잡담을 위한 소박한 장소

While calculating a specific problem, I succeeded in proving a more general problem. Proposition. For $0 < r < 1$ and $r < s$, the following holds:[1] \begin{equation} \label{eq_wts} \int_{-1}^{1} \frac{1}{x} \sqrt{\frac{1+x}{1-x}} \log \left| \frac{1 + 2rsx + (r^{2} + s^{2} - 1)x^{2}}{1 - 2rsx + (r^{2} + s^{2} - 1)x^{2}} \right| \, dx = 4\pi \arcsin r. \end{equation} Proof. We divide the proof ..

나흘 연휴의 절반이 지나가는 동안 한 거라곤 폐인짓밖에 없네…. 공부해야지~ Proposition. The following holds:[1] \begin{equation*} \tag{1} \int_{0}^{\frac{\pi}{2}} \arctan ( r \sin\theta ) \arctan ( s \sin\theta ) \, d\theta = \pi \chi_{2}(\alpha \beta), \end{equation*} where \begin{align*} \alpha = \frac{\sqrt{r^{2} + 1} - 1}{r}, \quad \beta = \frac{\sqrt{s^{2} + 1} - 1}{s} \end{align*} and $\chi_{2}$ is the Legendre chi funct..

Take Home Exam을 풀어야 하는데, 나는 이런 거나 계산하고 있을 뿐이고…. Proposition. The following holds:[1] \begin{align*} \sum_{n=1}^{\infty} \frac{H_{n}^{2}}{n^{2}} = \frac{17\pi^{4}}{360}. \end{align*} Proof. See the reference [1] below. Remark. This identity was first conjectured by Enrico Au-Yeung, a student of Jonathan Borwein, using computer search and the PSLQ algorithm, in 1993.[2] It is subsequently solved b..