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## Homework Statement

Given the definition of the spherical Bessel function,

[tex]j_{\ell}(\rho)=(-\rho)^{\ell} \left(\frac{1}{\rho}\frac{d}{d\rho}\right)\frac{Sin{\rho}}{\rho}[/tex]

Prove the recurrence relation:

[tex]j_{\ell+1}(\rho)=-j_{\ell}^{'}(\rho)+\frac{\ell}{\rho}j_{\ell}(\rho)[/tex]

## Homework Equations

[See

**a**]

## The Attempt at a Solution

The method to prove the recursion relation should be completed using proof by induction. This really comes down to the formalities involved in completing a proof of this nature: proof by induction implies that if the to-be-proved relation holds for one value (say, [itex]k[/itex]) then it may be induced that it holds for subsequent values ([itex]k+1[/itex]). My question: must I demonstrate that the relation holds for any

*arbitrary*[itex]k[/itex], or can I just pick one (say, 1) and then show that the relation holds for what you would expect from the original equation (for 1+1=2)?

Thanks

IHateMayonnaise

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