잡담을 위한 소박한 장소

이번 포스팅에서는 푸리에 사인 급수와 관련된 적분을 다뤄보겠습니다. Theorem. Let $f:(0,\infty)\to\mathbb{C}$ be a locally bounded Lebesgue-measurable function such that the improper integral $$ \int_{0}^{\infty} \frac{f(x)}{x} \, \mathrm{d}x = \lim_{\substack{a\to 0^+ \\ b\to\infty}} \int_{a}^{b} \frac{f(x)}{x} \, \mathrm{d}x $$ exists in $\mathbb{C}$. Then, for any $(a_n)_{n\geq 1}\subset\mathbb{C}$ satisfying $\sum_{n\geq1..

MSE를 뒤적이다가 슬쩍 본 계산인데, 예뻐서 가져와봅니다. We evaluate the integral $$ I = \int_{0}^{1} \int_{0}^{1} \frac{x}{1-(1-y^2) x^2} \, \mathrm{d}x\mathrm{d}y $$ in two ways. Integrating with respect to $x$ first, we get \begin{align*} I &= \int_{0}^{1} \frac{\log(1/y)}{1 - y^2} \, \mathrm{d}y = \sum_{n=0}^{\infty} \int_{0}^{1} y^{2n} \log(1/y) \, \mathrm{d}y = \sum_{n=0}^{\infty} \frac{1}{(2n+1)^2}. \end{align*..

In this posting, we compute the reproducing kernels of the Sobolev space $H^1([0,1])$ equipped with the inner product $$ \langle f, g \rangle = \int_{0}^{1} [ \overline{f(x)}g(x) + \overline{f'(x)}g'(x)] \,\mathrm{d}x. $$ In other words, we aim to find a function $K_a \in H^1([0, 1])$, for each $a \in [0, 1]$, such that $\langle K_a, f \rangle = f(a)$ for all $f \in H^1([0, 1])$. Assume there ex..