잡담을 위한 소박한 장소

Residue 계산하면 금방 나오는 쉬운 계산입니다. Proposition 1. We have \begin{equation*} \int_{0}^{\infty} \frac{x^3}{(x^4 + 1)(e^{2\pi x} - 1)} \, \mathrm{d}x = \frac{\gamma}{2} + \frac{\pi}{4} \frac{\sin \pi\sqrt{2}}{\cosh \pi\sqrt{2} - \cos \pi\sqrt{2}} - \frac{1}{8} \sum_{\omega^{4} = -1} H_{\omega}, \end{equation*} where $H_s = \gamma + \psi_{0}(1+s)$ is the analytic extension of the harmonic number.

간단한 스탈링 공식을 유도해보았습니다.. Theorem 1. We have \begin{equation*} n! \sim \sqrt{2\pi n} \, n^{n} e^{-n}. \tag{1} \end{equation*} For the proof of this theorem, the following observations are quite useful. Lemma 2. (i) For $u \geq 0$, let $f_n (u)$ by \begin{equation*} f_n (u) = \left(1 + \frac{u}{\sqrt{n}} \right)^{n} e^{-\sqrt{n} u}, \quad n = 1, 2, 3, \cdots. \tag{2} \end{equation*} Then $f_{n+1}(u)..

그냥 간만에 Croft's Lemma의 아주 사소한 일반화를 올려볼까 합니다. 왠지 어딘가 쓸 수 있을 것 같은 두근거림이 느껴져요. Theorem #. Let $A \subset \Bbb{R}$ be a subset of $\Bbb{R}$ whose interior $U = \operatorname{int}(A)$ is not bounded above, i.e., $\sup U = \infty$. Also assume that $(c_n)$ is an increasing sequence of real numbers such that $c_n \to \infty$ and $c_{n+1} - c_{n} \to 0$ as $n \to \infty$. Then the set $$ D = \{ x \in \Bb..