Monday, September 25, 2017

Proving Gamma function relation




Prompt.
Using integration by parts, show that the gamma function Γ(t)=0xt1exdx satisfies the relation tΓ(t)=Γ(t+1) for t>0.



My solution. Let u=ex, du=exdx, v=1txt , dv=xt1dx. Then after applying integration by parts, we get Γ(t)=1txtex+1t0xtexdx and subsequently tΓ(t)=xtex+0xtexdx . Now, Γ(t+1)=0xtexdx.



We can rewrite tΓ(t)=Γ(t)+Γ(t+1) .



Am I doing something wrong? Am I on the right track?


Answer



I'm sorry to say, but those are the wrong substitutiongs. By integration by parts, setu=xtdu=txt1dxAnd therefore, we haveΓ(z+1)=0ettzdt=tzet|0+0zettz1dtThe first term evaluates to zero. You can see this by taking the limit, and then using L'Hopital's. Therefore, we're left withΓ(z+1)=z0ettz1dt=zΓ(z)



No comments:

Post a Comment

analysis - Injection, making bijection

I have injection f:AB and I want to get bijection. Can I just resting codomain to f(A)? I know that every function i...