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An analytic function f(z) of complex variable z = x + iy may be written as f(z) = u(x,y) + iv(x,y). Then, u(x,y) and v(x,y) must satisfy

1. ∂u/∂x = ∂v/∂y and ∂u/∂y = ∂v/∂x
2. ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x
3. ∂u/∂x = −∂v/∂y and ∂u/∂y = ∂v/∂x
4. ∂u/∂x = −∂v/∂y and ∂u/∂y = −∂v/∂x

Correct answer: ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x

Solution

The correct option is right because it reflects the Cauchy-Riemann equations, which are necessary conditions for a function to be analytic. These equations establish a relationship between the partial derivatives of the real part u and the imaginary part v of the function, ensuring that the function is differentiable in the complex sense.

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