Evaluating integral in terms of Gamma functions

[arex]

Homework Statement

The Gamma function is given by $\Gamma$(x) = ₀∫^∞ t^x-1e^-t dt

Evaluate ₀∫^∞ exp(-αy²)dy in terms of Gamma functions.

Homework Equations

n/a

The Attempt at a Solution

Honestly, I don't know where to start. I have knowledge of parametric differentiation (we use alpha as the variable in class which may hint at its usage here) but I don't immediately see how it would help or what it even means to evaluate the integral "in terms of gamma functions."

I'm not just looking for the answer, I'd like a starting hint or tip to get going.

 

[Ray Vickson]

Homework Statement

The Gamma function is given by $\Gamma$(x) = ₀∫^∞ t^x-1e^-t dt

Evaluate ₀∫^∞ exp(-αy²)dy in terms of Gamma functions.

Homework Equations

n/a

The Attempt at a Solution

Honestly, I don't know where to start. I have knowledge of parametric differentiation (we use alpha as the variable in class which may hint at its usage here) but I don't immediately see how it would help or what it even means to evaluate the integral "in terms of gamma functions."

I'm not just looking for the answer, I'd like a starting hint or tip to get going.

You obviously need to find some alternative expression for the Gamma function that involves an integration with exp(-y^2) in it.

RGV

 

[Dick]

You obviously need to find some alternative expression for the Gamma function that involves an integration with exp(-y^2) in it.

RGV

I think arex can work with that definition. Try the change of variable x=ay^2.