Euler reflection formula

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Problem 3845 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Use the Euler integrals to compute the integral

∫0+∞x4(1+x)2 dx. \int _{0}^{+\infty }\dfrac {\sqrt[4]{x}}{(1+x)^{2}}\, dx .
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Problem 3846 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Use the Euler integrals to compute the integral

∫0+∞dx1+x3. \int _{0}^{+\infty }\dfrac {dx}{1+x^{3}} .
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Problem 3847 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Use the Euler integrals to compute the integral

∫0+∞x2 dx1+x4. \int _{0}^{+\infty }\dfrac {x^{2}\, dx}{1+x^{4}} .
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Problem 3849 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Use the Euler integrals to compute the integral

∫01dx1−xnn(n>1). \int _{0}^{1}\dfrac {dx}{\sqrt[n]{1-x^{n}}} \quad (n>1) .
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Problem 3851 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Determine the domain of existence of the integral

∫0+∞xm−11+xn dx(n>0) \int _{0}^{+\infty }\dfrac {x^{m-1}}{1+x^{n}}\, dx \quad (n>0)

and express this integral using the Euler integrals.

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Problem 3857 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Determine the domain of existence of the integral

∫0π2tan⁡nx dx \int _{0}^{\frac{\pi }{2}}\tan ^{n}x\, dx

and express this integral using Euler integrals.

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Problem 3863 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Determine the domain of existence of the integral

∫0+∞xp−1ln⁡x1+x dx(p>0) \int _{0}^{+\infty }\dfrac {x^{p-1}\ln x}{1+x}\, dx \quad (p>0)

and express this integral using Euler integrals.

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Problem 3864 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Determine the domain of existence of the integral

∫0+∞xp−1ln⁡2x1+x dx \int _{0}^{+\infty }\dfrac {x^{p-1}\ln ^{2}x}{1+x}\, dx

and express this integral using Euler integrals.

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Problem 3865 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Determine the domain of existence of the integral

∫0+∞xp−1−xq−1(1+x)ln⁡x dx \int _{0}^{+\infty }\dfrac {x^{p-1}-x^{q-1}}{(1+x)\ln x}\, dx

and express this integral using Euler integrals.

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Problem 3866 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Determine the domain of existence of the integral

∫01xp−1−x−p1−x dx(0<p<1) \int _{0}^{1}\dfrac {x^{p-1}-x^{-p}}{1-x}\, dx \quad (0<p<1)

and express this integral using Euler integrals.

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Remark.
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This integral can be regarded as

lim⁡ε→+0[B(p,ε)−B(1−p,ε)]. \lim _{\varepsilon \to +0}\left[B(p,\varepsilon )-B(1-p,\varepsilon )\right] .
Problem 3867 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Use Euler integrals to evaluate the integral

∫0+∞sh⁡αxsh⁡βx dx(0<α<β). \int _{0}^{+\infty }\dfrac {\operatorname {sh}\alpha x}{\operatorname {sh}\beta x}\, dx \quad (0<\alpha <\beta ) .
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Problem 3868 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Use Euler integrals to evaluate the integral

∫01ln⁡Γ(x) dx. \int _{0}^{1}\ln \Gamma (x)\, dx .
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Problem 3870 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Use the Euler integrals to evaluate the integral

∫01ln⁡Γ(x)sin⁡πx dx. \int _{0}^{1}\ln \Gamma (x)\sin \pi x\, dx.
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Problem 3864.1 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Problem 3864 "[NO TITLE]"

Determine the domain of existence of the integral

∫0∞xln⁡x1+x3 dx \int _{0}^{\infty }\dfrac {x\ln x}{1+x^{3}}\, dx

and express this integral using Euler integrals.

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Problem 3864.2 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Problem 3864 "[NO TITLE]"

Determine the domain of existence of the integral

∫0∞ln⁡2x1+x4 dx \int _{0}^{\infty }\dfrac {\ln ^{2}x}{1+x^{4}}\, dx

and express this integral using Euler integrals.

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Problem 2.24.72 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.24.2 "Problems"

Prove that Γ(x)Γ(1−x)=π/sin⁡πx\Gamma (x) \Gamma (1-x) = \pi / \sin \pi x.

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Problem 24.10 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 24 "Integrals Reducible to the Gamma Function"

Prove that for 0<Re(α)<10 < \mathrm{Re}\left(\alpha \right) < 1 the formula holds

∫−π/4π/4(cos⁡φ−sin⁡φcos⁡φ+sin⁡φ)2α−1dφ=π2sin⁡πα \int _{-\pi / 4}^{\pi / 4}\left(\frac{\cos \varphi -\sin \varphi }{\cos \varphi +\sin \varphi }\right)^{2 \alpha -1} d \varphi = \frac{\pi }{2 \sin \pi \alpha }
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Problem 24.12 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 24 "Integrals Reducible to the Gamma Function"

Denote by Dσ,ρD_{\sigma , \rho } the half-plane {z:Re(z)<σ}\left\{ z: \mathrm{Re}\left(z\right) < \sigma \right\} from which the disks {z:∣z+n∣<ρ},n=1,2,…\left\{ z:\left|z+n\right| < \rho \right\} , n = 1,2, \ldots have been removed. Prove that for any fixed values of the constants ρ>0,σ\rho > 0, \sigma, and mm the inequality holds

∣Γ(z)∣⩽M(1+∣z∣)−m(z∈Dσ,ρ) \left|\Gamma (z)\right| \leqslant M(1+\left|z\right|)^{-m} \quad \left(z \in D_{\sigma , \rho }\right)

with some constant MM depending on the choice of the numbers ρ,σ,m\rho , \sigma , m.

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Problem 24.13 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 24 "Integrals Reducible to the Gamma Function"

Prove that

∣Γ(12+iξ)∣=πch⁡πξ,−∞<ξ<∞. \left|\Gamma \left(\frac{1}{2}+i \xi \right)\right| = \sqrt{\frac{\pi }{\operatorname {ch} \pi \xi }}, \quad -\infty < \xi < \infty .
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References

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.
[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Интегралы. Ряды. 2-е изд., ФИЗМАТЛИТ, 2003.
[ШПК15] Шабунин М.И., Половинкин Е.С., Карлов М.И. Сборник задач по теории функций комплексного переменного. 4-е изд., БИНОМ. Лаборатория знаний, 2015.