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Uz find v the conjugate harmonic functions of u a u x y x y...

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Uz find v the conjugate harmonic functions of u a u x y x y c u x y sinh r cos y b u x y x 3ry d u x y 1 y Solution a Use Cauchy Riemanns equations page 83 b Use Cauchy Riemanns equations page 83 u x y x y du dx dy dv dx dy v x y x y d Use u x y x 3 xy du 3 r 3y dx 6xy dy dv 6xy dx 3x 3y dy v x y 3x y y c Use Cauchy Riemanns equations page 83 u r y sinh r cosy du cosh x cos y dx sinhr sin y dy du sinh r sin y dx cosh x cos y dy v x y cosh r sin y Cauchy Riemanns equations page 83 u x y x y du 2 y dx 1 y y 2 dy x y 2xy dx 2 y au x y 2 dy

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