User talk: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)