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One-definition-of-x-1-is-x-1-0-e-t-t-x-dx-According-to-WolframAlpha-another-definition-is-x-1-1-e-2ipix-1-L-e-t-t-x-dx-Can-someone-explian-to-me-where-this-comes-f




Question Number 10469 by FilupSmith last updated on 11/Feb/17
One definition of  Γ(x+1)  is:  Γ(x+1)=∫_0 ^( ∞) e^(−t) t^x dx     According to WolframAlpha, another  definition is:  Γ(x+1)=(1/(e^(2iπx) −1))∮_L e^(−t) t^x dx  Can someone explian to me where this  comes from and what it means.     Also, its been a long time since I learnt  contour integrals, so what does ∮_L  mean?
$$\mathrm{One}\:\mathrm{definition}\:\mathrm{of}\:\:\Gamma\left({x}+\mathrm{1}\right)\:\:\mathrm{is}: \\ $$$$\Gamma\left({x}+\mathrm{1}\right)=\int_{\mathrm{0}} ^{\:\infty} {e}^{−{t}} {t}^{{x}} {dx} \\ $$$$\: \\ $$$$\mathrm{According}\:\mathrm{to}\:\mathrm{WolframAlpha},\:\mathrm{another} \\ $$$$\mathrm{definition}\:\mathrm{is}: \\ $$$$\Gamma\left({x}+\mathrm{1}\right)=\frac{\mathrm{1}}{{e}^{\mathrm{2}{i}\pi{x}} −\mathrm{1}}\oint_{{L}} {e}^{−{t}} {t}^{{x}} {dx} \\ $$$$\mathrm{Can}\:\mathrm{someone}\:\mathrm{explian}\:\mathrm{to}\:\mathrm{me}\:\mathrm{where}\:\mathrm{this} \\ $$$$\mathrm{comes}\:\mathrm{from}\:\mathrm{and}\:\mathrm{what}\:\mathrm{it}\:\mathrm{means}. \\ $$$$\: \\ $$$$\mathrm{Also},\:\mathrm{its}\:\mathrm{been}\:\mathrm{a}\:\mathrm{long}\:\mathrm{time}\:\mathrm{since}\:\mathrm{I}\:\mathrm{learnt} \\ $$$$\mathrm{contour}\:\mathrm{integrals},\:\mathrm{so}\:\mathrm{what}\:\mathrm{does}\:\oint_{{L}} \:\mathrm{mean}? \\ $$

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