Explicit Cartesian expressions for the real solid harmonics

4 EXPLICIT CARTESIAN EXPRESSIONS FOR THE REAL SOLID HARMONICS 

The term [j + sgn(wi) iy]l l can be separated into real and imaginary parts as 

To highlight the symmetry between the real and imaginary parts, it is convenient to use i + 1/2 as the summation index in the imaginary part  

Substituting the binomial expansions (6.4.43), (6.4.45) and (6.4.46) in the real and imaginary parts of the complex solid harmonics (6.4.14), we arrive at the following explicit expression for the real 

Note that all terms in the solid harmonics of order / in (6.4.47) are of degree / exactly. The reader may wish to verify that (6.4.47) reproduces the real solid harmonics in Table 6.3. 

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