Proving Absolute Convergence of Gamma and Beta Integrals in Complex Analysis

[Ted123]

Homework Statement

Let [tex]z,p,q \in \mathbb{C}[/tex] be complex parameters.

Determine that the Gamma and Beta integrals:
[tex]\displaystyle \Gamma (z) = \int_0^{\infty} t^{z-1} e^{-t}\;dt[/tex]
[tex]\displaystyle B(p,q) = \int^1_0 t^{p-1} (1-t)^{q-1}\;dt[/tex]
converge absolutely for [tex]\text{Re}(z)>0[/tex] and [tex]p,q>0[/tex] respectively and explain why they do.

The Attempt at a Solution

How do I show that they converge absolutely and why do they?

 

[lanedance]

how about using a convergence technique such as a comparison method

 

[Ted123]

how about using a convergence technique such as a comparison method

I'm aquainted with such techniques for series but not integrals...

 

[lanedance]

well what's your definition of absolute convergence for an integral?

 

[Ted123]

well what's your definition of absolute convergence for an integral?

[tex]\int_A f(x)\;dx[/tex] where f(x) is a real or complex-valued function, converges absolutely if [tex]\int_A |f(x)|\;dx<\infty[/tex] where [tex]A=[a,b][/tex] is a closed bounded interval.

 

[lanedance]

Ok so for the first one, can you convince yourself that the integral over t from 1 to infinity converges?

That leaves you with the portion from 0 to 1 to prove. For that, take the absolute value.

The comparison test says
[tex] |g(x)|>|f(x)| \ \forall x \in I[/tex]

[tex] \implies \int_I|g(x)|>\int_I |f(x)| [/tex]

hence if the integral over |g| converges, so does the integral over |f|

you should be able to use this for both portions if need be

 

[lanedance]

for the 2nd the issue is the possibility each blows up too quickly at the boundaries, so I would again separate into 2 and consider each endpoint separately

 

[lanedance]

also note both these are actually improper integrals, so to be totally rigorous you should actually be considering the limit as the end point tends to the poorly defined region

http://en.wikipedia.org/wiki/Improper_integral