Question

Calculate $\int_0^{1/2}\, x\sin(\pi x)\Gamma(\frac{d-1}{2} + x)\Gamma(\frac{d-1}{2} - x) \mathrm dx$ for even $d\ge 4$

score 270 · Answer 1 · 2026-04-16 17:02:39

270

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Calculate $\int_0^{1/2}\, x\sin(\pi x)\Gamma(\frac{d-1}{2} + x)\Gamma(\frac{d-1}{2} - x) \mathrm dx$ for even $d\ge 4$

Published on 16 Apr 2026 - 17:02

score 154 · Answer 2 · 2026-04-11 13:15:40

154

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A good practise book suggestion

Published on 11 Apr 2026 - 13:15

score 166 · Answer 3 · 2026-04-14 20:16:29

166

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Sum with central trinomial coefficients

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score 102 · Answer 4 · 2026-04-16 10:25:15

102

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Name of "divided difference" transform $\frac{f(x)-f(x_0)}{x-x_0}$ and special case $\frac{e^x - 1}{x}$?

Published on 16 Apr 2026 - 10:25

score 137 · Answer 5 · 2026-04-12 19:56:01

137

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Deriving exponential generating function for central trinomial coefficients

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score 150 · Answer 6 · 2026-04-10 20:02:13

150

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Integral of multiplication of error function and cosine

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score 41 · Answer 7 · 2026-04-10 12:03:14

41

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How can I solve an integral by using hypergeometric function of the second kind?

Published on 10 Apr 2026 - 12:03

score 95 · Answer 8 · 2026-04-17 05:41:50

95

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Published on 17 Apr 2026 - 5:41

score 174 · Answer 9 · 2026-04-14 10:20:29

174

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Meijer G explicit form

Published on 14 Apr 2026 - 10:20

score 33 · Answer 10 · 2026-04-08 03:18:35

33

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How can I solve an integral by using hypergeometric function of the second kind?

Published on 08 Apr 2026 - 3:18

score 41 · Answer 11 · 2026-04-17 05:11:25

41

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Let $x=\cos\theta$. Show that $\frac{d^{n-1}\sin^{2n-1}\theta}{dx^{n-1}}=\frac{(-1)^{n-1}}{n}\frac{(2n)!}{2^{n}n!}\sin n\theta$

Published on 17 Apr 2026 - 5:11

score 129 · Answer 12 · 2026-04-17 11:50:00

129

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Convergence of $\int_0^{\infty} s^{2-2\alpha}e^{is^2}ds$

Published on 17 Apr 2026 - 11:50

score 128 · Answer 13 · 2026-04-14 11:35:49

128

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How to evaluate $\int\limits_0^\infty {\frac{{{e^{ - x - \frac{1}{x}}}}}{{1 + x}}dx}$ in form of special function?

Published on 14 Apr 2026 - 11:35

score 96 · Answer 14 · 2026-04-18 11:59:58

96

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For what $a$ and $b$ are there explicit expressions for $I(a, b) =\int_0^1 \int_0^1 \dfrac{dx\,dy}{1-x^ay^b} $?

Published on 18 Apr 2026 - 11:59

score 208 · Answer 15 · 2026-04-30 02:14:28

208

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How can I derive the Legendre function of first kind in terms of the hypergeometric function?

Published on 30 Apr 2026 - 2:14

List Question

Calculate $\int_0^{1/2}\, x\sin(\pi x)\Gamma(\frac{d-1}{2} + x)\Gamma(\frac{d-1}{2} - x) \mathrm dx$ for even $d\ge 4$

A good practise book suggestion

Sum with central trinomial coefficients

Name of "divided difference" transform $\frac{f(x)-f(x_0)}{x-x_0}$ and special case $\frac{e^x - 1}{x}$?

Deriving exponential generating function for central trinomial coefficients

Integral of multiplication of error function and cosine

How can I solve an integral by using hypergeometric function of the second kind?

Special Functions prerequisites

Meijer G explicit form

How can I solve an integral by using hypergeometric function of the second kind?

Let $x=\cos\theta$. Show that $\frac{d^{n-1}\sin^{2n-1}\theta}{dx^{n-1}}=\frac{(-1)^{n-1}}{n}\frac{(2n)!}{2^{n}n!}\sin n\theta$

Convergence of $\int_0^{\infty} s^{2-2\alpha}e^{is^2}ds$

How to evaluate $\int\limits_0^\infty {\frac{{{e^{ - x - \frac{1}{x}}}}}{{1 + x}}dx}$ in form of special function?

For what $a$ and $b$ are there explicit expressions for $I(a, b) =\int_0^1 \int_0^1 \dfrac{dx\,dy}{1-x^ay^b} $?

How can I derive the Legendre function of first kind in terms of the hypergeometric function?

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