Definition and properties of the Gamma function

See https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/14%3A_Analytic_Continuation_and_the_Gamma_Function/14.02%3A_Definition_and_properties_of_the_Gamma_function

余元公式（Euler’s reflection formula）

勒让德倍元公式（Legendre duplication formula）如何证明？

见知乎 https://www.zhihu.com/question/403116146