Complex Variables (ME)

If $C$ is the unit circle in the complex plane with its center at the origin, then the value of $n$ in the equation given below is _______ (rounded off to 1 decimal place). $$ \oint_c \frac{z^3}{\left(z^2+4\right)\left(z...

Let f(z) be an analytic function, where z = x + iy . If the real part of f(z) is cosh x cos y , and the imaginary part of f(z) is zero for y = 0 , then f(z) is

The value of k that makes the complex-valued function 𝑓(𝑧) = 𝑒 −𝑘𝑥 (cos 2𝑦 − 𝑖 sin 2𝑦) analytic, where 𝑧 = 𝑥 + 𝑖𝑦, is _________. (Answer in integer)

Given z = x +iy, i = √-1 C is a circle of radius 2 with the centre at the origin. If the contour C is traversed anticlockwise, then the value of the integral $\frac{1}{2\pi}\int_c\frac{1}{(z-i)(z+4i)}dZ$ is ________ (rou...

The value of the integral $\rm \oint \left( \frac{6z}{2z^4 - 3z^3 + 7 z^2 - 3z + 5} \right) dz$ evaluated over a counter-clockwise circular contour in the complex plane enclosing only the pole z = i, where 𝑖 is the imag...

If $$f\left( z \right) = \left( {{x^2} + a{y^2}} \right) + ibxy$$ is a complex analytic function of $$z=x+iy,$$ where $${\rm I} = \sqrt { - 1} ,$$ then

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 $$\left( {4\pi i} \right),$$ where $$z=x+iy$$ and $$i = \sqrt { - 1} ....

$$f\left( z \right) = u\left( {x,y} \right) + i\,\,\,\,v\left( {x,y} \right)$$ is an analytic function of complex variable $$z=x+iy$$ , where $$i = \sqrt { - 1} $$ If $$u(x,y)=2xy,$$ then $$v(x,y)$$ may be expressed as

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

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 $$k,$$ for which the function is analytic, is __________...

Solutions of Laplace's equation having continuous second-order partial derivatives are called

Given two complex numbers $${z_1} = 5 + \left( {5\sqrt 3 } \right)i$$ and $${z_2} = {2 \over {\sqrt 3 }} + 2i,$$ the argument of $${{{z_1}} \over {{z_2}}}$$ in degrees $$i$$

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 $$f\left( z \right) = u\left( {x + y} \right) + iv\left( {x,y} \right),$$ where $$i = \sqrt { - 1} .$$ If $$u(x, y)=$$ $${x^3} - {y^2}$$ then expr...

An analytic function of a complex variable $$z=x+iy,$$ where $$i = \sqrt { - 1} $$ is expressed as $$f\left( z \right) = u\left( {x,y} \right) + i\,v\left( {x,y} \right).\,$$ If $$u(x,y)=2xy,$$ then $$v(x,y)$$ must be

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 $$f\left( z \right) = u\left( {x,y} \right) + i\,\,v\,\,\left( {x,y} \right)$$ where $$i = \sqrt { - 1} .$$ If $$u=xy$$ then the expression for...

The integral $$\oint {f(z)dz} $$ evaluated around the unit circle on the complex plane for $$f(z) = {{\cos z} \over z}$$ is

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

$${i^i}$$, where $$i\, = \,\sqrt { - 1} $$ is given by