Gamma function to be calculated by integration
The gamma function Γ(a) is computed using a double exponential (DE) numerical integration.
| \(\normalsize \Gamma(a)={\large\int_{\tiny 0}^ {\tiny \infty}}t^{a-1}e^{-t}dt,\hspace{20px} Re(a)\gt 0\\\) |
| \(\normalsize Gamma\ function\ \Gamma(a)\\ (1)\ \Gamma(a)={\large\int_{\tiny 0}^ {\tiny \infty}}t^{a-1}e^{-t}dt,\hspace{20px} Re(a)\gt 0\\ (2)\ \Gamma(a)={\large\frac{\Gamma(a+1)}{a}},\hspace{20px} \Gamma(a)\Gamma(1-a)={\large\frac{\pi}{\sin(\pi a)}}\\ (3)\hspace{5px}{\large\int_{\tiny 0}^{\tiny\infty}}f(x)dx={\large\int_{\tiny -\infty}^{\tiny\infty}}f(x(t))x'(t)dt\\ \hspace{20px}f(x)=x^{a-1}e^{-x}\\ \hspace{20px}x(t)={\large e^{t-e^{-t}}},\hspace{10px}x'(t)=(1+{\large e^{-t}}){\large e^{t-e^{-t}}}\\ \) |
No customer feedback for this calculation yet.