evaluate-csc-pis-ipi-C-t-s-1-e-t-dt-path-C-1-s-2pii-C-t-s-1-e-t-1-dt-path-C-

Tinku Tara May 4, 2025 None 0 Comments

Question Number 220066 by SdC355 last updated on 04/May/25

evaluate −((csc(πs))/(iπ))∫_( C) (−t)^(s−1) e^(−t) dt , path C;(−∞,∞) −((𝚪(1−s))/(2πi)) ∫_( C) (((−t)^(s−1) )/(e^t −1)) dt , path C;(−∞,∞)

$$\mathrm{evaluate} \\ $$$$−\frac{\mathrm{csc}\left(\pi{s}\right)}{\boldsymbol{{i}}\pi}\int_{\:\boldsymbol{\mathcal{C}}} \:\left(−{t}\right)^{{s}−\mathrm{1}} {e}^{−{t}} \:\mathrm{d}{t}\:,\:\mathrm{path}\:\boldsymbol{\mathcal{C}};\left(−\infty,\infty\right) \\ $$$$−\frac{\boldsymbol{\Gamma}\left(\mathrm{1}−{s}\right)}{\mathrm{2}\pi\boldsymbol{{i}}}\:\int_{\:\boldsymbol{\mathcal{C}}} \:\frac{\left(−{t}\right)^{{s}−\mathrm{1}} }{{e}^{{t}} −\mathrm{1}}\:\mathrm{d}{t}\:,\:\mathrm{path}\:\boldsymbol{\mathcal{C}};\left(−\infty,\infty\right) \\ $$

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