Euler-Gauss limit and Weierstrass product for the gamma function

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Problem 3105 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 5.9 "Infinite Products"

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

According to Euler's definition, the gamma function $\Gamma (x)$ is determined by the following formula:

\Gamma (x)=\lim _{n \rightarrow \infty } \frac{n! \, n^{x}}{x(x+1)\cdots (x+n)}.

Starting from this formula:

(a)

Express the function $\Gamma (x)$ in the form of an infinite product;

(b)

Prove that $\Gamma (x)$ is meaningful for all real $x$ that are not negative integers;

(c)

Deduce the following property

\Gamma (x+1)=x\Gamma (x);

(d)

Find the value of $\Gamma (n)$ for positive integer $n$ .

Problem 5533 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 14.5 "Expansion of Functions into Partial Fractions"

Prove the identity:

\prod _{n=1}^{\infty } \frac{n(a+b+n)}{(a+n)(b+n)} = \frac{\Gamma (a+1)\Gamma (b+1)}{\Gamma (a+b+1)}.

Problem 2.24.48 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"

Determine for which values of $x$ the infinite products converge. $\prod_{n = 1}^{\infty }\left(1+\frac{x}{n}\right) e^{-x / n}$ .

Problem 2.24.70 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"

(1)(2)(3)

Prove that:

(1)

the infinite product $\Gamma (x) = \frac{1}{x} \prod_{n = 1}^{\infty } \frac{(1+1 / n)^{x}}{1+x / n}$ absolutely converges for all $x \neq m, m = 0,-1,-2, \ldots$ ;

(2)

$\Gamma (x) = \lim_{n \rightarrow \infty } \frac{n ! n^{x}}{x(x+1)(x+2) \ldots (x+n)}$ ;

(3)

$\Gamma (x+1) = x \Gamma (x)$ (Euler).

Problem 2.24.71 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 $\frac{1}{\Gamma (x+1)} = e^{C x} \prod_{n = 1}^{\infty }\left(1+\frac{x}{n}\right) e^{-x / n}$ , where $C$ is Euler's constant (Weierstrass).

Problem II.1.12 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section II.1 "Computing Limits"

Prove that

\prod _{n \geqslant 0}\left(1-\frac{a^{2}}{(n+b)^{2}}\right)=\frac{\Gamma ^{2}(b)}{\Gamma (b+a) \Gamma (b-a)}

$(a, b \in \mathbb {R}, b \neq 0, b \pm a \neq 0,-1,-2,-3, \ldots )$ . Deduce from this that the fraction $\frac{\Gamma \left(x+\frac{1}{2}\right)}{\sqrt{x} \Gamma (x)}$ is increasing on the interval (0, $+\infty$ ).

Problem 2.31 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 4 "Improper Integrals Between Infinite Limits"

The $\Gamma$ -function is defined for $\alpha > 0$ (or $\Re \alpha > 0$ ) by the integral

\Gamma (\alpha ) = \int _{0}^{\infty } e^{-x} x^{\alpha -1} \, dx .

Using Problem 89 of Part I prove that

\Gamma (\alpha ) = \lim _{n \to \infty } \frac{n^{\alpha -1} n!}{\alpha (\alpha +1) \cdots (\alpha +n-1)}, \qquad \alpha > 0 .

References

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.

[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Интегралы. Ряды. 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.

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.