Project ForgeEngineering Mathematics
Calculus
GATE 2018
Medium
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?
- A0
- 1
- C−1
- Dy
Explanation
F(z) = i(x+iy) + kx + iy = (kx−y) + i(x+y), so u=kx−y, v=x+y. Cauchy-Riemann requires u_x=v_y: k=1. Check u_y=−v_x: −1=−1 ✓. So k=1.
Source: GATE ME 2018 Session 1 (Set 1) (2018)
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