The Cauchy--Riemann Equations

Find the analytic function $w(z)$ from the following conditions: $u = x^{2}-y^{2}+xy$; $w(0) = 0$.

Find the analytic function $w(z)$ from the following conditions: $u = x^{3}+6x^{2}y-3xy^{2}-2y^{3}$; $w(0) = 0$.

Find the analytic function $w(z)$ from the following conditions: $v = \frac{y}{x^{2}+y^{2}}$; $w(2) = 0$.

Find the analytic function $w(z)$ from the following conditions: $u = e^{x}(x\cos y - y\sin y)$; $w(0) = 0$.

Find the analytic function $w(z)$ from the following conditions: $u = \frac{2\sin 2x}{e^{2y}+e^{-2y}-2\cos 2x}$.

Find the analytic function $w(z)$ from the following conditions: $u = \varphi \left(\frac{y}{x}\right)$, where $\varphi$ is some function.

Find the analytic function $w(z)$ from the following conditions: $u = F(x+\sqrt{x^{2}+y^{2}})$.

Find the analytic function $w(z)$ from the following conditions: $v = f(x^{2}+y^{2})$.

Prove that the equation $u(x,y) = C$ can represent a family of lines of equal potential only when the ratio of the quantities

is a function of $u$ alone.

Prove that the curves $x^{n}+y^{n} = C$ can be lines of equal potential only for $n = 1$ and for $n = 2$.

Find the analytic function $w(z)$ from the conditions: the lines $\mathrm{Re}\left(w(z)\right) = c$ are the circles $x^{2}+y^{2} = a^{2}$.

Find the analytic function $w(z)$ from the conditions: the lines $\left|w(z)\right| = c$ are the circles $x^{2}+y^{2} = ax$.

Find the analytic function $w(z)$ from the conditions: the lines $\left|w(z)\right| = c$ are the lemniscates $r^{2} = a^{2}\sin 2\varphi$.

Find the analytic function $w(z)$ from the conditions: the lines $\arg w(z) = c$ are the circles $x^{2}+y^{2} = a^{2}$.

Prove the identity:

The function $w = f(z)$ is regular in a simply connected domain bounded by the curve $\left|w(z)\right| = c$. Show that $\arg w$ varies monotonically as $z$ traverses the curve $\left|w(z)\right| = c$ in the positive direction.