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Unformatted text preview: ) with z = x + iy the CauchyRiemann 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 CauchyRiemann equations are now r r u = v , u =rv r 5b) Now suppose that u ( r, ) and v ( r, ) obey the CauchyRiemann 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.
 Winter '06
 staff
 Physics, Work

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