I Relation Between Beta and Gamma functions

Click For Summary
The discussion centers on understanding a specific proof involving the relationship between the Beta and Gamma functions. The key point is the integral expression, which states that the integral from 0 to infinity of e^{-x(1+y)} multiplied by x raised to the power of (m+n-1) equals the Gamma function divided by (1+y) raised to the power of (m+n). Participants are seeking clarification on how to apply the relevant formulas to derive this result. The conversation emphasizes the need for a step-by-step explanation of the proof to grasp the underlying concepts. Understanding this relationship is crucial for further studies in advanced mathematics.
PLAGUE
Messages
35
Reaction score
2
TL;DR
Proof of Beta Gamma function relation
Screenshot 2024-01-02 173019.png

So, my teacher showed me this proof and unfortunately it is vacation now. I don't understand what just happened in the marked line. Can someone please explain?
 
Physics news on Phys.org
The line asserts that <br /> \int_0^\infty e^{-x(1+ y)}x^{m+n-1}\,dx = \frac{\Gamma(m + n)}{(1 + y)^{m + n}}. How would you apply the given formulae to arrive at that?
 

Similar threads

  • · Replies 6 ·
Replies
6
Views
3K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 200 ·
7
Replies
200
Views
28K
Replies
12
Views
2K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 9 ·
Replies
9
Views
4K
  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 3 ·
Replies
3
Views
4K
  • · Replies 1 ·
Replies
1
Views
3K
  • · Replies 1 ·
Replies
1
Views
2K