冪級数
Contents
冪級数とは
冪級数 (英：power series) とは、次式の形の級数のこと。
\[ \sum_{k=1}^\infty a_k(z-z_0)^k \quad (z\in\Complex) \]
冪級数の演算
\[ \begin{aligned} \text{加減算} &: f(z)\pm g(z) = \sum_{k=0}^\infty (a_k\pm b_k)(z-z_0) \\ \text{乗算} &: f(z)g(z) = \sum_{k=0}^\infty\left(\sum_{\ell=0}^k a_\ell b_{k-\ell}\right)(z-z_0)^k \\ \end{aligned} \]
冪級数の加減算の導出：
\[ \begin{aligned} f(z)\pm g(z) &= \sum_{k=0}^\infty a_k(z-z_0)\pm\sum_{k=0}^\infty b_k(z-z_0) \\ &= \sum_{k=0}^\infty\big(a_k(z-z_0)\pm b_k(z-z_0)\big) \\ &= \sum_{k=0}^\infty (a_k\pm b_k)(z-z_0) \\ \\ \therefore f(z)\pm g(z) &= \sum_{k=0}^\infty (a_k\pm b_k)(z-z_0) \end{aligned} \]
冪級数の乗算の導出：
\[ \begin{aligned} f(z)g(z) &= \left[\sum_{k=0}^\infty a_k(z-z_0)^k\right]\left[\sum_{k=0}^\infty b_k(z-z_0)^k\right] \\ &= a_0b_0(z-z_0)^0 + a_0b_1(z-z_0)^1 + \cdots + a_0b_\infty(z-z_0)^\infty \\ &\quad + a_1b_0(z-z_0)^1 + a_1b_1(z-z_0)^2 +\cdots + a_1b_\infty(z-z_0)^{\infty+1} \\ &\qquad\qquad\vdots \\ &\quad\quad + a_\infty b_0(z-z_0)^\infty + a_\infty b_1(z-z_0)^{\infty+1} +\cdots + a_\infty b_\infty(z-z_0)^{\infty+\infty} \\ &= \sum_{k=0}^\infty\left(\sum_{\ell=0}^k a_\ell b_{k-\ell}\right)(z-z_0)^k \\ \\ \therefore f(z)g(z) &= \sum_{k=0}^\infty\left(\sum_{\ell=0}^k a_\ell b_{k-\ell}\right)(z-z_0)^k \end{aligned} \]
収束半径
収束半径 (英：radius of convergence) とは、冪級数において下記条件を満たす $\rho$ のこと。
\[ \begin{cases} |z-z_0|\lt\rho & \displaystyle\cdots\sum_{k=1}^\infty a_k(z-z_0)^k\text{ は収束する} \\ |z-z_0|\gt\rho & \displaystyle\cdots\sum_{k=1}^\infty a_k(z-z_0)^k\text{ は発散する} \\ \end{cases} \]