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joshmccraney

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I was wondering if anyone could shed some light on my understanding of arriving at the coefficients of Bessel Equations? Namely, why do we use the indicial equation to determine coefficients?

As an example, if we have to solve $$s^2 \alpha'' + 2 s \alpha ' - \frac{1}{4} \gamma^2 s^2 \alpha = 0$$ we gues a solution as $$\alpha(s) = \sum_{k=0} a_k s^{n+k}$$

Plugging this guess into the above ode yields $$\sum_{k=0} a_k (n+k)(n+k+1) s^{n+k} - \frac{1}{4} \gamma^2 \sum_{k=2} a_{k-2} s^{n+k} = 0$$

Now for solving this, I know we let ##k=0 \implies n(n+1) = 0## but what about when ##k=1##? Can someone help me know why we look only at the indicial equation? I can provide more work if you need it.

Thanks