Project ForgeEngineering Mathematics
Calculus
GATE 2007
Medium
If φ(x,y) and ψ(x,y) are functions with continuous second derivatives, then φ(x,y)+iψ(x,y) can be expressed as an analytic function of x+iy (i=√−1), when
- A∂φ/∂x = −∂ψ/∂x ; ∂φ/∂y = ∂ψ/∂y
- ∂φ/∂y = −∂ψ/∂x ; ∂φ/∂x = ∂ψ/∂y
- C∂²φ/∂x²+∂²φ/∂y² = ∂²ψ/∂x²+∂²ψ/∂y² = 1
- D∂φ/∂x+∂φ/∂y = ∂ψ/∂x+∂ψ/∂y = 0
Explanation
The Cauchy-Riemann equations for f=φ+iψ to be analytic are φx=ψy and φy=−ψx, matching option (B) exactly.
Source: GATE ME 2007 (2007)
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