Smoothness and convexity of the gamma function

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Problem 3841 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Prove that the $\Gamma$ function $\Gamma (x)$ is continuous on the domain $x>0$ , and has continuous derivatives of every order.

Problem 3.16.2 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.16.2 "Problems"

Prove that $\Gamma (p)$ is an infinitely differentiable function, and moreover

\Gamma ^{(m)}(p) = \int _{0}^{+\infty } x^{p-1}(\ln x)^{m} e^{-x} d x

Problem 3.16.3 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.16.2 "Problems"

Prove that

$\Gamma (\alpha ) \sim 1 / \alpha$ as $\alpha \rightarrow +0$ and $\Gamma (\alpha ) \rightarrow +\infty$ as $\alpha \rightarrow +\infty$ .

Problem 3.16.5 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.16.2 "Problems"

Prove that the function $\Gamma (\alpha )$ is strictly convex upward on the interval $(0 ;+\infty )$ .

Problem VII.1.30 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section VII.1 "Convexity"

(a)(b)(c)(d)

(a)

Let $a_{n}, \lambda_{n} \geqslant 0 \quad (n \in \mathbb {N}), f(x)=\sum a_{n} e^{-\lambda_{n} x}$ . Prove that $\ln f$ is a convex function on $\mathbb {R}$ .

(b)

A positive function $f$ is called logarithmically convex if the function $\ln f$ is convex. Prove that a logarithmically convex function is convex and that the sum of logarithmically convex functions is logarithmically convex.

(c)

Prove that the Euler Gamma function is logarithmically convex on $(0,+\infty )$ .

(d)

Let $f$ be a logarithmically convex solution on $(0,+\infty )$ of the functional equation $f(x+1)=x f(x)(x>0)$ . Prove that $f(x)=f(1) \Gamma (x)$ for all $x>0$ .

Problem 18.18 from Billingsley P. "Probability and Measure" [Bil95]More problems in Section 18 "Product Measure and Fubini's Theorem"

(a)(b)(c)(d)

Euler's gamma function is defined for positive $t$ by $\Gamma (t)=\int_{0}^{\infty } x^{t-1} e^{-x} d x$ .

(a)

Prove that $\Gamma^{(k)}(t)=\int_{0}^{\infty } x^{t-1}(\log x)^{k} e^{-x} d x$ .

(b)

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

(c)

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

(d)

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

V_{k}=\frac{\pi ^{k / 2}}{\Gamma ((k / 2)+1)}. \tag {18.19}

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. Функции многих переменных. 2-е изд., ФИЗМАТЛИТ, 2003.

[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.

[Bil95] Billingsley P. Probability and Measure. 3rd ed., Wiley-Interscience, 1995.