Yes, it’s Tuesday. Busy weekend, including a Relay for Life (you should find your local one and donate!). Today we’ll get back on schedule a bit and make this one a rather more utilitarian than usual Sunday Function, playing into the last step we’ll need in order to complete the Bose-Einstein condensation discussion.

In the course of our journey we will meet this somewhat alarming creature:

It’s a function that takes two parameters: a more traditional variable z, and a number (lowercase Greek letter nu) denoting the particular subspecies of function g. The capital Greek letter gamma is itself a function.

We’re only going to have to deal with this for nu = 3/2, so we may as well plot the stuff under the integral sign. Taking z = 1/2 just for the random heck of it:

As z changes that curve will change shape and correspondingly the area under the curve will change as well. Is there some z that will maximize that area? Sometimes you can save yourself trouble if you just plot the graph, and you can save the rigorous proof work for later. We’ll do that. This is the actual graph of g(z):

It attains a maximum at z = 1, and we shall see that in our case the physical meaning of z requires that it not be able to go above 1 anyway.

Mystery upon mystery at the moment, I know. Unless you’ve done this before, of course. But we’ll put it all together shortly and I think it’ll be worth the wait.