GATE ME 2007 | Complex Variable Question 20 | Engineering Mathematics

If $$\phi (x,y)$$ and $$\psi (x,y)$$ are function with continuous 2^nd derivatives then $$\phi (x,y)\, + \,i\psi (x,y)$$ can be expressed as an analytic function of x +iy ($$i = \sqrt { - 1} $$) when

$${{\partial \phi } \over {\partial x}} = - {{\partial \psi } \over {\partial x}},\,{{\partial \phi } \over {\partial y}} = {{\partial \psi } \over {\partial y}}$$

$${{\partial \phi } \over {\partial y}} = - {{\partial \psi } \over {\partial x}},\,{{\partial \phi } \over {\partial x}} = {{\partial \psi } \over {\partial y}}$$

$${{{\partial ^2}\phi } \over {\partial {x^2}}} + {{{\partial ^2}\phi } \over {\partial {y^2}}} = {{{\partial ^2}\psi } \over {\partial {x^2}}} + {{{\partial ^2}\psi } \over {\partial {y^2}}} = 1$$

$${{\partial \phi } \over {\partial x}} + {{\partial \phi } \over {\partial y}} = {{\partial \psi } \over {\partial x}} + {{\partial \psi } \over {\partial y}} = 0$$

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