Project ForgeEngineering Mathematics
Calculus
GATE 2016
Hard
f(z) = u(x,y) + i v(x,y) is an analytic function of complex variable z = x + iy. If u(x,y) = 2xy, then v(x,y) may be expressed as
- -x² + y² + constant
- Bx² - y² + constant
- Cx² + y² + constant
- D-(x² + y²) + constant
Explanation
By the Cauchy-Riemann equations, ∂v/∂y = ∂u/∂x = 2y → v = y² + g(x). Also ∂v/∂x = -∂u/∂y = -2x → g'(x) = -2x → g(x) = -x² + C. So v = -x² + y² + constant.
Source: GATE ME 2016 Set-1 (2016)
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