Example 7. Find the first few terms of the expansion in a power series of the function $f(z)=\operatorname{tg} z$ and find the radius of convergence of the series.
Solution. Let the desired series have the form

$$
f(z)=c_{0}+c_{1} z+c_{2} z^{2}+c_{3} z^{3}+\ldots
$$

where

$$
c_{n}=\frac{f^{(n)}(0)}{n!} \quad(n=0,1,2, \ldots), \quad f^{(0)}(0)=f(0)=0
$$

To find the values of the derivatives $f^{(n)}(z)$ at the point $z=0$, we differentiate the given function. We have

$$
\begin{gathered}
f^{\prime}(z)=\frac{1}{\cos ^{2} z} \text { or } f^{\prime}(z)=1+f^{2}(z) \\
\left\{\begin{array}{l}
f^{\prime \prime}(z)=2 f(z) f^{\prime}(z) \\
f^{\prime \prime \prime}(z)=2\left[f^{\prime 2}(z)+f(z) f^{\prime \prime}(z)\right] \\
f^{\prime V}(z)=2\left[3 f^{\prime}(z) f^{\prime \prime}(z)+f(z) f^{\prime \prime \prime}(z)\right] \\
f^{\mathrm{V}}(z)=2\left[3 f^{\prime \prime 2}(z)+4 f^{\prime}(z) f^{\prime \prime \prime}(z)+f(z) f^{\prime V}(z)\right] \\
\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots
\end{array}\right.
\end{gathered}
$$

Substituting $z=0$ into (13) and (14), we find

$$
f^{\prime}(0)=1 ; \quad f^{\prime \prime}(0)=0 ; \quad f^{\prime \prime \prime}(0)=2 ; \quad f^{\prime v}(0)=0 ; \quad f^{V}(0)=16
$$

Substituting the found values of the derivatives into the series, we get

$$
\operatorname{tg} z=z+\frac{2}{3!} z^{3}+\frac{16}{5!} z^{5}+\ldots
$$

The nearest singular point to the point $z=0$ is the point $\zeta=\frac{\pi}{2}$. Therefore, the radius of convergence of the obtained series is $R=\frac{\pi}{2}$.

## Problems for Independent Solution

In the following problems, expand the given functions into Taylor series using ready-made expansions, and find the radii of convergence of the series:

210. $\sin (2 z+1)$ in powers of $z+1$.
211. $\cos z$ in powers of $z+\frac{\pi}{4}$.
212. $e^{z}$ in powers of $2 z-1$.
213. $\frac{1}{3 z+1}$ in powers of $z+2$.
214. $\frac{z+1}{z^{2}+4 z-5}$ in powers of $z$.
215. $\frac{z}{z^{2}+i}$ in powers of $z$.
216. $\cos ^{2} \frac{i z}{2}$ in powers of $z$.
217. $\operatorname{sh}^{2} \frac{z}{2}$ in powers of $z$.
218. $\ln (2-z)$ in powers of $z$.
219. $\ln \left(2+z-z^{2}\right)$ in powers of $z$.

Find the first few terms of the expansion in a series in powers of $z$ of the following functions. Find the radii of convergence of the series:
220. $\frac{1}{1+\mathrm{e}^{2}}$.
221. $\frac{1}{2+\sin z}$.
222. $\frac{1}{e^{-x}+5}$.
223. $\ln \left(1+e^{-3}\right)$.

224. $\ln \cos z . \quad 225 . \ln (1+\cos z)$ 226. $e^{1 /(1-x)}$
225. Find the function $f(z)$, analytic in the disk $|z| \leqslant 1$ and taking on the circle $|z|=1$ the value

$$
\frac{a-\cos \theta+i \sin \theta}{a^{2}-2 a \cos \theta+1}, \quad a>1, \quad \theta=\arg z
$$

228. Let the function $f(z)=\sum_{k=0}^{\infty} a_{k} z^{k}$ be analytic in the disk $|z| \leqslant 1$. Prove that the average value of the function $\frac{f(z)}{z^{n}}$ on the circle $|z|=1$ is $a_{n}$.

Let the series be given

$$
\frac{c_{-1}}{z-z_{0}}+\frac{c_{-2}}{\left(z-z_{0}\right)^{2}}+\ldots+\frac{c_{-n}}{\left(z-z_{0}\right)^{n}}+\ldots=\sum_{n=1}^{\infty} \frac{c_{-n}}{\left(z-z_{0}\right)^{n}}
$$

If $c_{-n} \neq 0$ and there exists a finite limit

$$
r=\lim _{n \rightarrow \infty} \frac{\left|c_{-n-1}\right|}{\left|c_{-n}\right|}
$$

then this series converges in the region

$$
\left|z-z_{0}\right|>r
$$