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36 ) n = 1 the define then we can ask for the best choice of coefficients a , so that
we get the " best " representation of the function ( 9 ) . If " best " is defined as
minimizing the mean square error My : My – L \ 50 – 3 a , U _ c8 ) * a ( 2 . 37 ) n =
1 it ...
34 ) This is just a Legendre series of the form ( 3 . 23 ) , so that the coefficients A ,
are : P2 ( cos O ) sin 0 do ( 3 . 35 ) 2a ' JO A - 21 + 1 If , for example , V ( O ) is that
of Section 2 . 8 , with two hemispheres at equal and opposite potentials , .
The coefficients Aim in ( 16 . 35 ) are not completely arbitrary . The divergence
condition V . B = 0 must be satisfied . Since the radial functions are linearly
independent , the condition V . B = 0 must hold for the two sets of terms in ( 16 .
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Introduction to Electrostatics
References and suggested reading
Multipoles Electrostatics of Macroscopic Media
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