Ramprax

Welcome! Some math code copy-pasted for reference. - Ramprax (talk) 11:42, 4 February 2020 (UTC)

$D_{n}(x)=\sum _{k=-n}^{n}e^{2\pi ikx}={\frac {\sin \left(\left(2n+1\right)\pi x\right)}{\sin(\pi x)}}$

be the Dirichlet kernel.

This is clearly symmetric about zero, that is,

$D_{n}(-x)=D_{n}(x)$

$0=\int _{\gamma }g(z)\,dz=\int _{-R}^{R}{\frac {e^{ix}}{x+i\varepsilon }}\,dx+\int _{0}^{\pi }{\frac {e^{i(Re^{i\theta }+\theta )}}{Re^{i\theta }+i\varepsilon }}iR\,d\theta .$

The second term vanishes as R goes to infinity. As for the first integral, one can use one version of the Sokhotski–Plemelj theorem for integrals over the real line: for a complex-valued function f defined and continuously differentiable on the real line and real constants $a$ and $b$ with $a<0<b$ one finds

$\lim _{\varepsilon \to 0^{+}}\int _{a}^{b}{\frac {f(x)}{x\pm i\varepsilon }}\,dx=\mp i\pi f(0)+{\mathcal {P}}\int _{a}^{b}{\frac {f(x)}{x}}\,dx,$

where ${\mathcal {P}}$ denotes the Cauchy principal value. Back to the above original calculation, one can write

$0={\mathcal {P}}\int {\frac {e^{ix}}{x}}\,dx-\pi i.$

${\vec {r}}\cdot {\hat {n}}=d={\vec {a}}\cdot {\hat {n}}.$

$\int e^{kx}\sin(nx)\,dx={\frac {e^{kx}}{k^{2}+n^{2}}}\left(k\sin(nx)-n\cos(nx)\right)$

$\int e^{kx}\cos(nx)\,dx={\frac {e^{kx}}{k^{2}+n^{2}}}\left(k\cos(nx)+n\sin(nx)\right)$

Ramprax (talk) 11:42, 4 February 2020 (UTC)