Question

What is $\lim_{x \rightarrow 0}\left[\psi\left(1+\frac{a}{2x}\right)-\psi\left(\frac12+\frac{a}{2x}\right)\right]$?

score 92 · Answer 1 · 2023-06-28 12:33:16

92

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What is $\lim_{x \rightarrow 0}\left[\psi\left(1+\frac{a}{2x}\right)-\psi\left(\frac12+\frac{a}{2x}\right)\right]$?

Published on 28 Jun 2023 - 12:33

score 49 · Answer 2 · 2026-03-27 23:23:37

49

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Computing $\int_{0}^{\infty }x^a\,e^{-bx}\,Q(x)\,dx$, where $Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty }e^{-t^2/2}\,dt$.

Published on 27 Mar 2026 - 23:23

score 30 · Answer 3 · 2023-07-01 07:57:51

30

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How to solve $x^2\left(x_{min}!\right)^{3}\leq \left(x!\right)^{2}\left(\frac{1}{x+y}\right)!$ ? for $x,y>0$

Published on 01 Jul 2023 - 7:57

score 84 · Answer 4 · 2023-07-05 09:47:16

84

Views

How to compute the following integral

Published on 05 Jul 2023 - 9:47

score 103 · Answer 5 · 2026-03-26 04:53:52

103

Views

Generalization of Gauss multiplication formula for $\Gamma(jm+kn+a);j,k\in\Bbb N$?

Published on 26 Mar 2026 - 4:53

score 22 · Answer 6 · 2026-03-27 04:23:30

22

Views

Need help in detailing the proof of the existence of the test function

Published on 27 Mar 2026 - 4:23

score 62 · Answer 7 · 2026-03-27 04:16:53

62

Views

How to compute $\int_{-\infty}^{\infty} 1/(x^2+1)^s dx$?

Published on 27 Mar 2026 - 4:16

score 153 · Answer 8 · 2023-07-09 13:34:18

153

Views

Asymptotic behaviour of an integral with power and exponential functions

Published on 09 Jul 2023 - 13:34

score 90 · Answer 9 · 2023-07-09 21:09:06

90

Views

Estimating integral along arc in the contour integral $\int_{c}z^{a-1}e^{-c z}\,dz$

Published on 09 Jul 2023 - 21:09

score 108 · Answer 10 · 2026-03-25 14:25:32

108

Views

Does $\sum_{r = 1}^n \ln\left(\frac{1 + r}{r}\right) = \ln (\Gamma(n + 2)) - \ln (\Gamma(n+1))$? If so, why?

Published on 25 Mar 2026 - 14:25

score 42 · Answer 11 · 2026-03-26 03:11:17

42

Views

Asymptotic Convergence Rate and limit of Poisson Probability and Gamma function

Published on 26 Mar 2026 - 3:11

score 50 · Answer 12 · 2023-07-14 20:41:41

50

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$[(1+\delta)e^{-\delta}]^{t-1} \int_{\delta}^{\infty}(z+1)e^{-z}dz \leqslant 2\exp{ \{1-\frac{t \delta^2}{2} + \frac{\delta^3}{3(1-\delta)} \}}$

Published on 14 Jul 2023 - 20:41

score 90 · Answer 13 · 2026-03-26 02:58:30

90

Views

Asymptotic behavior of Gamma cdf, gamma function, and incomplete gamma function

Published on 26 Mar 2026 - 2:58

score 177 · Answer 14 · 2026-03-27 04:17:38

177

Views

How to solve this definite integral? $\int_0^1\frac{1}{\sqrt{1+x^4}}\,{\rm d}x$

Published on 27 Mar 2026 - 4:17

score 96 · Answer 15 · 2026-03-30 06:48:43

96

Views

Evaluate $\sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(2b)_n} \frac{\psi^{(0)}\left (n+ \frac{a}2+1 \right ) }{\Gamma\left (n+ \frac{a}2+1 \right ) }$

Published on 30 Mar 2026 - 6:48

List Question

What is $\lim_{x \rightarrow 0}\left[\psi\left(1+\frac{a}{2x}\right)-\psi\left(\frac12+\frac{a}{2x}\right)\right]$?

Computing $\int_{0}^{\infty }x^a\,e^{-bx}\,Q(x)\,dx$, where $Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty }e^{-t^2/2}\,dt$.

How to solve $x^2\left(x_{min}!\right)^{3}\leq \left(x!\right)^{2}\left(\frac{1}{x+y}\right)!$ ? for $x,y>0$

How to compute the following integral

Generalization of Gauss multiplication formula for $\Gamma(jm+kn+a);j,k\in\Bbb N$?

Need help in detailing the proof of the existence of the test function

How to compute $\int_{-\infty}^{\infty} 1/(x^2+1)^s dx$?

Asymptotic behaviour of an integral with power and exponential functions

Estimating integral along arc in the contour integral $\int_{c}z^{a-1}e^{-c z}\,dz$

Does $\sum_{r = 1}^n \ln\left(\frac{1 + r}{r}\right) = \ln (\Gamma(n + 2)) - \ln (\Gamma(n+1))$? If so, why?

Asymptotic Convergence Rate and limit of Poisson Probability and Gamma function

$[(1+\delta)e^{-\delta}]^{t-1} \int_{\delta}^{\infty}(z+1)e^{-z}dz \leqslant 2\exp{ \{1-\frac{t \delta^2}{2} + \frac{\delta^3}{3(1-\delta)} \}}$

Asymptotic behavior of Gamma cdf, gamma function, and incomplete gamma function

How to solve this definite integral? $\int_0^1\frac{1}{\sqrt{1+x^4}}\,{\rm d}x$

Evaluate $\sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(2b)_n} \frac{\psi^{(0)}\left (n+ \frac{a}2+1 \right ) }{\Gamma\left (n+ \frac{a}2+1 \right ) }$

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