Stack Exchange

Sangchul Lee

Los Angeles, CA, USA

http://sos440.blogspot.kr/

I received my PhD at University of California, Los Angeles. I am currently interested in probability theory and analytic combinatorics.

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Top Questions
180 votes

Generalizing $\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{\operatorname dx}{x^{2}+1} = \frac{5\pi^{2}}{96}$

integration definite-integrals special-functions asked Nov 25 '13 at 12:51
math.stackexchange.com

63 votes

How much does symbolic integration mean to mathematics?

integration numerical-methods symbolic-computation asked May 11 '11 at 15:25
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52 votes

Integral $\int_{-1}^{1} \frac{1}{x}\sqrt{\frac{1+x}{1-x}} \log \left( \frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} - sx + 1} \right) \, \mathrm dx$

calculus integration definite-integrals closed-form asked Nov 16 '13 at 1:48
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50 votes

Extending the result $\int_{0}^{\infty} \left( ( 1 - 2C(x))^{2} + (1-2S(x))^{2} \right) \, dx = \frac{4}{\pi} $

special-functions improper-integrals asked May 25 '13 at 15:06
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48 votes

Polynomial equations $p(A, B) = 0$ for matrices that ensure $AB = BA$

linear-algebra matrices asked Jul 30 '11 at 20:53
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34 votes

Patterns of the zeros of the Faulhaber polynomials (modified)

polynomials roots asked Mar 1 '13 at 0:58
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29 votes

Is there a closed form for this sum?

integration summation definite-integrals special-functions asked Nov 12 '13 at 16:39
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22 votes

References to integrals of the form $\int_{0}^{1} \left( \frac{1}{\log x}+\frac{1}{1-x} \right)^{m} \, dx$

integration reference-request riemann-zeta asked Mar 25 '13 at 15:17
math.stackexchange.com

21 votes

Probability on entering direction of a simple random walk

probability probability-theory random-walk asked Mar 22 '15 at 20:50
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20 votes

Showing that $a_{n+1}=\frac{n}{a_n}-a_n-a_{n-1}$ with $a_0 = 0$ and $a_1=2\Gamma(\frac34)\big/\Gamma(\frac14)$ stays positive for $n\geq1$.

sequences-and-series analysis asked Jun 5 '19 at 0:54
math.stackexchange.com

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Top Answers
197
A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$
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166
Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$
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137
Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$
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115
Prove that $\frac{100!}{50!\cdot2^{50}} \in \Bbb{Z}$
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96
How can a “proper” function have a vertical slope?
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66
Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$
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61
How to show that $\int_0^1 \left(\sqrt[3]{1-x^7} - \sqrt[7]{1-x^3}\right)\;dx = 0$
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56
An integral involving Fresnel integrals $\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$
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52
All real numbers in $[0,2]$ can be represented as $\sqrt{2 \pm \sqrt{2 \pm \sqrt{2 \pm \dots}}}$
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50
Evaluating $\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2}$
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46
An interesting definite integral $\int_0^1(1+x+x^2+x^3+\cdot\cdot\cdot+x^{n-1})^2 (1+4x+7x^2+\cdot\cdot\cdot+(3n-2)x^{n-1})~dx=n^3$
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46
Can we calculate $ i\sqrt { i\sqrt { i\sqrt { \cdots } } }$?
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45
Integral Contest
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43
Compute $ I_{n}=\int_{-\infty}^\infty \frac{1-\cos x \cos 2x \cdots \cos nx}{x^2}\,dx$
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40
Finding the limit of a sequence with an undesirable $\ln k$
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40
Infinite Series $\sum\limits_{n=1}^\infty\left(\frac{H_n}n\right)^2$
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37
How to evaluate $\int_0^\infty\operatorname{erfc}^n x\ \mathrm dx$?
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36
Prove the divergence of the sequence $\left\{ \sin(n) \right\}_{n=1}^{\infty}$.
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36
Find the value of $\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$
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35
Integral $\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$
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35
Ways to show that $\int_{0}^{1}((1-x^r)^{1/r}-x)^{2k}dx=\frac{1}{2k+1}$
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34
Integral $\int_0^{\pi/2}\arctan^2\left(\frac{6\sin x}{3+\cos 2x}\right)\mathrm dx$
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34
Prove $\int_0^\infty \frac{\ln \tan^2 (ax)}{1+x^2}\,dx = \pi\ln \tanh(a)$
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33
$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!} = \frac{1}{2}$ - basic methods
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33
Does the sum $\sum_{n \geq 1} \frac{2^n\operatorname{mod} n}{n^2}$ converge?
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32
Maximal ideals in the ring of real functions on $[0,1]$
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32
Calculate sums of inverses of binomial coefficients
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32
Integral $\int_0^\infty\frac{1}{\sqrt[3]{x}}\left(1+\log\frac{1+e^{x-1}}{1+e^x}\right)dx$
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31
Integral $\int_0^\infty\frac{\operatorname{arccot}\left(\sqrt{x}-2\,\sqrt{x+1}\right)}{x+1}\mathrm dx$
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30
Integral $\int_0^\infty\frac{\ln\left(1+x+\sqrt{x^2+2\,x}\right)\,\ln\left(1+\sqrt{x^2+2\,x+2}\right)}{x^2+2x+1}dx$
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