f(z) = u(x, y) + iv(x, y) is an analytic function of complex variable z = x + iy. If v = xy then u(x, y) equals

Concept:

if f(z) = u(x, y) + iv(x, y) is an analytic function then Cauchy-Riemann condition will be satisfied.

i.e., \(\frac{{\partial u}}{{\partial x}} = \frac{{\partial v}}{{\partial y}}~and~\frac{{\partial u}}{{\partial y}} = - \frac{{\partial v}}{{\partial x}}\)

Calculation:

Given:

v = xy​

\(\frac{{\partial v}}{{\partial y}} = x \Rightarrow \frac{{\partial u}}{{\partial x}} = x\)

\( \frac{{\partial v}}{{\partial x}} = y, \frac{{\partial u}}{{\partial y}} = - \frac{{\partial v}}{{\partial x}} = -y\)

If u = f(x, y)

\(du = \frac{{\partial u}}{{\partial x}}dx + \frac{{\partial u}}{{\partial y}}dy\)

du = xdx - ydy

Integrating both sides

\(\smallint du = \smallint \left( { x} \right)dx - \smallint ydy\)

\(u = \frac{1}{2}\left( {{x^2} - {y^2}} \right)\)