$$z = {{2 - 3i} \over { - 5 + i}}$$ can be expressed as

For an analytic function $$f\left( {x + i\,y} \right) = u\left( {x,y} \right) + i\,v\left( {x,y} \right),\,u$$ is given by $$u = 3{x^2} - 3{y^2}.$$ The expression for $$v,$$ considering $$k$$ is to be constant is

The modulus of the complex number $${{3 + 4\,i} \over {1 - 2\,i}}$$ is

The analytical function has singularities at, where $$f(z) = {{z - 1} \over {{z^2} + 1}}$$