
The spherical harmonics

Answer to Problem 4.7P
The spherical harmonics
Explanation of Solution
Write the general expression for the spherical harmonics
Here,
Write the expression for the associated Legendre function.
Here,
Write the expression for the Legendre function.
Use equation (III) in (II) to solve for
The
Then the equation (IV) becomes,
The expansion of
Use equation (VI) in (V) and it becomes,
Use equation (VII) in (I) to solve for the spherical harmonics
Use equation (VIII) to find
Consider
Differentiate equation (X) with respect to
Take the second derivative of
The sum of the equations (XII) and (XI) can be written as
The equation (XIII) satisfies with the equation given in (4.18).
Consider
Take the partial derivative of equation (XIV) with respect to
Use equation (XV) in (XIII) and it can be written as
Take the second partial derivative of
Use equation (XVII) and (XVI) in (XIII) and compare,
In equation (XVIII), where
Conclusion:
Therefore, The spherical harmonics
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