Abstract
For proofs of the following results, see for example Chapters 12 and 13 of Whittaker and Watson (A Course of Modern Analysis. Cambridge University Press, Cambridge, 1996).
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For proofs of the following results, see for example Chapters 12 and 13 of [2].
For all \(z \in \mathbb {C}\) with \(\Re (z)>0\), we define
For all \(z\in {{\mathbb {C}}}\) ,
where γ = 0.5772156649… is the Euler constant.
Thus the function Γ is meromorphic on \({{\mathbb {C}}}\) , has no zero, and admits simple poles at z = 0, −1, −2, ….
FormalPara Proposition 4.3For \(z\in {{\mathbb {C}}}\setminus (-{{\mathbb {N}}})\) , we have
For \(n\in {{\mathbb {N}}}\) , we have Γ(n + 1) = n!.
FormalPara Proposition 4.5For \(z\in {{\mathbb {C}}}\setminus {{\mathbb {Z}}}\) ,
We have \(\varGamma ({{\textstyle \frac {1}{2}}}) = \sqrt {\pi }\).
FormalPara Proposition 4.7 (Legendre Duplication Formula)For \(z\in {{\mathbb {C}}}\setminus (-{{\textstyle \frac {1}{2}}}{{\mathbb {N}}})\) ,
For \(z\in {{\mathbb {C}}}\setminus (2{{\mathbb {Z}}})\) , we have
For fixed \(0\leqslant \vartheta < \pi \) and \(\left | arg(z)\right | < \vartheta \) , we have
For fixed \(0\leqslant \vartheta < \pi \) and \(\left | arg(z)\right | < \vartheta \) , we have
See [1, footnote, p.57].
References
A.E. Ingham, The Distribution of Prime Numbers. Cambridge Mathematical Library (Cambridge University Press, Cambridge, 1990); Reprint of the 1932 original, With a foreword by R. C. Vaughan
E.T. Whittaker, G.N. Watson, A Course of Modern Analysis. Cambridge Mathematical Library (Cambridge University Press, Cambridge, 1996); An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, Reprint of the fourth (1927) edition
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Rivat, J. (2018). Euler’s Gamma Function. In: Ferenczi, S., Kułaga-Przymus, J., Lemańczyk, M. (eds) Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics. Lecture Notes in Mathematics, vol 2213. Springer, Cham. https://doi.org/10.1007/978-3-319-74908-2_4
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