Question

Show that the identity $\int_0^\infty\sin(t)t^{z-1}\ dt = \Gamma(z)\sin\left(\pi{z\over 2}\right)$ holds on $-1<\operatorname{Re}(z)<1$

score 194 · Answer 1 · 2022-11-17 06:36:06

194

Views

Show that the identity $\int_0^\infty\sin(t)t^{z-1}\ dt = \Gamma(z)\sin\left(\pi{z\over 2}\right)$ holds on $-1<\operatorname{Re}(z)<1$

Published on 17 Nov 2022 - 6:36

score 201 · Answer 2 · 2022-11-19 03:27:32

201

Views

$\sum\limits_{n=1}^\infty\frac{(-1)^\frac{(5+\sqrt3i)n}6x^n}{n!}(\frac ns )^{(n-1)}\,_2\text F_1\left(1-n,-sn;2-sn;s\right)$ closed form or integral.

Published on 19 Nov 2022 - 3:27

score 556 · Answer 3 · 2022-11-27 19:26:42

556

Views

Integral representation for series of any order

Published on 27 Nov 2022 - 19:26

score 156 · Answer 4 · 2022-12-04 22:59:56

156

Views

Explicit Taylor Expansions of Zeta Function

Published on 04 Dec 2022 - 22:59

score 36 · Answer 5 · 2022-12-06 05:13:29

36

Views

Analytic continuation of the sum of the reciprocals of the $n$-bonacci sequences.

Published on 06 Dec 2022 - 5:13

score 91 · Answer 6 · 2022-12-09 13:18:53

91

Views

Proof check: derivative of function is analytic continuation

Published on 09 Dec 2022 - 13:18

score 51 · Answer 7 · 2022-12-10 13:19:18

51

Views

Extending functions from integers to reals. What properties (generally) matter?

Published on 10 Dec 2022 - 13:19

score 153 · Answer 8 · 2022-12-13 05:58:11

153

Views

Computation on converging alternating zeta-like series by analytic continuation

Published on 13 Dec 2022 - 5:58

score 224 · Answer 9 · 2022-12-19 16:18:28

224

Views

Numerical analytic continuation for $\tanh$-like function

Published on 19 Dec 2022 - 16:18

score 32 · Answer 10 · 2023-01-02 02:11:48

32

Views

The relationship between interior and exterior Riemann maps

Published on 02 Jan 2023 - 2:11

score 63 · Answer 11 · 2023-01-04 17:26:24

63

Views

Deducing that an analytic function $f$ in a simply connected domain $\Omega$ is the derivative of an analytic function $F:\Omega\to\mathbb{C}$

Published on 04 Jan 2023 - 17:26

score 127 · Answer 12 · 2026-03-26 04:51:22

127

Views

New $ \overset {s}{\underset{k=x}{ \lower 3pt { \LARGE\Xi}}} $ operator and Möbius function

Published on 26 Mar 2026 - 4:51

score 55 · Answer 13 · 2023-01-19 17:35:02

55

Views

Can the complex function $f(z) = \sqrt[n]{r^n - z^n}$ be analytically continued to fill the complex plane?

Published on 19 Jan 2023 - 17:35

score 207 · Answer 14 · 2023-01-20 22:25:36

207

Views

Roots and analytic continuation of $T(s)=\sum_{n>0} (n^s + n^{-s})^{-1} $?

Published on 20 Jan 2023 - 22:25

score 115 · Answer 15 · 2026-02-23 02:15:30

115

Views

Analytic continuation of $f(z) = \sum_{n=1}^{\infty} \frac{1}{2^n (z-e^{2\pi i r_n})}$

Published on 23 Feb 2026 - 2:15

List Question

Show that the identity $\int_0^\infty\sin(t)t^{z-1}\ dt = \Gamma(z)\sin\left(\pi{z\over 2}\right)$ holds on $-1<\operatorname{Re}(z)<1$

$\sum\limits_{n=1}^\infty\frac{(-1)^\frac{(5+\sqrt3i)n}6x^n}{n!}(\frac ns )^{(n-1)}\,_2\text F_1\left(1-n,-sn;2-sn;s\right)$ closed form or integral.

Integral representation for series of any order

Explicit Taylor Expansions of Zeta Function

Analytic continuation of the sum of the reciprocals of the $n$-bonacci sequences.

Proof check: derivative of function is analytic continuation

Extending functions from integers to reals. What properties (generally) matter?

Computation on converging alternating zeta-like series by analytic continuation

Numerical analytic continuation for $\tanh$-like function

The relationship between interior and exterior Riemann maps

Deducing that an analytic function $f$ in a simply connected domain $\Omega$ is the derivative of an analytic function $F:\Omega\to\mathbb{C}$

New $ \overset {s}{\underset{k=x}{ \lower 3pt { \LARGE\Xi}}} $ operator and Möbius function

Can the complex function $f(z) = \sqrt[n]{r^n - z^n}$ be analytically continued to fill the complex plane?

Roots and analytic continuation of $T(s)=\sum_{n>0} (n^s + n^{-s})^{-1} $?

Analytic continuation of $f(z) = \sum_{n=1}^{\infty} \frac{1}{2^n (z-e^{2\pi i r_n})}$

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