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The real part of an analytic function f(z) where z = x + jy is given by e^(−y)cos(x). The imaginary part of f(z) is

1. e^y cos(x)
2. e^(−y)sin(x)
3. −e^y sin(x)
4. −e^(−y)sin(x)

Correct answer: e^(−y)sin(x)

Solution

With u=e^(-y)cos x, Cauchy-Riemann give v_x=-u_y=e^(-y)cos x and v_y=u_x=-e^(-y)sin x, integrating to v=e^(-y)sin x. Then f(z)=e^(-y)cos x + i e^(-y)sin x = e^(iz). The imaginary part is +e^(-y)sin(x), so the stored '-e^(-y)sin x' is wrong.

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