Hypersphere Volume - Fractional Dimensions

[Old Guy]

Homework Statement

I've completed the derivation of the hypersphere volume for integer dimensions, and my solution matches what's on Wikipedia. How can I generalize it to fractional dimensions?

Homework Equations

The Attempt at a Solution

Not a clue; my only guess at this point is that the multidimensional "radius", which is a sum for integer dimensions, certainly becomes an integral, but how?

 

[jdwood983]

Wikipedia's entry on n-sphere's says that you can use the same formula because it is a continuous function up until $n\sim5.26$. Beyond this value, I cannot say what the generalization would be.


Hope this helps.

 

[Old Guy]

Well, I read that, but I don't really see what makes it so. I (sort of) followed the argument about why that produces the maximum volume, but what happens there? If it's some kind of discontinuity, what causes it?

 

[jdwood983]

I don't know what causes the discontinuity, perhaps you can plot the volume versus n and see what happens around then?