Gamma and Beta Integrals

[Ted123]

Homework Statement

The Gamma and Beta integrals are defined respectively as

$$\Gamma (z) = \int^{\infty}_0 t^{z-1} e^{-t}\;dt$$
$$B(p,q) = \int^1_0 t^{p-1} (1-t)^{q-1}\;dt.$$
Determine for what values of the complex parameters z, p, q the integrals converge absolutely and explain why.

The Attempt at a Solution

How would I do this? Is the answer: $$\text{Re}(z) > 0,\;p,\,q>0\;?$$
If so, how do I show it and explain why?

 

[mighty2000]

I would first like to clarify that the Gamma and Beta integrals are not necessarily defined for complex parameters. They are typically defined for real parameters, but can be extended to complex parameters under certain conditions.

For the Gamma integral, it is well known that it converges absolutely for all real values of z greater than 0. This can be shown using the comparison test, as the integrand can be bounded by a convergent integral, such as \int^{\infty}_0 t^{x-1}\;dt. However, for complex values of z, the integral may not converge absolutely. In fact, it is only defined for values of z with positive real part, as you have correctly stated. This can be seen by considering the convergence of the integral for large values of t, where the exponential term dominates and the integrand behaves like t^{Re(z)-1}. For the integral to converge, this term must approach 0 as t approaches infinity, which is only possible if Re(z) is positive.

Similarly, for the Beta integral, it is well known that it converges absolutely for all real values of p and q greater than 0. This can also be shown using the comparison test, as the integrand can be bounded by a convergent integral, such as \int^1_0 t^{x-1}\;dt. However, for complex values of p and q, the integral may not converge absolutely. It is only defined for values of p and q with positive real parts, as can be seen by considering the convergence of the integral for large values of t, where the integrand behaves like t^{Re(p)-1}(1-t)^{Re(q)-1}. For the integral to converge, both of these terms must approach 0 as t approaches 1, which is only possible if Re(p) and Re(q) are positive.

In summary, the Gamma and Beta integrals converge absolutely for real values of z, p, and q greater than 0, and they are only defined for complex values of z, p, and q with positive real parts. This can be explained by considering the behavior of the integrands for large values of t and the necessary conditions for convergence.