How to find the coefficients of this expansion?

[LCSphysicist]

Homework Statement

z.

Relevant Equations

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I have solved a differential equation whose solutions is $$u = B + \sum_{n=1} C_{n }e^{-\lambda_{n}² q² t} J_{0}(\lambda_{n}r)$$

Where $(\lambda_{n}r)$ is such that $J_{0}'(\lambda_{n}a) = 0$. So i should now try to satisfy the condition that, at t=0, u = $f(r)$.

The problem is that i don't know what is the orthogonality here. If $\lambda_{n}$ were such that $J_{0}(\lambda_{n}a)=0$, i would use the normal orthogonaltiy generally used, namely $\int_{0}^{a} r J_{0}(\lambda_{n}r)J_{0}(\lambda_{q}r)$

But this doesn't work here, since $\lambda_{q}a$ is not a zero of J0, but it is a zero in fact of its derivative.

Not just it, what orthogornality i would use to find the B?

 

[Twigg]

Believe it or not, orthogonality, exactly the way you wrote it, is already guaranteed. My source is Jackson Classical Electrodynamics 3rd Edition, bottom of page 115:

But it will be noted that an alternative expansion is possible in a series of functions $\sqrt{\rho} J_\nu (y_{\nu n}\rho / a)$ where $y_{\nu n}$ is the $n$th root of the equation $[dJ_\nu(x)]/dx = 0$. The reason is that, in proving the orthogonality of functions, all that is demanded is that thte quantity $[\rho J_\nu (k\rho) (d/d\rho) J_\nu (k' \rho) - \rho J_\nu (k' \rho) (d/d\rho) J_\nu (k\rho)]$ vanish at the endpoints $\rho = 0$ and $\rho = a$.