F(z) is a function of the complex variable z = x + iy given by F(z) = iz + k Re(z) + i Im(z). For what value of k will F(z) satisfy the Cauchy-Riemann equations?

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Accepted Answer

1. F=iz+k Re(z)+i Im(z)=(kx-y)+i(x+y), so u=kx-y, v=x+y. Cauchy-Riemann requires u_x=v_y => k=1 (and u_y=-v_x gives -1=-1, satisfied). Hence k=1.

F(z) is a function of the complex variable z = x + iy given by F(z) = iz + k Re(z) + i Im(z). For what value of k will F(z) satisfy the Cauchy-Riemann equations?

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