Power Series
(Lecture 17 of Mathematical Methods II.)
A power series is a series of the form:
\begin{equation} \tag{1} a_0+a_1(z-z_0)+a_2(z-z_0)^2+\cdots+a_n(z-z_0)^n+\cdots \end{equation}
where $z$ is the (complex) variable, $a_i$ are constants called the coefficients of the series, and $z_0$ is also a constant, called the center of the series.
The convergence of Power Series is ruled by the following concept of radius of convergence:
- Every power series converges at its center.
- If a power series converges at the point $z_1$, it converges absolutely for all points in the disk of center $z_0$ and radius $|z_1-z_0|$.
- If a power series diverges at the point $z_2$, it diverges for all points farther from $z_0$ than $z_2$.
The radius of convergence $R$ is that of the smallest circle that contains all the points where the series converges. No general statement can be made for points on the circle itself (cf. exercise 1).
The Cauchy--Hadamard formula allows to determine the Radius of convergence of a series from its coefficients:
\begin{equation} \mathrm{if\ } |a_{n+1}/a_n|\rightarrow L,\quad \mathrm{then\ }R=1/L. \end{equation}
For the following, with no loss of generality, we can assume the center is at $z_0=0$.
A power series has a unique representation, i.e., if $\sum a_iz^i=\sum b_iz^i$ for all $z$, then $a_i=b_i$ for all $i$.
- Termwise addition or subtraction of two power series with
radius of convergence $R_1$ and $R_2$ gives a power series with radius of convergence at least $\min(R_1,R_2)$.
- Termwise multiplication of $f(z)=\sum a_kz^k$ and $g(z)=\sum b_mz^m$ results in the Cauchy product:
\begin{equation} \tag{2} f(z)g(z)=\sum_{n=0}^\infty(a_0b_n+a_1b_{n-1}+\cdots+a_{n-1}b_1+a_nb_0)z^n \end{equation}
that converges in the smallest radius of convergence of the two series (unproven in class).
- Termwise Differentiation of a power series:
\begin{equation} \tag{3} \left(\sum_{n\ge0} a_nz^n\right)'=\sum_{n\ge1} na_n z^{n-1}=\sum_{n\ge0}(n+1)a_{n+1}z^n\,, \end{equation}
with the same radius of convergence.
- Termwise Integration of a power series:
\begin{equation} \tag{4} \int\left(\sum_{n\ge0} a_nz^n\right)\,dz=\sum_{n\ge0}\frac{a_n}{n+1}z^{n+1}\,, \end{equation}
with the same radius of convergence.
We will later see that Power Series are the analytic functions.
