_17-36-26.png

remark:

An interesting way to sum infinite series:

$\begin{align*} \sum_{n=-\infty}^{\infty} f(n) &=-(\operatorname{sum} \quad \text { of } \quad \text { residues } \quad \text { of } \quad \pi \cot \pi z f(z)) \\ \sum_{n=-\infty}^{\infty}(-1)^{n} f(n) &=-(\operatorname{sum} \quad \text { of } \quad \text { residues } \quad \text { of } \quad \pi \csc \pi z f(z)) \\ \sum_{n=-\infty}^{\infty} f\left(\frac{2 n+1}{2}\right) &=(\operatorname{sum} \quad \text { of } \quad \text { residues of } \quad \pi \tan \pi z f(z)) \\ \sum_{n=-\infty}^{\infty}(-1)^{n} f\left(\frac{2 n+1}{2}\right) &=(\operatorname{sum} \quad \text { of residues of } \pi \sec \pi z f(z)) . .\end{align*}$

exercise (Sum formulas: 1/(n-a)^2):

Show that $\begin{align*} \sum_{k\in {\mathbb{Z}}}{1\over (z-k)^2} = (\pi \csc(\pi z))^2 .\end{align*}$

#complex/exercise/work

exercise (Sum formulas:

$1/n^2$ ):

Show $\begin{align*} \sum_{n \geq 1} \frac{1}{n^{2}}=\frac{\pi^{2}}{6} \end{align*}$ by integrating $\pi \cot(\pi z)z^{-2}$ .

#complex/exercise/work

exercise (Sum formulas:

$1/n^2+a^2$ ):

Show that $\begin{align*} \sum_{k\in {\mathbb{Z}}} {1\over k^2 + a^2} = {\pi \coth(\pi a) \over a} \qquad\text{for } a>0 .\end{align*}$

#complex/exercise/work

exercise (?):

Show that $\begin{align*} \sum_{k\geq 1}{1\over k^2 + a^2} = {1\over 2}{\pi \coth(\pi a)\over a} - {1\over 2a^2} \qquad a>0 .\end{align*}$

#complex/exercise/work

exercise (?):

Show that $\begin{align*} \sum_{k\in {\mathbb{Z}}} {1\over \qty{ k - {1\over 2}}^2 } = \pi^2 .\end{align*}$

#complex/exercise/work

exercise (?):

Show that $\begin{align*} \sim_{k\in {\mathbb{Z}}} { (-1)^k \over (k+a)^2} = \pi^2 \cos(\pi a)\csc^2(\pi a) && \text{for } a\in {\mathbb{R}}\setminus{\mathbb{Z}} .\end{align*}$

#complex/exercise/work

exercise (Computing

$\zeta(2)$ by integration):

By computing $\begin{align*} {1\over 2\pi i}\oint {\cot(\pi z)\over z^2}\,dz ,\end{align*}$ say using a large rectangle, show that $\begin{align*} \zeta(2) = \sum_{k\geq 0} {1\over k^2} = {\pi^2\over 6} .\end{align*}$

#complex/exercise/work

Special Functions

Transforms

The Gamma Function

The Beta Function

Riemann Zeta

Weierstrass ℘\wp

Elliptic Functions

Infinite Series and Products

Weierstrass $\wp$