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Chem Differential Eq HW Solutions Fall 2011 154

# Chem Differential Eq HW Solutions Fall 2011 154 - radius...

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154 Chapter 12 Green’s Functions and Conformal Mappings Solutions to Exercises 12.4 1. The function u ( r,θ ) = 1 is harmonic in the unit disk and has boundary values f ( θ ) = 1. Use (10) with f ( θ ) = 1 and you get, for all 0 r< 1, 1 = 1 2 π 2 π 0 1 - r 2 1 + r 2 - 2 r cos( θ - φ ) dθ. 5. For n = 1 , 2 ,... , the function u ( r,θ ) = r n cos is harmonic in the disk of
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Unformatted text preview: radius R > 0 and has boundary values (when r = R ) f ( θ ) = R n cos nθ . Use (10) with f ( θ ) = R n cos nθ and you get, for all 0 ≤ r < R , r n cos nθ = R n ( R 2-r 2 ) 2 π ± 2 π cos nθ R 2 + r 2-2 rR cos( θ-φ ) dθ....
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