Project ForgeEngineering Mathematics
Complex Variables
GATE 2015
Hard
A complex-valued function f(z)=u(x,y)+iv(x,y), z=x+iy, is analytic in domain D. Which of the following is NOT correct?
- df/dz=∂v/∂y+i∂u/∂y
- Bdf/dz=∂u/∂x+i∂v/∂x
- Cdf/dz=∂v/∂y−i∂u/∂y
- Ddf/dz=∂v/∂y+i∂v/∂x
Explanation
Using Cauchy-Riemann (∂u/∂x=∂v/∂y, ∂u/∂y=−∂v/∂x), df/dz=∂u/∂x+i∂v/∂x=∂v/∂y−i∂u/∂y — matching option C, not option A which has the wrong sign on the second term.
Source: GATE CH 2015 (2015)
This question is part of a 15-day revision cycle — sign in and Project Forge will bring it back the day before you'd forget it.