잡담을 위한 소박한 장소

Problem 1. Show that[각주:1] the product \begin{equation*} P = \left( \frac{2}{1} \right)^{\frac{1}{8}} \left( \frac{2 \cdot 2}{1 \cdot 3} \right)^{\frac{3}{16}} \left( \frac{2 \cdot 2 \cdot 2 \cdot 4}{1 \cdot 3 \cdot 3 \cdot 3} \right)^{\frac{6}{32}} \left( \frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot 4 \cdot 4 \cdot 4 \cdot 4}{1 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 5} \right)^{\frac{10..

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..

이번 만우절 장난은 잘 즐기셨나요? 이번의 관전 포인트는, 역시 포토샵으로 만든 책 이미지가 아닐까 싶습니다. 이번 작업에 사용된 책은 Bartle & Sherbert 의 Introduction to Real Analysis 라는 책입니다. 간단한 스펀지질을 통해 제목과 그림을 지우고 대신 아래와 같은 가짜 저자와 제목을 덮어씌웠죠. 원본 책 지못미... 당연히, 저런 책도 저런 저자도 존재하지 않습니다. 저자 이름은 스리니바사 라마누잔(Srinivasa Ramanujan)을 거꾸로 썼고, 부제는 Spivak의 The Joy of TeX을 베꼈습니다. 수식들도 블로그 어딘가에 잘 숨어 있습니다. 제가 지금까지 어떻게 이런 이상한 계산들을 할 수 있었는지 궁금해하시는 분들이 있을 것 같습니다. 하지만 저는..

Motivated by a problematic exercise in an analysis textbook, I decided to prove the following proposition. 1. The statement Theorem. There exists an ordered field $F$ such that it satisfies the nested interval property but does not satisfy the completeness axiom. Before the proof, we should explain what a non-principal ultrafilter means. Given a set $X$, a subset $\mathcal{U}$ of $\mathcal{P}(X)..

최근의 포스팅 「오늘의 계산 53」과 관련하여, 다음과 같은 관찰을 하였습니다. \begin{align*} \int_{0}^{1}\left( \frac{1}{\log x} + \frac{1}{1-x} \right) \, dx &= \lim_{s\to 1} \left( \zeta(s) - \frac{1}{s-1} \right), \\ \int_{0}^{1}\left( \frac{1}{\log x} + \frac{1}{1-x} \right)^{2} \, dx &= -2 \zeta'(0) -\frac{3}{2}, \\ \int_{0}^{1}\left( \frac{1}{\log x} + \frac{1}{1-x} \right)^{3} \, dx &= -6 \zeta '(-1) -\frac{19}{24}, \\ \..

요즘 StackExchange에서 노닥거리다가 줏은 문제 몇 개를 조금씩 각색해서 올려봅니다. Problem 1.Let $(a_n)$ and $(b_n)$ be sequences of non-negative real numbers which are not identically zero. Also, let $(c_n)$ be the Cauchy product of these sequences: $$ c_n = (a \ast b)_{n} = \sum_{k=0}^{n} a_k b_{n-k}. $$ Prove that $$ \limsup_{n\to\infty} c_{n}^{1/n} = \max \left\{ \limsup_{n\to\infty} a_{n}^{1/n}, \limsup_{n\to\infty} b_{..

Here we want to calculate an integral related to our earlier calculation. Problem. Prove that \begin{equation} \label{eqn:wts} \int_{0}^{1} \left( \frac{1}{\log x} + \frac{1}{1-x} \right)^2 \, dx = \log (2\pi) - \frac{3}{2}. \end{equation} 1st Proof. Note that $$ \frac{1}{1-x} + \frac{1}{\log x} = \int_{0}^{1} \frac{1 - x^{t}}{1-x} \, dt. $$ Then by Tonelli's theorem, we have \begin{align*} \int..

이번 계산에서는, 마지막 부분에 제가 깔끔하게 처리하지 못한 부분을 다른 분의 도움을 얻어 깔끔하게 처리해보았습니다. 어떤 분은 복소로 풀어내기도 했는데, 이 풀이 역시 상당히 마음에 와닿더군요. Problem. Let $(u_n)$ be the sequence defined by \begin{equation} \label{eqn:recur} \frac{u_1}{n} + \frac{u_2}{n-1} + \cdots + \frac{u_{n-1}}{2} + u_n = 1. \end{equation} Find the asymptotic behavior of $u_n$. Solution. We introduce the generating function $U(x)$ of $(u_n)$ given by $$U(x..

