Homework Help: Hypersphere Volume - Fractional Dimensions

1. Feb 14, 2010

Old Guy

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

2. Relevant equations

3. 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?

2. Feb 14, 2010

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.

3. Feb 14, 2010

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?

4. Feb 14, 2010

jdwood983

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

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