Gamma function obtained by integration Calculator

Calculates "Gamma function Γ(a)" by the Double Exponential(DE) integration.

 $\normal \Gamma(a)={\large\int_{\tiny 0}^ {\hspace{25}\tiny \infty}}t^{a-1}e^{-t}dt,\hspace{20} Re(a)\gt 0\\$
 integrand: f(t,a) variable a interval( , )
 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt
 $\normal Gamma\ function\ \Gamma(a)\\[10](1)\ \Gamma(a)={\large\int_{\tiny 0}^ {\hspace{25}\tiny \infty}}t^{a-1}e^{-t}dt,\hspace{20} Re(a)\gt 0\\(2)\ \Gamma(a)={\large\frac{\Gamma(a+1)}{a}},\hspace{20} \Gamma(a)\Gamma(1-a)={\large\frac{\pi}{sin(\pi a)}}\\(3)\hspace{5}{\large\int_{\tiny 0}^{\hspace{25}\tiny\infty}}f(x)dx={\large\int_{\tiny -\infty}^{\hspace{25}\tiny\infty}}f(x(t))x'(t)dt\\\hspace{20}f(x)=x^{a-1}e^{-x}\\\hspace{20}x(t)={\large e^{t-e^{-t}}},\hspace{10}x'(t)=(1+{\large e^{-t}}){\large e^{t-e^{-t}}}\\$

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