202102190301 Homework 1 (Q3)

Find the analytic functions $w(z)=u(x,y)+\mathrm{i}v(x,y)$ if

(a) $u(x,y)=x^3-3xy^2$ ;

(b) $v(x,y)=e^{-y}\sin x$ .

Attempts.

(a)

$\displaystyle{\frac{\partial u}{\partial x}}=3x^2-3y^2$ .

From (first of the Cauchy-Riemann equations)

$\displaystyle{\frac{\partial u}{\partial x}}=\displaystyle{\frac{\partial v}{\partial y}}$ ,

we have

$v=\displaystyle{\int} (3x^2-3y^2)\,\mathrm{d}y+g(x)$

where $g$ is a function independent of $y$ . Then

$v=3x^2y-y^3+g(x)$ .

Now

$\displaystyle{\frac{\partial v}{\partial x}}=6xy+g'(x)$ ;

$\displaystyle{\frac{\partial u}{\partial y}}=-6xy$

From (second of the Cauchy-Riemann equations)

$\displaystyle{\frac{\partial u}{\partial y}}=-\displaystyle{\frac{\partial v}{\partial x}}$ ,

we have

$-6xy=-6xy-g'(x)$

i.e.,

$g'(x)=0$ ;

$g(x)=C$ .

Thus $v(x,y)=3x^2y-y^3+C$ for some constant $C\in\mathbb{R}$ .

Part (b) is left to the reader.