by-using-residus-theorem-prove-that-0-t-a-1-1-t-dt-pi-sin-pia-with-0-lt-a-lt-1-

Tinku Tara June 4, 2023 Integration 0 Comments

Question Number 33069 by abdo imad last updated on 09/Apr/18

by using residus theorem prove that ∫_0 ^∞ (t^(a−1) /(1+t)) dt = (π/(sin(πa))) with 0<a<1 .

$${by}\:\:{using}\:{residus}\:{theorem}\:{prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}^{{a}−\mathrm{1}} }{\mathrm{1}+{t}}\:{dt}\:=\:\frac{\pi}{{sin}\left(\pi{a}\right)}\:{with}\:\:\mathrm{0}<{a}<\mathrm{1}\:. \\ $$

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by-using-residus-theorem-prove-that-0-t-a-1-1-t-dt-pi-sin-pia-with-0-lt-a-lt-1-

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