Example 5. Find the analytic function $w=f(z)$ given its imaginary part $v(x, y)=3 x+2 x y$ under the condition that $f(-i)=2$.
Solution. We will use formula (6). In our example, $v(x, y)=3 x+2 x y, z_{0}=-i, C_{0}=2$, so

$$
f(z)=2 i\left(3 \frac{z+i}{2}+2 \frac{z+i}{2} \cdot \frac{z-i}{2 i}\right)+2=3 i z+z^{2}
$$

## Problems for Independent Solution

Reconstruct the function $f(z)$, analytic in a neighborhood of the point $z_{0}$, given the real part $\boldsymbol{u}(\boldsymbol{x}, \boldsymbol{y})$ or the imaginary part $v(x, y)$ and the value $f\left(z_{0}\right)$:

114. a) $u=\frac{x}{x^{2}+y^{2}}, \quad f(\pi)=\frac{1}{\pi}$

b) $v=\operatorname{arctg} \frac{y}{x}(x>0), f(1)=0$

c) $u=x^{2}-y^{2}+2 x, \quad f(i)=2 i-1$.

115. a) $v=2(\operatorname{ch} x \sin y-x y) ; \quad f(0)=0$;

b) $u=2 \sin x \operatorname{ch} y-x, \quad f(0)=0$

c) $v=2(2 \operatorname{sh} x \sin y+x y), \quad f(0)=3$.

116. a) $v=-2 \sin 2 x \operatorname{sh} 2 y+y, \quad f(0)=2$;

b) $v=2 \cos x \operatorname{ch} y-x^{2}+y^{2}, \quad f(0)=2$.

Definition 3. A function $\varphi(x, y)$ is called harmonic in a domain $D$ if it has continuous partial derivatives up to the second order inclusive in this domain and satisfies the Laplace equation in this domain

$$
\frac{\partial^{2} \varphi}{\partial x^{2}}+\frac{\partial^{2} \varphi}{\partial y^{2}}=0
$$

If the function $f(z)=u+i v$ is analytic in some domain $D$, then its real part $u(x, y)$ and imaginary part $v(x, y)$ are harmonic functions in this domain.

However, if $u_{1}(x, y)$ and $v_{1}(x, y)$ are any two harmonic functions, the function $f_{1}(z)=u_{1}(x, y)+i v_{1}(x, y)$ is not necessarily an analytic function: for $f_{1}(z)$ to be analytic, the functions $u_{1}$ and $v_{1}$ must additionally satisfy the Cauchy-Riemann conditions.

Two harmonic functions that satisfy conditions (2) are called conjugate harmonic functions (the order of the functions in the pair is significant).

## Problems for Independent Solution

117. Show that the following functions are harmonic:
a) $u=x^{2}+2 x-\dot{y}^{2}$
b) $u=2 e^{x} \cos y$;
c) $u=\frac{x}{x^{2}+y^{2}}$, d) $u=-\frac{y}{x^{2}+y^{2}}$;

e) $u=\operatorname{arctg} \frac{y}{x} ;$ f) $u=\ln \left(x^{2}+y^{2}\right)$.

118. Can the following functions be the real or imaginary parts of an analytic function $f(z)=u(x, y)+i v(x, y)$:
a) $u=x^{2}-y^{2}+2 x y$
b) $u=x^{2}$;
c) $v=\ln \left(x^{2}+y^{2}\right)$;
d) $v=\frac{1}{2}\left(x^{2}+1\right) y^{2}$ ?
119. Under what conditions is the quadratic form $u=a x^{2}+2 b x y+c y^{2}$ a harmonic function?

In the following examples, pairs of $u(x, y), v(x, y)$ harmonic functions are given. Find among them pairs of conjugate harmonic functions.

120. a) $u=3\left(x^{2}-y^{2}\right), v=3 x^{2} y-y^{3} ; \quad$ b) $u=\frac{x}{x^{2}+y^{2}}, v=-\frac{y}{x^{2}+y^{2}}$;

c) $u=x, v=-y ;$ d) $u=e^{x} \cos y+1, v=1+e^{x} \sin y$.

121. Let the function $u(x, y)$ be harmonic in the domain $D$, where it has continuous partial derivatives of any order. Show that the latter will also be harmonic functions in the domain $D$.
122. Let the function $u=u(x, y)$ be harmonic in the domain $D$. Find all functions $f$ for which the function $f(u)$ will be harmonic in the domain $D$.
123. Let the function $w=f(z)$ be analytic in the domain $D$. Which of the functions
a) $|w|$;
b) $\arg w$;
c) $\ln |w|$

will be harmonic in the domain $D$?

124. Prove that the product of conjugate harmonic functions $u(x, y), v(x, y)$ in the domain $D$ will be a function that is harmonic in the domain $D$.
125. Let $u=u(x, y), v=v(x, y)$ be a pair of conjugate harmonic functions in the domain $D$. Which of the following pairs of functions will be conjugate harmonic functions in the domain $D$:
a) $A u-B v, \quad B u+A v, \quad A, B$ - const;

b) $u^{2}-v^{2}, u v$
c) $e^{v} \cos v, e^{x} \sin v$?