# Absolute Convergence

1. Jun 27, 2011

### Ted123

1. The problem statement, all variables and given/known data

Let $$z,p,q \in \mathbb{C}$$ be complex parameters.

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

3. The attempt at a solution

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

2. Jun 27, 2011

### lanedance

how about using a convergence technique such as a comparison method

3. Jun 28, 2011

### Ted123

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

4. Jun 28, 2011

### lanedance

well whats your definition of absolute convergence for an integral?

5. Jun 28, 2011

### Ted123

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

Last edited: Jun 28, 2011
6. Jun 28, 2011

### 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
$$|g(x)|>|f(x)| \ \forall x \in I$$

$$\implies \int_I|g(x)|>\int_I |f(x)|$$

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

7. Jun 28, 2011

### 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

8. Jun 28, 2011