Functional equation and special values of the gamma function

[6/50%]
Show
LaTeX
Problem 2348 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 4.4 "Improper integrals"

Using a reduction formula, evaluate the following improper integral (nn is a natural number): In=∫0+∞xne−xdx\displaystyle I_{n}=\int_{0}^{+\infty }x^{n}e^{-x} dx.

?
Problem 2.12.32 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.12.2 "Problems"

Compute the integral. ∫0+∞xne−xdx,n∈N\int_{0}^{+\infty } x^{n} e^{-x} d x, n \in N.

?
Problem 10.3.5 from Vestrup E.M. "The Theory of Measures and Integration" [Ves03]More problems in Section 10.3 "The Fubini Theorems in Euclidean Space"

Verify the following facts concerning the gamma function:

?
(a)

Γ(α)=(α−1)Γ(α−1)\Gamma (\alpha )=(\alpha -1)\Gamma (\alpha -1) for all α>1\alpha >1.

(b)

Γ(n)=n!\Gamma (n)=n! for every positive integer nn.

Remark.
?

Part (a)'s hint: integrate by parts.

For positive real α\alpha and β\beta, the gamma and beta functions are defined by

Γ(α)=∫0∞xα−1e−x dλˉ(x)andB(α,β)=∫01xα−1(1−x)β−1 dλˉ(x). \Gamma (\alpha )=\int _{0}^{\infty } x^{\alpha -1} e^{-x} \, \mathrm{d}\bar\lambda (x) \quad \text{and} \quad B(\alpha ,\beta )=\int _{0}^{1} x^{\alpha -1}(1-x)^{\beta -1} \, \mathrm{d}\bar\lambda (x) .
Problem 18.18 from Billingsley P. "Probability and Measure" [Bil95]More problems in Section 18 "Product Measure and Fubini's Theorem"

Euler's gamma function is defined for positive tt by Γ(t)=∫0∞xt−1e−xdx\Gamma (t)=\int_{0}^{\infty } x^{t-1} e^{-x} d x.

?
(a)

Prove that Γ(k)(t)=∫0∞xt−1(log⁡x)ke−xdx\Gamma^{(k)}(t)=\int_{0}^{\infty } x^{t-1}(\log x)^{k} e^{-x} d x.

(b)

Show by partial integration that Γ(t+1)=tΓ(t)\Gamma (t+1)=t \Gamma (t) and hence that Γ(n+1)=n!\Gamma (n+1) =n! for integral nn.

(c)

From (18.10) deduce Γ(12)=π\Gamma \left(\frac{1}{2}\right)=\sqrt{\pi }.

(d)

Show that the unit sphere in RkR^{k} has volume (see Example 18.5)

Vk=πk/2Γ((k/2)+1).(18.19) V_{k}=\frac{\pi ^{k / 2}}{\Gamma ((k / 2)+1)}. \tag {18.19}
Problem I-42 from Cacoullos T. "Exercises in Probability" [Cac89]More problems in Appendix I "Miscellaneous Exercises"

Using the fact that if XX follows the normal distribution N(0,1)\mathscr {N}\left(0, 1\right), then

P(X2≤x)=(2/π)1/2∫0xe−u2/2 du, \mathbb {P}\left(X^{2} \leq x\right) = (2/\pi )^{1/2} \int _{0}^{\sqrt{x}} e^{-u^{2}/2} \, du,

conclude that

Γ(1/2)=∫0∞x(1/2)−1e−x dx=π. \Gamma (1/2) = \int _{0}^{\infty } x^{(1/2)-1} e^{-x} \, dx = \sqrt{\pi }.
?
Problem 4.4.1 from Grimmett G.R., Stirzaker D.R. "Probability and Random Processes" [GrSt20]More problems in Section 4.4 "Examples of continuous variables"

Prove that the gamma function satisfies Γ(t)=(t−1)Γ(t−1)\Gamma (t) = (t-1) \Gamma (t-1) for t>1t > 1, and deduce that Γ(n)=(n−1)!\Gamma (n) = (n-1)! for n=1,2,…n = 1,2, \ldots Show that Γ(12)=π\Gamma \left(\frac{1}{2}\right) = \sqrt{\pi } and deduce a closed form for Γ(n+12)\Gamma \left(n+\frac{1}{2}\right) for n=0,1,2,…n = 0,1,2, \ldots.

?

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.
[Ves03] Vestrup E.M. The Theory of Measures and Integration. Wiley-Interscience, 2003.
[Bil95] Billingsley P. Probability and Measure. 3rd ed., Wiley-Interscience, 1995.
[Cac89] Cacoullos T. Exercises in Probability. Springer-Verlag, 1989.
[GrSt20] Grimmett G.R., Stirzaker D.R. Probability and Random Processes. 4th ed., Oxford University Press, 2020.