Example 1. Investigate the analytical properties of the function $w=e^{z}$ and find its derivative.
Solution.

1. Find the real and imaginary parts of the function $e^{z}$, using the definition of the exponential function

$$
e^{z}=e^{x}(\cos y+i \sin y)
$$

We obtain

$$
u(x, y)=e^{x} \cos y, \quad v(x, y)=e^{x} \sin y
$$

2. Find the partial derivatives

$$
\begin{aligned}
\frac{\partial u}{\partial x}=e^{x} \cos y, & \frac{\partial v}{\partial y}=e^{x} \cos y \\
\frac{\partial u}{\partial y}=-e^{x} \sin y, & \frac{\partial v}{\partial x}=e^{x} \sin y
\end{aligned}
$$

The partial derivatives are clearly continuous everywhere, and therefore the functions $u(x, y)$ and $v(x, y)$ are differentiable everywhere.

3. Cauchy-Riemann conditions

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

are satisfied at any point $(x, y) \in \mathbb{R}^{2}$, and therefore the function $e^{z}$ is differentiable and analytic at all points $z \in \mathbb{C}$.

4. Compute the derivative of the function $e^{z}$, for example, using the formula

$$
f^{\prime}(z)=\frac{\partial u}{\partial x}+i \frac{\partial v}{\partial x}=e^{x}(\cos y+i \sin y)=e^{z}
$$

Answer. The function $f(z)=e^{z}$ is analytic at all points $z$ and $f^{\prime}(z)=e^{z}$.

Remark. All elementary functions of a complex variable are analytic in their domain of definition, and the table of derivatives and differentiation rules known for functions of a real variable are valid.