Prompt.
Using integration by parts, show that the gamma function Γ(t)=∫∞0xt−1e−xdx satisfies the relation tΓ(t)=Γ(t+1) for t>0.
My solution. Let u=e−x, du=−e−xdx, v=1txt , dv=xt−1dx. Then after applying integration by parts, we get Γ(t)=1txte−x+1t∫∞0xte−xdx and subsequently tΓ(t)=xte−x+∫∞0xte−xdx . Now, Γ(t+1)=∫∞0xte−xdx.
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=txt−1dxAnd therefore, we haveΓ(z+1)=∞∫0e−ttzdt=−tze−t|∞0+∞∫0ze−ttz−1dtThe 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)=z∞∫0e−ttz−1dt=zΓ(z)
No comments:
Post a Comment