How to prove that Γ(z+1)=zΓ(z)?

  • Thread starter Aesops
  • Start date
  • #1
2
0

Homework Statement



I'm not sure how to prove that the Gamma function (the extension of the factorial) has the property that Γ(z+1) = zΓ(z).

Homework Equations



[itex]Γ(z)=\int_0^\infty t^{z-1} e^{-t}\,{\rm d}t[/itex]

The Attempt at a Solution



Looking online, I should try integration by parts, but I'm still unsure as to how to attack the problem.
 

Answers and Replies

  • #2
Curious3141
Homework Helper
2,850
87
Integrate the integrand in the expression for [itex]\Gamma(z)[/itex] using integration by parts ([itex]\int udv = uv - \int vdu[/itex]). Use [itex]u = e^{-t}[/itex] and [itex]dv = t^{z-1}[/itex]. Remember that [itex]t[/itex] is the variable of integration and [itex]z[/itex] is a constant.

After getting the expression, apply the bounds. The [itex]uv[/itex] term is bounded by zero and infinity, and the upper bound can be resolved with L' Hopital's Rule (the limit is zero). The lower bound is also zero, so this term vanishes. You're left with an expression for the [itex]\int vdu[/itex] term that's equivalent to [itex]\frac{1}{z}\Gamma(z+1)[/itex]. Now just do the simple algebra to finish off the proof.

EDIT: Can't believe I used zeta in place of gamma!
 
Last edited:

Related Threads on How to prove that Γ(z+1)=zΓ(z)?

  • Last Post
Replies
14
Views
1K
Replies
1
Views
1K
  • Last Post
Replies
10
Views
6K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
11
Views
2K
  • Last Post
Replies
4
Views
3K
Replies
1
Views
2K
  • Last Post
Replies
12
Views
851
Replies
5
Views
9K
Top