GATE ME 2016 Set 1 | Complex Variable Question 10 | Engineering Mathematics

GATE ME 2016 Set 1

MCQ (Single Correct Answer)

-0.6

The value of the integral $$\int\limits_{ - \infty }^\infty {{{\sin x} \over {{x^2} + 2x + 2}}} dx$$
evaluated using contour integration and the residue theorem is

$$ - \pi {{\sin \left( 1 \right)} \over e}$$

$$ - \pi {{\cos \left( 1 \right)} \over e}$$

$${{\sin \left( 1 \right)} \over e}$$

$${{\cos \left( 1 \right)} \over e}$$

GATE ME 2014 Set 4

MCQ (Single Correct Answer)

-0.6

If $$z$$ is a complex variable, the value of $$\int\limits_5^{3i} {{{dz} \over z}} $$ is

$$ - 0.511 - 1.57i$$

$$-0.511+1.57i$$

$$0.511-1.57i$$

$$0.511+1.57i$$

GATE ME 2014 Set 3

MCQ (Single Correct Answer)

-0.6

An analytic function of a complex variable $$z = x + iy$$ is expressed as
$$f\left( z \right) = u\left( {x + y} \right) + iv\left( {x,y} \right),$$ where $$i = \sqrt { - 1} .$$ If $$u(x, y)=$$ $${x^3} - {y^2}$$
then expression for $$v(x,y)$$ in terms of $$x,y$$ and a general constant $$c$$ would be

$$xy+c$$

$${{{x^2} + {y^2}} \over 2} + c$$

$$2xy+c$$

$${{{{\left( {x - y} \right)}^2}} \over 2} + c$$

GATE ME 2007

MCQ (Single Correct Answer)

-0.6

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

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

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

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

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