GATE ME
Engineering Mathematics
Complex Variable
Previous Years Questions

## Marks 1

Solutions of Laplace's equation having continuous second-order partial derivatives are called
$$f\left( z \right) = u\left( {x,y} \right) + i\,\,\,\,v\left( {x,y} \right)$$ is an analytic function of complex variable $$z=x+iy$$ , where $$i = \... Given two complex numbers$${z_1} = 5 + \left( {5\sqrt 3 } \right)i$$and$${z_2} = {2 \over {\sqrt 3 }} + 2i,$$the argument of$${{{z_1}} \over {{z_...
An analytic function of a complex variable $$z=x+iy,$$ where $$i = \sqrt { - 1}$$ is expressed as $$f\left( z \right) = u\left( {x,y} \right) + i\,... The argument of the complex number$${{1 + i} \over {1 - i}},$$where$$i = \sqrt { - 1} ,$$is The product of two complex numbers$$1 + i\,\,\,\,\& \,\,\,\,2 - 5\,i$$is An analytic function of a complex variable$$z = x + i\,y$$is expressed as$$f\left( z \right) = u\left( {x,y} \right) + i\,\,v\,\,\left( {x,y} \rig...
The integral $$\oint {f(z)dz}$$ evaluated around the unit circle on the complex plane for $$f(z) = {{\cos z} \over z}$$ is
$${i^i}$$, where $$i\, = \,\sqrt { - 1}$$ is given by

## Marks 2

If $$f\left( z \right) = \left( {{x^2} + a{y^2}} \right) + ibxy$$ is a complex analytic function of $$z=x+iy,$$ where $${\rm I} = \sqrt { - 1} ,$$ th...
The value of $$\oint\limits_\Gamma {{{3z - 5} \over {\left( {z - 1} \right)\left( {z - 2} \right)}}dz}$$ along a closed path $$\Gamma$$ is equal to...
A function $$f$$ of the complex variable $$z=x+iy,$$ is given as $$f(x,y)=u(x,y)+iv(x,y),$$ Where $$u(x,y)=2kxy$$ and $$v(x,y)$$ $$= {x^2} - {y^2}.... The value of the integral$$\int\limits_{ - \infty }^\infty {{{\sin x} \over {{x^2} + 2x + 2}}} dx$$evaluated using contour integration and the res... If$$z$$is a complex variable, the value of$$\int\limits_5^{3i} {{{dz} \over z}} $$is An analytic function of a complex variable$$z = x + iy$$is expressed as$$f\left( z \right) = u\left( {x + y} \right) + iv\left( {x,y} \right),$$w... If$$\phi (x,y)$$and$$\psi (x,y)$$are function with continuous 2nd derivatives then$$\phi (x,y)\, + \,i\psi (x,y) can be expressed as an analyti...
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