Question

Gauss's proof that the Digamma function equals $\int_0^{\infty}(\frac{e^{-t}}{t} - \frac{e^{-zt}}{1-e^{-t}})dt$.

score 436 · Answer 1 · 2026-03-25 12:54:46

436

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Gauss's proof that the Digamma function equals $\int_0^{\infty}(\frac{e^{-t}}{t} - \frac{e^{-zt}}{1-e^{-t}})dt$.

Published on 25 Mar 2026 - 12:54

score 497 · Answer 2 · 2026-03-25 16:00:06

497

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How one can show that the gamma function is a strictly increasing function on the interval $(1.4616,+∞)$.

Published on 25 Mar 2026 - 16:00

score 648 · Answer 3 · 2026-03-25 11:09:52

648

Views

Find Harmonic Numbers for Imaginary and Complex Values

Published on 25 Mar 2026 - 11:09

score 355 · Answer 4 · 2026-03-25 11:09:59

355

Views

Is there a closed form for the polygamma function?

Published on 25 Mar 2026 - 11:09

score 337 · Answer 5 · 2026-03-25 07:45:23

337

Views

Evaluation of $\int\limits ^{\infty }_{0}\frac{x}{\left( x^{2} +1\right)^2\left( e^{tx} +1\right)} dx$

Published on 25 Mar 2026 - 7:45

score 126 · Answer 6 · 2026-03-25 11:13:27

126

Views

Does $\sum_{n=1}^{\infty} (-1)^n \zeta(n + 1)$ converge to $-1$?

Published on 25 Mar 2026 - 11:13

score 52 · Answer 7 · 2026-03-25 11:05:19

52

Views

Sum involving 2nd antiderivative of the digamma function.

Published on 25 Mar 2026 - 11:05

score 32 · Answer 8 · 2026-03-25 12:50:17

32

Views

Is there any possibility that there exists a formula for the partial sums of the harmonic series? If not, why?

Published on 25 Mar 2026 - 12:50

score 57 · Answer 9 · 2026-03-25 11:10:46

57

Views

Why does $\lim_{k \to \infty}\psi(k)$ diverge?

Published on 25 Mar 2026 - 11:10

score 83 · Answer 10 · 2026-03-25 12:50:57

83

Views

Is the following ratio of gamma functions increasing: $\frac{\Gamma(2n - \frac{1.25506n}{\ln n})}{\Gamma(n)^2}$?

Published on 25 Mar 2026 - 12:50

score 115 · Answer 11 · 2026-03-25 11:16:46

115

Views

Is this a correct way to use the digamma to analyze whether a ratio of gamma functions is increasing?

Published on 25 Mar 2026 - 11:16

score 327 · Answer 12 · 2026-03-25 12:50:15

327

Views

Understanding this property of the digamma function: $\psi(n + \frac{1}{2}) = -\gamma - 2\ln 2 + \sum\limits_{k=1}^n\frac{2}{2k-1}$

Published on 25 Mar 2026 - 12:50

score 358 · Answer 13 · 2026-03-25 11:12:23

358

Views

On a log-gamma definite integral

Published on 25 Mar 2026 - 11:12

score 59 · Answer 14 · 2026-03-25 11:16:43

59

Views

The antiderivative of $\sum_{n\gt 0}\frac{x}{n(x+n)}$

Published on 25 Mar 2026 - 11:16

score 578 · Answer 15 · 2026-03-25 14:26:05

578

Views

Alternative approaches to showing that $\Gamma'(1/2)=-\sqrt\pi\left(\gamma+\log(4)\right)$

Published on 25 Mar 2026 - 14:26

List Question

Gauss's proof that the Digamma function equals $\int_0^{\infty}(\frac{e^{-t}}{t} - \frac{e^{-zt}}{1-e^{-t}})dt$.

How one can show that the gamma function is a strictly increasing function on the interval $(1.4616,+∞)$.

Find Harmonic Numbers for Imaginary and Complex Values

Is there a closed form for the polygamma function?

Evaluation of $\int\limits ^{\infty }_{0}\frac{x}{\left( x^{2} +1\right)^2\left( e^{tx} +1\right)} dx$

Does $\sum_{n=1}^{\infty} (-1)^n \zeta(n + 1)$ converge to $-1$?

Sum involving 2nd antiderivative of the digamma function.

Is there any possibility that there exists a formula for the partial sums of the harmonic series? If not, why?

Why does $\lim_{k \to \infty}\psi(k)$ diverge?

Is the following ratio of gamma functions increasing: $\frac{\Gamma(2n - \frac{1.25506n}{\ln n})}{\Gamma(n)^2}$?

Is this a correct way to use the digamma to analyze whether a ratio of gamma functions is increasing?

Understanding this property of the digamma function: $\psi(n + \frac{1}{2}) = -\gamma - 2\ln 2 + \sum\limits_{k=1}^n\frac{2}{2k-1}$

On a log-gamma definite integral

The antiderivative of $\sum_{n\gt 0}\frac{x}{n(x+n)}$

Alternative approaches to showing that $\Gamma'(1/2)=-\sqrt\pi\left(\gamma+\log(4)\right)$

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