GATE Questions & Answers of Complex variables Mechanical Engineering

$ F\left(z\right) $ is a function of the complex variable $ z=x+iy $ given by

                              $ F\left(z\right)=i\;z\;+\;k\;Re\;\left(z\right)+i\;Im\left(z\right) $.

For what value of k will $ F\left(z\right) $ satisfy the Cauchy-Riemann equations?

Let $z$ be a complex variable. For a counter-clockwise integration around a unit circle $C$ , centred at origin,

                                                              $ \oint\limits_c\;\frac1{5z-4}\;dz=A\pi i\;, $

the value of $A$ is

The argument of the complex number 1+i1-i , where i=-1,,is

An analytic function of a complex variable z=x+iy is expressed as fz=ux,y+ivx,y,, where i=-1 . If ux,y=2xy, then vx,y must be

An analytic function of a complex variable z = x + i y is expressed as f (z) = u(x, y) + i v(x, y) ,where i = -1 . If u(x, y) = x2 − y 2 , then expression for v(x, y) in terms of x, y and a general constant c would be

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

The product of two complex numbers 1 + i and 2 - 5i is

The modulus of the complex number 3+4i1-2i is

An analytic function of a complex variable z = x + iy is expressed as f(z) = u(x,y) + i v(x,y) where i = -1. If u = xy, the expression for v should be

In the Taylor series expansion of ex about x = 2, the coefficient of (x- 2)4 is

The integral $\oint f\left(z\right)dz$ evaluated around the unit circle on the complex plane for fz=coszz is

If φx,y and ψx,y are functions with continuous second derivatives, then φx,y+iψx,y can be expressed as an analytic function of x+iyi=-1, when