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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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