그동안 여기저기 싸지른 잡다한 계산들을 한 번 정리해볼까 합니다. Problem #. Show that \begin{equation*} \int_{0}^{\frac{\pi}{2}} \log \left(x^{2} + \log^{2}(\cos x) \right) \, \mathrm{d}x = \pi \log \log 2. \tag{1} \end{equation*} This problem is from [IS1]. Solution. Let $I$ denote the integral in $\text{(1)}$. By recalling the identity \begin{align*} x^{2} + \log^{2} (\cos x) = \left| \log \left( \frac{1+e^{2ix}}{2} \rig..
Here I am going to introduce some easy results on some criteria for interchanging the order of integration which are not covered by the classical Fubini theorem. Though both statements and proofs are weak and easy, it often reduces our burden to large extent. Let $f$ be a locally integrable function on $(0, \infty)$. That is, $f$ is a measurable function which is integrable on any compact subset..
- Total
- Today
- Yesterday
- 유머
- 편미방
- 이항계수
- Euler integral
- Beta function
- 수학
- 해석학
- 보컬로이드
- binomial coefficient
- Coxeter
- Zeta function
- 무한급수
- 제타함수
- 대수기하
- 렌
- 오일러 적분
- 미쿠
- 감마함수
- 푸리에 변환
- 계산
- 적분
- Euler constant
- Fourier Transform
- 오일러 상수
- Gamma Function
- 루카
- 노트
- Integral
- 린
- infinite summation
일 | 월 | 화 | 수 | 목 | 금 | 토 |
---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | |
7 | 8 | 9 | 10 | 11 | 12 | 13 |
14 | 15 | 16 | 17 | 18 | 19 | 20 |
21 | 22 | 23 | 24 | 25 | 26 | 27 |
28 | 29 | 30 |