Power Series from MTW 19.1: Explained?

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The forum discussion centers on the equations 19.3b and 19.3c from Chapter 19 of "Mathematical Methods for Physicists" (MTW). The user seeks clarification on these equations, which are related to the concept of multipole expansion. Equation 19.3a is recognized as a familiar formula, while the other two equations require further explanation. The discussion emphasizes the importance of understanding these equations in the context of multipole expansion in physics.

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zn5252
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hello
Please see attachment which is a snapshot from MTW first page of chapter 19. Can someone please elaborate on how the equations 19.3b and c can be explained ?
I know that equation 19.3a is a familiar formula but not so much the other two. I'm just confused.
Thank you,
Long live & clear skies.
 

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Look up "Multipole expansion"
 

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