Quiz2solns-1 - 2 Since dh dy = 3 y 2 , then h ( y ) = y 3 +...

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Quiz 2 Solutions Math 322 – Fall, 2009. September, 4 2009. NAME: Instructor: Bole Yang, Erica McEvoy Please show ALL of your work. 1. Consider the function f ( z )= z 2 +5 . Use the Cauchy-Riemann equations to check if f ( z ) is an analytic function. For z = x + iy , we have f ( z )=( x + iy ) 2 + 5 = ( x 2 - y 2 + 5) + (2 xy ) i = u ( x, y )+ iv ( x, y ) . Computing partial derivatives, we get that ∂u ∂x =2 x , ∂u ∂y = - 2 y , ∂v ∂x =2 x , and ∂v ∂y =2 y . Since ∂u ∂x = ∂v ∂y and ∂u ∂y = - ∂v ∂x , then f ( z ) is analytic. 2. Consider the function u ( x, y )=3 y 2 x - x 3 . Is u ( x, y ) a harmonic function? If yes, ±nd its harmonic conjugate. Yes, u ( x, y ) is a harmonic function, since it satis±es Laplace’s equation: 2 u ∂x 2 + 2 u ∂y 2 = - 6 x +6 x =0 . The harmonic conjugate, v ( x, y ) , can be found from using Cauchy’s Equations. Since ∂u ∂x =3 y 2 - 3 x 2 and ∂u ∂y =6 xy , then ∂v ∂y =3 y 2 - 3 x 2 (1) ∂v ∂x = - 6 xy (2) Integrating both sides of equation [2], gives v ( x, y )= - 3 x 2 y + h ( y ) . Taking this expression for v , differen- tiate (with respect to y)and plug into equation [1] to solve for h ( y ) : - 3 x 2 + dh dy =3 y 2 - 3 x 2 (3)
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MATH 322. QUIZ 1 ————————————————————————————
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Unformatted text preview: 2 Since dh dy = 3 y 2 , then h ( y ) = y 3 + c . Now we have the harmonic conjugate, v ( x, y ) =-3 x 2 y + y 3 + c (where c is just a constant). For fun, we can write down an expression for the corresponding analytic function, f ( z ) : f ( z ) = u + iv = 3 xy 2-x 3-i (3 x 2 y ) + i ( y 3 ) + c , which can be condensed as f ( z ) =-z 3 + c for z = x + iy . (Hint on condensing: since our expression for f ( z ) has terms that go like x 3 and y 3 , a natural place to start is to compare our expression to that of z 3 . If you factor out z 3 carefully, youll get z 3 = z 2 * z = (( x 2-y 2 ) + i (2 xy )) * ( x + iy ) = x 3-xy 2 + i ( x 2 )-y 3 ) + i (2 x 2 y )-2 xy 2 = x 3-3 xy 2-i ( y 3 ) + i (3 x 2 y ) , which is almost exactly our expression for f ( z ) , but just off by a - sign.)...
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Quiz2solns-1 - 2 Since dh dy = 3 y 2 , then h ( y ) = y 3 +...

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