complex variables

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)

$F(z)$ is a function of the complex variable $z = x+iy$ given by $F(z) = i z + k Re(z) + i Im(z)$. For what value of $k$ will $F(z)$ satisfy the Cauchy-Riemann equations?

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

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

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

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...

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...

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 $$...

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

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

If $$\phi (x,y)$$ 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 = \s...