Project ForgeEngineering Mathematics
Calculus
GATE 2009
Medium
An analytic function of a complex variable z=x+iy is expressed as f(z)=u(x,y)+iv(x,y), where i=√-1. If u=xy, the expression for v should be
- A(x+y)²/2+k
- B(x²-y²)/2+k
- (y²-x²)/2+k
- D(x-y)²/2+k
Explanation
Cauchy-Riemann: ∂u/∂x=∂v/∂y and ∂u/∂y=-∂v/∂x. With u=xy: ∂u/∂x=y=∂v/∂y ⟹ v=y²/2+g(x). ∂u/∂y=x=-∂v/∂x ⟹ ∂v/∂x=-x ⟹ v=-x²/2+h(y). Combining: v=(y²-x²)/2+k.
Source: GATE ME 2009 (2009)
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