Mathematical Methods in the Physical Sciences · Problem 5.11

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · radial-ode-powers

Problem

The following two R(r)R(r) equations arise in various separation of variables problems in polar, cylindrical, or spherical coordinates: rddr(rdRdr)=n2R,r\frac{d}{dr}\left(r\frac{dR}{dr}\right) = n^2 R, ddr(r2dRdr)=l(l+1)R.\frac{d}{dr}\left(r^2\frac{dR}{dr}\right) = l(l+1)R. There are various ways of solving them: They are a standard kind of equation (often called Euler or Cauchy equations---see Chapter 8, Section 7d); you could use power series methods; given the fact that the solutions are just powers of rr, it is easy to find the powers. Choose any method you like, and solve the two equations for future reference. Consider the case n=0n = 0 separately. Is this necessary for l=0l = 0?

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