Gamma function in the complex plane

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Problem 3.16.4 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 the formula

\Gamma (\alpha ) = \sum _{k = 0}^{\infty } \frac{(-1)^{k}}{k !(k+\alpha )}+\int _{1}^{+\infty } x^{\alpha -1} e^{-x} d x, \quad \alpha > 0

Problem 2.143 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 4 "Singular Integrals. Weierstrass' Approximation Theorem"

The $\Gamma$ -function

\Gamma (s) = \lim _{n \to \infty } \frac{n^{s} n!}{s(s+1)\cdots (s+n)}

can be written as an integral (Problem 31). Use this fact to prove that $\Gamma (s)$ does not have any zeroes.

Remark.

$\Gamma (s+1) = s\Gamma (s)$ (Problem 142).

Problem 3.198 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 4 "Rouché's Theorem"

The entire function $\dfrac {1}{\Gamma (z)}$ assumes each value infinitely often in the half-strip

-d<\mathrm{Im}\left(z\right)<d, \quad \mathrm{Re}\left(z\right)<0 \quad (d \text{ arbitrary}).

Problem 9.18 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 9 "The Uniqueness Theorem. Regular Continuation."

Euler's gamma function $\Gamma (z)$ is defined for real $z > 0$ by the equality

\Gamma (z) = \int _{0}^{\infty } t^{z-1} e^{-t} d t

Prove that the function $\Gamma (z)$ can be regularly continued to the entire complex plane, with the exception of the points $z = 0,-1,-2, \ldots$ .

References

[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Функции многих переменных. 2-е изд., ФИЗМАТЛИТ, 2003.

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

[ШПК15] Шабунин М.И., Половинкин Е.С., Карлов М.И. Сборник задач по теории функций комплексного переменного. 4-е изд., БИНОМ. Лаборатория знаний, 2015.