Project ForgeEngineering Mathematics
Calculus
GATE 2020
Hard
The function f(z) of complex variable z = x+iy, where i=√−1, is given as f(z) = (x³−3xy²) + iv(x,y). For this function to be analytic, v(x,y) should be
- A(3xy²−y³) + constant
- B(3x²y²−y³) + constant
- C(x³−3x²y) + constant
- (3x²y−y³) + constant
Explanation
u=x³−3xy². By Cauchy-Riemann, ∂v/∂y=∂u/∂x=3x²−3y², so v=3x²y−y³+g(x). Also −∂v/∂x=∂u/∂y=−6xy, i.e. −(6xy+g′(x))=−6xy, giving g′(x)=0, so g=constant. Hence v=3x²y−y³+constant — option (D).
Source: GATE ME 2020 Session 2 (ME-2) (2020)
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