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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

1. ∂φ/∂x = −∂ψ/∂x; ∂φ/∂y = ∂ψ/∂y
2. ∂φ/∂y = −∂ψ/∂x; ∂φ/∂x = ∂ψ/∂y
3. ∂²φ/∂x² + ∂²φ/∂y² = ∂²ψ/∂x² + ∂²ψ/∂y² = 1
4. ∂φ/∂x + ∂φ/∂y = ∂ψ/∂x + ∂ψ/∂y = 0

Correct answer: ∂φ/∂y = −∂ψ/∂x; ∂φ/∂x = ∂ψ/∂y

Solution

The correct option states the Cauchy-Riemann equations, which are necessary conditions for a function to be analytic. These equations ensure that the real and imaginary parts of the function are related in a way that allows for differentiability in the complex sense.

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