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When I solve the diffusion equation for a spherically symmetric geometry in spherical coordinates I obtain the following general solution (after application of the boundary conditions).

[tex] T(r,t) = \sum_{n=1}^{\infty}\, \frac{A_n}{r}\sin(\lambda_nr)\exp(-\alpha\lambda_n^2t)[/tex]

So to determine [tex]A_n[/tex] I apply the Initial condition [tex]T(r,0) = f(r)[/tex] which gives me

[tex] T(r,0) = f(r) = \sum_{n=1}^{\infty} \frac{A_n}{r} \sin(\lambda_nr)[/tex]

My problem is the [tex]1/r[/tex] term in the above equation. If this term wasn't there this is exactly the same as the half-range Fourier series.

However I have seen a document (please find it at the bottom) online which still writes A_n as

[tex] A_n = \frac{2}{b}\int_0^b\, f(r')\sin\left(\frac{n\phi{r'}}{b}\right)r'dr'[/tex]

this looks very much like the Fourier coefficient of a half-range Fourier series, however, notice the extra [tex]r[/tex] term in there, which is not in the standard Fourier coefficient of a half-range Fourier series. I don't understand how this particular form of Fourier series is obtained?

http://www.docstoc.com/docs/22184820/Heat-Conduction-in-Cylindrical-and-Spherical-Coordinates-I

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# Fourier series representation

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