Question

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?

Answer 1

Correct Answer - Option 2 : 1

Concept:

Cauchy Riemann equations:

\(\frac{{\partial u}}{{\partial x}} = \frac{{\partial v}}{{\partial y}};\frac{{\partial v}}{{\partial x}} = - \frac{{\partial u}}{{\partial y}}\)

Calculation:

z = x + iy

Let F(z) = u + iv

F(z) = iz + k Re(z) + i Im(z) = i (x + iy ) + kx + iy = i (x + y) + (kx – y)

u = (kx – y), v = (x + y)

\(\frac{{\partial u}}{{\partial x}} = \frac{{\partial v}}{{\partial y}} \Rightarrow k = 1\)