이전에 비슷한 적분을 실해석적으로 푼 적이 있는데, 이번에는 깔끔하게 복소로 계산해보았습니다. 제 복소적분 실력도 못 써먹을 수준은 아니군요. Problem. Prove that[1] $$ \int_0^{\pi/2} \frac{\log(\cos x)}{x^2+\log^2(\cos x)} \, dx = \frac{\pi}{2}\left(1-\frac{1}{\log 2}\right) $$ Proof. Note that for $|x| < \frac{\pi}{2}$, we have $$ \frac{\log\cos x}{\log^2 \cos x + x^2} = \Re \frac{1}{\log\left( \frac{1+e^{2ix}}{2} \right)}. $$ Thus if $I$ denotes the give..

p-adic number에 익숙하다면 그냥 몇 줄 끄적이면 바로 따라나오는 정리인데, 오랜만에 다시 보니깐 그냥 싱숭생숭해서 다시 적어봅니다. 당분간은 유학 준비 - 특히 회화 준비 - 로 인해 수학 공부는 좀 쉴 것 같으니, 이렇게 간단한 내용들이나 스크랩해서 정리해볼까 하네요. 그나저나 역시 저는 뭘 해도 해석학적인 계산으로 때려박아야 직성이 풀리나봅니다. =ㅁ= Wolstenholme's Theorem. Let $p > 3$ be a prime number. If we write \begin{align*} 1 + \frac{1}{2} + \cdots + \frac{1}{p-1} = \frac{r}{q} \end{align*} in lowest term, then $r \equiv 0 \ (\math..

Here are chosen ones from my 50+ε calculations from the Today's Calculation archive, which are polished and supplemented. Any feedback is greatly welcomed. Table of Contents Preliminary Some materials on analysis Mathematical Constants Riemann Zeta function Gamma function and Polygamma functions Beta function Polylogarithms and related functions Today’s Calculation Derivation of $\zeta(2n)$ An i..

Enjoy this calculation and have a good day! Problem. For $\alpha > 0$, examine the following limit \begin{equation}\label{eqn:wts} \lim_{n\to\infty} e^{-\alpha\sqrt{n}} \sum_{k=0}^{n-1} 2^{-n-k} \binom{n-1+k}{k} \sum_{m=0}^{n-1-k}\frac{(\alpha\sqrt{n})^m}{m!}. \end{equation} Solution. Let $A_n$ denote the formula inside the limit \eqref{eqn:wts}. By noting that the double summation is taken for ..

Today I want to deal with a problem which does not lie in analysis. It would be easy for those who are adapted to algebra, but for those with algebra trauma like me, it was rather a challenge. Problem. Let $m$ be an integer and $(u_n)$ be a sequence defined by \begin{equation}\label{eqn:recur} u_0 = 1, \quad u_1 = m \quad \text{and} \quad u_{n+2} = mu_{n+1} + u_n. \end{equation} Prove that for a..

Recently I was astonished by the following problem. Question.Let $f$ be a continuous function such that \begin{equation}\label{eqn:cond} \int_{-\infty}^{\infty} x^n f(x) \; dx = 0 \end{equation} for all $n = 0, 1, 2, \cdots$. Then is $f = 0$ everywhere? Answer. It might seem at the first glance that this is true. Surprisingly, however, this is not true. Let $$ g(x) = \exp\left(-x^2-\frac{1}{x^2}..

Problem 1. Evaluate the following integral.[1] \begin{equation}\label{prob:wts} \int_{0}^{\infty} \frac{u \log(a^2+u^2)}{e^{2\pi u} - 1} \; du \end{equation} Proof. Let $I(a)$ denote the given integral. Then we have \begin{align*}I(0) & = 2 \int_{0}^{\infty} \frac{u \log u}{e^{2\pi u} - 1} \; du \\ & = \frac{1}{2\pi^2} \int_{0}^{\infty} \frac{x (\log x - \log (2\pi))}{e^{x} - 1} \; dx \qquad (x ..

쉬운 것도 어렵게 하라! \begin{align*} & \int_{-\infty}^{\infty}\frac{\cos x}{1+x^2}\;dx \\ &=\int_{-\infty}^{\infty}\cos x\left(\int_{0}^{\infty}e^{-(1+x^2)t}\;dt\right)dx\\ &=\int_{0}^{\infty}e^{-t}\left(\int_{-\infty}^{\infty}\cos x\,e^{-tx^2}\;dx\right)dt\\ &=\int_{0}^{\infty}e^{-t}\left(\int_{-\infty}^{\infty}e^{-tx^2+ix}\;dx\right)dt\\ &=\int_{0}^{\infty}e^{-t-\frac{1}{4t}}\left(\int_{-\infty}^{\inf..

Problem. Prove that[1] \begin{equation}\label{prob:wts} \int_{0}^{\frac{\pi}{2}} \frac{x^2}{x^2 + \log^2(2\cos x)} \; dx = \frac{\pi}{8}\left( 1 - \gamma + \log(2\pi) \right). \end{equation} Proof. Here I refer to the following identity \begin{equation}\label{eq:01} \binom{\alpha}{\omega} = \frac{1}{2\pi} \int_{-\pi}^{\pi} \left(1 + e^{i\theta}\right)^{\alpha} e^{-i\omega \theta} \; d\theta, \en..

Problem. Find the limit \begin{equation}\label{prob:lim} \lim_{n\to\infty} \frac{1}{n^{3/2}} \int_{-\infty}^{\infty} \frac{1 - \prod_{k=1}^{n} \cos (kx)}{x^2} \; dx. \end{equation} To find the value of \eqref{prob:lim}, we need some preliminaries. Lemma 1. For any $\alpha \in \mathbb{R}$, we have \begin{equation*} \int_{-\infty}^{\infty} \frac{1 - \cos (\alpha x)}{x^2} \; dx = \pi |\alpha|. \end..