IIT JAM & CSIR NET - Mathematical Sciences

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Q1.
[Linear Algebra & Real Analysis Problem #1] Regarding Cauchy-Riemann equations for an analytic function f(z) = u(x,y) + i v(x,y): Which of the following analytical formulations is correct?

(A)
∂^2u/∂x^2 + ∂^2v/∂y^2 = 0 strictly
(B)
∂u/∂x + ∂v/∂y = 0 and ∂u/∂y + ∂v/∂x = 0
(C)
∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x
(D)
∂u/∂x = -∂v/∂y and ∂u/∂y = ∂v/∂x
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Q2.
[Linear Algebra & Real Analysis Problem #2] Regarding Cauchy-Riemann equations for an analytic function f(z) = u(x,y) + i v(x,y): Which of the following analytical formulations is correct?

(A)
∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x
(B)
∂^2u/∂x^2 + ∂^2v/∂y^2 = 0 strictly
(C)
∂u/∂x + ∂v/∂y = 0 and ∂u/∂y + ∂v/∂x = 0
(D)
∂u/∂x = -∂v/∂y and ∂u/∂y = ∂v/∂x
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