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ans3 - Math 185 Sample Answers to Problem Set#3 Page 68 1c...

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Math 185. Sample Answers to Problem Set #3 Page 68 1c. Here u ( x, y ) = 2 x and v ( x, y ) = xy 2 . Then u x = 2 and v y = 2 xy , which are not equal unless xy = 1 . Also u y = 0 and - v x = - y 2 , which are unequal unless y = 0 . These two equations ( xy = 1 and y = 0 ) have no simultaneous solution, so f ( x ) is nowhere differentiable, by the theorem in Section 20. 4b. Here u ( r, θ ) = r cos θ/ 2 and v ( r, θ ) = r sin θ/ 2 . We have ru r = r 1 2 r cos θ/ 2 = r 2 cos θ/ 2 = v θ and rv r = r 1 2 r sin θ/ 2 = r 2 sin θ/ 2 = - u θ , so u and v satisfy the polar Cauchy-Riemann equations; hence f ( z ) is analytic on its domain. By (7), we then have f ( z ) = e - ( u r + iv r ) = e - 1 2 r cos θ/ 2 + i 1 2 r sin θ/ 2 = e iθ/ 2 2 re = 1 2 re iθ/ 2 . 6. We have u ( x, y ) = x 3 - 3 xy 2 x 2 + y 2 when z = 0 0 when z = 0 and v ( x, y ) = y 3 - 3 x 2 y x 2 + y 2 when z = 0 0 when z = 0 . We have u (0 , y ) = 0 for all y and v ( x, 0) = 0 for all x , so u y (0 , 0) = - v x (0 , 0) = 0 . Also u x (0 , 0) = lim h 0 h 3 h 2 +0 h = 1 and v y (0 , 0) = lim h 0 h 3 0+ h 2 h = 1 , so u x (0 , 0) = v y (0 , 0) . Therefore the Cauchy-Riemann equations are satisfied at the origin. 1
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2 8. First of all, we need an expression for v x . Solving equations (3) on page 66 for v x gives rv r cos θ - v θ sin θ = r ( v x cos θ + v y sin θ ) cos θ - ( - v x r sin θ + v y r cos θ ) sin θ = rv x (cos 2 θ + sin 2 θ ) = rv x , so v x = v r cos θ - v θ sin θ r .
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