homework3 - ) with z = x + iy the Cauchy-Riemann equations...

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Due 01-30-09 Physics 132 Winter 2009 P. Kraus HOMEWORK #3 Hand into box marked “132” outside 1-707D PAB by 3pm Fri. —————————————————————————————————————————— 1) For the following functions state where (if anywhere) d dz f ( z ) exists, and compute d dz f ( z ) when it exists. a ) f ( z ) = 3 z 4 + 2 z ; b ) f ( z ) = 1 z 2 ; c ) f ( z ) = z 2 + z + 1 ; d ) 3 z 2 z 2 2a) Consider the function f ( z ) = x x 2 + y 2 - i y x 2 + y 2 . We want to see if d dz exists for this function. Do this in two ways: first by checking the Cauchy-Riemann equations, and second by rewriting the function in terms of z and z and seeing if the z ’s cancel out. Give the result for the complex derivative if it exists. 2b) Now do the same for the function f ( z ) = x + iy + x 2 - y 2 - 2 ixy . 3) BC p. 72 problem 5: Show that when f ( z ) = x 3 + i (1 - y ) 3 , it is legitimate to write d dz f ( z ) = x u + i∂ x v = 3 x 2 only when z = i . 4) Write the function f ( z ) = z z + 1 in the form f = u ( x,y ) + iv ( x,y ). After determining u ( x,y ) show that it is harmonic, i.e. that it obeys 2 x u ( x,y ) + 2 y v ( x,y ) = 0 except at z = - 1. 5a) In the form f ( z ) = u ( x,y )+ iv ( x,y
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Unformatted text preview: ) with z = x + iy the Cauchy-Riemann equations take the form x u = y v , y u =- x v . We can instead use polar coordinates and write z = re i . Then a complex function can always be written in the form f ( z ) = u ( r, )+ iv ( r, ). Show that the Cauchy-Riemann equations are now r r u = v , u =-rv r 5b) Now suppose that u ( r, ) and v ( r, ) obey the Cauchy-Riemann equations derived in (2a). Show that this implies that u ( r, ) obeys the polar form of Laplaces equation: r 2 2 r u ( r, ) + r r u ( r, ) + 2 u ( r, ) = 0 1 6) BC p. 92 problem 1: Show that a ) e 2 3 i =-e 2 ; b ) e 2+ i 4 = r e 2 (1 + i ) ; c ) e z + i =-e z 7) BC p. 92 problem 10: a) Show that if e z is real, then Im(z) = n ( n = 0 , 1 , 2 ,... ). b) If e z is pure imaginary, what restriction is placed on z ? 2...
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This note was uploaded on 03/30/2010 for the course PHYSICS 132 taught by Professor Staff during the Winter '06 term at UCLA.

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homework3 - ) with z = x + iy the Cauchy-Riemann equations...

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