Recovering an analytic function from its real or imaginary part

Let $u : \mathbb {R}^{2} \to \mathbb {R}$ be the function defined by $u(x,y) = x^{3} - 3xy^{2}$. Show that $u$ is harmonic and find $v : \mathbb {R}^{2} \to \mathbb {R}$ such that the function $f : \mathbb {C} \to \mathbb {C}$ defined by

is analytic.

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: $v = f(x^{2}+y^{2})$.

Find the analytic function of $z$ that vanishes for $z = 0$ and has the real part

We denote by $a$ and $b$, $a < b$, two fixed real numbers, by $z$ a variable point in the half-plane $\mathrm{Im}\left(z\right) > 0$ and by $\omega$ the variable angle under which the interval $[a,b]$ is seen from the point $z$. If possible find an analytic function the real part of which is $\omega$.

We assume that the function

Express the coefficients $a_{1},a_{2},\ldots ,a_{n},\ldots$ in terms of the real and the imaginary part resp.of $f(z)$ on the circle $\left|z\right|=r$, $0<r<R$.

In the following problems, one of the pair of conjugate harmonic functions $u$ or $v$ is given. Find the second function of the pair.

Let $P(z)$ be a polynomial. Set

Prove that the formulas hold

Recover the regular function $f(z)$ from the condition