Chem Differential Eq HW Solutions Fall 2011 162

Chem Differential Eq HW Solutions Fall 2011 162 - 162...

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Unformatted text preview: 162 Chapter 12 Greens Functions and Conformal Mappings 17. We map the region onto the upper half-plane using the mapping f ( z ) = z 2 (see Example 1). The transformed problem in the uv-plane is 2 U = 0 with boundary values on the u-axis given by U ( u, 0) = u if 0 < u < 1 and 0 otherwise. To solve the problem in the uv-plane, we apply the Poisson integral formula ((5), Section 7.5). We have U ( u, v ) = v - U ( s, 0) ( u- s ) 2 + v 2 ds = v 1 s ( u- s ) 2 + v 2 ds. Now use your calculus skills to compute this integral. We have v 1 s ( u- s ) 2 + v 2 ds = v 1 ( s- u ) ( s- u ) 2 + v 2 ds + v 1 u ( s- u ) 2 + v 2 ds = v 1 2 ln[( s- u ) 2 + v 2 ] 1 + 1- u- u u t 2 + v 2 dt ( s- u = t ) = v 2 ln (1- u ) 2 + v 2 u 2 + v 2 + u tan- 1 t v 1- u- u = v 2 ln (1- u ) 2 + v 2 u 2 + v 2 + u tan- 1 1- u v- tan- 1- u v = v 2 ln (1- u ) 2 + v 2 u 2 + v 2 + u tan- 1 1- u v + tan- 1 u v ....
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This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.

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