Complex Variable · Engineering Mathematics · GATE IN

Let $$z=x+iy$$ where $$i = \sqrt { - 1} .$$ Then $$\overline {\cos \,z} = $$

The value of the integral $${1 \over {2\pi j}}\int\limits_c {{{{z^2} + 1} \over {{z^2} - 1}}} dz$$
where $$z$$ is a complex number and $$C$$ is a unit circle with center at $$1+0j$$ in the complex plane is ____.

In the neighborhood of $$z=1,$$ the function $$f(z)$$ has a power series expansion of the form
$$f\left( z \right) = 1 + \left( {1 - z} \right) + {\left( {1 - z} \right)^2} + ..................$$
Then $$f(z)$$ is

The value of $$\oint\limits_c {{1 \over {{z^2}}}dz} $$ where the contour is the unit circle traversed clock - wise, is

For the function $${{\sin z} \over {{z^3}}}$$ of a complex variable z, the point z = 0 is

Let $$j\, = \,\sqrt { - 1} $$. Then one value of $${j^j}$$ is

Consider the circle $$\left| {z\, - 5\, - 5i} \right|\, = \,2$$ in the complex number plane (x, y) with z = x + iy. The minimum distance from the origin to the circle is

Let $${z^3}\, = \,\overline z $$, where z is a complex number not equal to zero. Then z is a solution of

The bilinear transformation $$w\, = \,{{z\, - \,1} \over {z\, + \,1}}$$

The complex number $$z\, = \,x\, + \,jy$$ which satisfy the equation $$\left| {z + 1} \right|\, = \,1$$ lie on

The real part of the complex number $$z\, = \,x\, + \,iy$$ is given by

$$\cos \phi $$ can be represented as