Skip to content
Related Articles
Open in App
Not now

Related Articles

Gamma Function

Improve Article
Save Article
  • Difficulty Level : Medium
  • Last Updated : 16 Jun, 2020
Improve Article
Save Article

Gamma function is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers.

Gamma function denoted by \Gamma\left (p \right) is defined as:
\Gamma\left(p \right) = \int_{0}^{\infty}e^{-t} t^{p-1} dt where p>0.
Gamma function is also known as Euler’s integral of second kind.
Integrating Gamma function by parts we get,
\Gamma\left (p+1 \right) = \int_{0}^{\infty}e^{-t} t^{p} dt
=-e^{-t} t^p \Biggr |_{0}^{\infty}+p\int_{0}^{\infty}e^{-t} t^{p-1} dt
=0+p\Gamma\left (p \right)
Thus \Gamma\left (p+1 \right) = p\Gamma\left (p \right)

Some standard results:

  1. \Gamma\left (1/2 \right) = \sqrt \pi
    We know that \Gamma\left(1/2 \right) = \int_{0}^{\infty}t^{-\frac{1}{2}}e^{-t} dt
    Put t=u^2
    Thus \Gamma\left(1/2 \right) = 2\int_{0}^{\infty}e^{{-u^2}}du
    \Gamma\left(1/2 \right) .\Gamma\left(p \right) = (2\int_{0}^{\infty}e^{{-u^2}}du)(2\int_{0}^{\infty}e^{{-u^2}}du)
    =4\int_{0}^{\infty} \int_{0}^{\infty}e^{-{u^2 + v^2}} du dv
    Now changing to polar coordinates by using u = r cosθ and v = r sinθ
    Thus {\Gamma\left(1/2 \right)}^2 = 4\int_{\theta=0}^{\pi/2}\int_{r=0}^{\infty}e^{-{r^2}} dr d\theta
    =4\int_{0}^{\pi/2} -\frac{1}{2}e^{-r^2}\Biggr|_{r=0}^{\infty}
    =2\int_{0}^{\pi/2}d\theta = 2.\theta \Biggr|_{0}^{\pi/2}=\pi
    Hence \Gamma\left (1/2 \right) = \sqrt \pi


  2. \Gamma\left(n+1 \right) = (m+1)^{n+1}(-1)^n \int_{0}^{1}x^m (ln x)^n dx
    Where n is a positive integer and m>-1
    Put x=e^-y such that dx=-e-ydy=-x dy
    \int_{0}^{1}x^m(ln x)^n dx= \int_{0}^{\infty}e^{-my} . (-y)^n e^{-y} dy
    (-1)^n \int_{0}^{\infty} y^n . e^{-(m+1)y} dy
    Put (m+1)y = u
    =(-1)^n \int_{0}^{\infty}\frac{u^n}{(m+1)^n}.e^{-u} .\frac{du}{m+1}
    =\frac{(-1)^n}{(m+1)^n+1}\int_{0}^{\infty}e^{-u} .u^n du = \frac{(-1)^n}{(m+1)^{n+1}}.\Gamma\left(n+1\right)



Example-1:
Compute \Gamma\left(4.5\right).

Explanation :
Using \Gamma\left(p+1\right)=p\Gamma\left(p\right)
\Gamma\left(4.5\right)=\Gamma\left(3.5+1 \right)=3.5\Gamma\left(3.5\right)
=(3.5)(2.5)\Gamma\left(2.5\right)
=(3.5)(2.5)(1.5)\Gamma\left(1.5\right)
=(3.5)(2.5)(1.5)(0.5)\Gamma\left(0.5\right)
We know \Gamma\left(0.5\right)=\sqrt\pi
Thus \Gamma\left(4.5\right)=6.5625\sqrt\pi



Example-2:
Evaluate I=\int_{0}^{\infty}x^4 e^-{x^4} dx

Explanation :
Put x4 = t, 4x3dx = dt, dx = ¼ t-3/4 dt
I=\int_{0}^{\infty}t.e^{-t} \frac{t^{-3/4}}{4}dt
= \frac{1}{4}\int_{0}^{\infty}e^{-t} t^{3/4} dt
= \frac{1}{4}\Gamma\left(1+\frac{1}{4}\right)
= \frac{1}{4}\Gamma\left(\frac{5}{4}\right)


My Personal Notes arrow_drop_up
Related Articles
1. Gamma Distribution Model in Mathematics
2. Counting Boolean function with some variables
3. Beta Function
4. Derivative of the Sigmoid Function
5. Euler Graph and Arbitrarily Traceable Graphs in Graph Theory
6. What is Link-Cut Tree?
7. What is Chordal Graphs?
8. Galois Fields and Its Properties
9. Plotting Functions in Gnuplot
10. Multiplot Command in Gnuplot
Previous
Beta Function
Next
Volume of solid of revolution
Article Contributed By :
https://media.geeksforgeeks.org/auth/avatar.png
mohitg593
@mohitg593
Vote for difficulty
Current difficulty : Medium
Article Tags :
Report Issue
We use cookies to ensure you have the best browsing experience on our website. By using our site, you acknowledge that you have read and understood our Cookie Policy & Privacy Policy
Lightbox

Start Your Coding Journey Now!