Chem Differential Eq HW Solutions Fall 2011 154

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

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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 &gt; 0 and has boundary values (when r = R ) f ( ) = R n cos n . Use (10) with f ( ) = R n cos n and you get, for all 0 r &lt; R , r n cos n = R n ( R 2-r 2 ) 2 2 cos n R 2 + r 2-2 rR cos( - ) d....
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