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M461solasstB

# M461solasstB - x(sin y-e-x(sin y y cos y C y v = e-x-x sin...

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MAT 461 Suggested Problems B (Posted September 17th 2007) (1) e x 2 - y 2 e i 2 xy = e x 2 - y 2 cos(2 xy ) + ie x 2 - y 2 sin(2 xy ) u ( x,y ) = e x 2 - y 2 cos(2 xy ) u x = 2 xe x 2 - y 2 cos(2 xy ) - 2 ye x 2 - y 2 sin(2 xy ) = e x 2 - y 2 (2 x cos(2 xy ) - 2 y sin(2 xy )) u y = e x 2 - y 2 ( - 2 y cos(2 xy ) - 2 x sin(2 xy )) v ( x,y ) = e x 2 - y 2 sin(2 xy ) v x = e x 2 - y 2 (2 x sin(2 xy ) + 2 y cos(2 xy )) = - u y v y = e x 2 - y 2 ( - 2 y sin(2 xy ) + 2 x cos(2 xy )) = u x The Cauchy-Riemann equations are satisfied and the partials are continuous for all z C , so f is entire. (2) f is analytic so u x = v y and u y = - v x . ¯ zf ( z ) = ( x - iy )( u ( x,y ) + iv ( x,y )) = ( xu ( x,y ) + yv ( x,y )) + i ( xv ( x,y ) - yu ( x,y )) ∂x ( xu + yv ) = u + xu x + yv x ∂y ( xv - yu ) = xv y - u - yu y ∂y ( xu + yv ) = xu y + v + yv y ∂x ( xv - yu ) = v - xv x - yu x ¯ zf ( z ) is analytic so u + xu x + yv x = xv y - u - yu y xu y + v + yv y = - ( v + xv x - yu x ) Sub in the C-R equations for f , u + xu x - yu y = xu x - u - yu y 2 u = 0 - xv x + v + yv y = - v - xv x + yv y 2 v = 0 f ( z ) = u + iv = 0 1

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(3) u ( x,y ) = e - x ( x cos y + y sin y ) u x = - e - x ( x cos y + y sin y ) + e - x cos y u xx = e - x ( x cos y + y sin y ) - 2 e - x cos y u y = e - x ( - x sin y + sin y + y cos y ) u yy = e - x ( - x cos y + 2 cos y - y sin y ) = - u xx v y = e - x ( - x cos y - y sin y + cos y ) v = e - x ( - x sin y + y cos y ) + C ( x ) v x = e - x ( - x sin y + sin
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Unformatted text preview: x )(sin y )-e-x (sin y + y cos y ) + C ( y ) v = e-x (-x sin y + y cos y ) (4) Clearly ze-z is analytic since z ,-z and e z are analytic, and compositions and products analytic functions are analytic. Since u = Re f , we know u is harmonic. (5) | e iz + e-iz | ≤ | e iz | + | e-iz | = | e ix e-y | + | e-ix e y | = | e-y | + | e y | (6) Branch cut is at π 2 (-1) i = e i log(-1) = e i log( e i ( π +2 πk ) ) = e i (ln 1+ i ( π )) = e-π + i (7) sin( iz ) = e i ( iz )-e-i ( iz ) 2 i = e-z-e z 2 i = i e z-e-z 2 = i sinh( z ) 2...
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