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I'm currently working through electrodynamics from Purcell supplemented by Jackson and online notes. I've read up the basic cases demonstrating the method of image charges, using the Green's function and its properties.

In the following notes (link below if you are interested, page 23), the lecturer first chooses to expand the Dirichlet Green’s function for a sphere of radius $a$ in terms of spherical orthonormal functions and represents it like a normal expansion with coefficients depending on the two vectors of position of observation and location of the unit point charge respectively, and two spherical angles as measured from the latter point.

So far, I got it. Then, taking into consideration the symmetry of the Green's function $G(r,r')$ which results from the very definition of said functions themselves as seen and verified previously for the case of the sphere, he splits the above-mentioned expansion coefficient into a new coefficient which is a function of the two positions and the complex conjugate of the spherical harmonics in terms of the two angles defining the original coefficient A. Then he goes on to define the Dirac delta and the completeness relation for spherical harmonics in spherical coordinates, solve the Poisson equation and the resulting azimuthal equation follows.

Here's my problem. How do we get the complex conjugate of the spherical harmonics from merely symmetry conditions during the second expansion? Has it to do with the fact that the harmonics themselves constitute a complex basis? I would be grateful for any help, for though this seems simple and obvious, I am too stuck to move on.

Here's the link: https://web.archive.org/web/20210507063944/https://courses.maths.ox.ac.uk/node/view_material/36494

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It is both the symmetry and the reality of the Green's function that implies this. For example, if I know $A\left(\theta\right)Y\left(\theta^{\prime}\right)$ is both symmetric and real, then \begin{align}A\left(\theta\right)Y\left(\theta^{\prime}\right) = A\left(\theta^{\prime}\right)Y\left(\theta\right) = A^{*}\left(\theta^{\prime}\right)Y^{*}\left(\theta\right) = Y^{*}\left(\theta\right)A^{*}\left(\theta^{\prime}\right)\end{align} Now since the two arguments may vary independently, it must be that $A\left(\theta\right)=Y^{*}\left(\theta\right)$. This argument can then be applied to each term in the expansion of the Green's function.

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