1. 常数项级数的概念和性质
1. 常数项级数的概念
设有数列 \(\left\{ {u}_{n}\right\} : {u}_{1},{u}_{2},\cdots ,{u}_{n},\cdots\) ,将其各项依次累加所得的式子 \({u}_{1} + {u}_{2} + \cdots + {u}_{n} + \cdots\) 称为 (常数项) 无穷级数,简称 (常数项) 级数,记作 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\) ,即 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n} = {u}_{1} + {u}_{2} + \cdots + {u}_{n} + \cdots\)
2. 常数项级数收敛的概念
设给定常数项级数\[\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n} = {u}_{1} + {u}_{2} + \cdots + {u}_{n} + \cdots\qquad①\]记 \({S}_{n} = \mathop{\sum }\limits_{{i = 1}}^{n}{u}_{i} = {u}_{1} + {u}_{2} + \cdots + {u}_{n}\) 为级数 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\) 的前 \(n\) 项部分和,若 \(\mathop{\lim }\limits_{{n \rightarrow \infty }}{S}_{n} = S\) (有限值),则称级数①收敛; 若 \(\mathop{\lim }\limits_{{n \rightarrow \infty }}{S}_{n}\) 不存在,则称级数①发散
3. 常数项级数的基本性质
(1) 级数 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\) 与 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }k{u}_{n}\) 有相同敛散性 ( \(k\) 是不为零的常数)
(2) 若级数 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n},\mathop{\sum }\limits_{{n = 1}}^{\infty }{v}_{n}\) 均收敛,则级数 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }\left( {{u}_{n} \pm {v}_{n}}\right)\) 亦收敛,且有\[\mathop{\sum }\limits_{{n = 1}}^{\infty }\left( {{u}_{n} \pm {v}_{n}}\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n} \pm \mathop{\sum }\limits_{{n = 1}}^{\infty }{v}_{n}\](3)在级数中去掉或添加有限项,不会影响级数的敛散性(但收敛时,级数和一般会改变)
(4)收敛级数任意加括号后所成的级数仍收敛. 如果正项级数加括号后所成的级数收敛, 则原级数收敛
(5) 级数收敛的必要条件: 若 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\) 收敛,则 \(\mathop{\lim }\limits_{{n \rightarrow \infty }}{u}_{n} = 0\)
4. 柯西收敛准则
级数 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\) 收敛的充分必要条件: 对任意给定的正数 \(\varepsilon\) ,总存在 \(N\) ,使得当 \(n > N\) 时,对于任意的自然数 \(p = 1,2,3,\cdots\) ,不等式\[\left| {{u}_{n + 1} + {u}_{n + 2} + \cdots + {u}_{n + p}}\right| < \varepsilon\]恒成立
2. 正项级数的审敛法
设 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n},\mathop{\sum }\limits_{{n = 1}}^{\infty }{v}_{n}\) 均为正项级数
1. 比较审敛法
(1)若 \(0 \leq {u}_{n} \leq {v}_{n},\mathop{\sum }\limits_{{n = 1}}^{\infty }{v}_{n}\) 收敛,则 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\) 收敛
(2)若 \(0 \leq {v}_{n} \leq {u}_{n},\mathop{\sum }\limits_{{n = 1}}^{\infty }{v}_{n}\) 发散,则 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\) 发散
比较审敛法的极限形式 若 \(\mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{{u}_{n}}{{v}_{n}} = \lambda \left( {0 < \lambda < + \infty }\right)\) ,则级数 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\) 与 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{v}_{n}\) 具有相同的敛散性
2. 比值审敛法
若 \(\displaystyle\mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{{u}_{n + 1}}{{u}_{n}} = \rho \left\{ \begin{array}{ll} < 1, & \text{ 则 }\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\text{ 收敛 } \\ > 1, & \text{ 则 }\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\text{ 发散 } \\ = 1, & \text{ 则 }\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\text{ 敛散性不定 } \end{array}\right.\)
3. 根值审敛法
若 \(\displaystyle\mathop{\lim }\limits_{{n \rightarrow \infty }}\sqrt[n]{{u}_{n}} = \rho \left\{ \begin{array}{ll} < 1, & \text{ 则 }\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\text{ 收敛 } \\ > 1, & \text{ 则 }\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\text{ 发散 } \\ = 1, & \text{ 则 }\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\text{ 敛散性不定 } \end{array}\right.\)
4. 对数审敛法
(1)若存在 \(\alpha > 0\) ,使当 \(n \geq {n}_{0}\) 时, \(\displaystyle\frac{\ln \frac{1}{{u}_{n}}}{\ln n} \geq 1 + \alpha\) ,则正项级数 \(\displaystyle\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\) 收敛
(2) 若 \(n \geq {n}_{0}\) 时, \(\displaystyle\frac{\ln \frac{1}{{u}_{n}}}{\ln n} \leq 1\) ,则正项级数 \(\displaystyle\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\) 发散
对数审敛法的极限形式 \(\displaystyle\mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{\ln\frac{1}{u_n}}{\ln{n}}=\rho \left\{ \begin{array}{ll} > 1, & \text{ 则 }\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\text{ 收敛 } \\ < 1, & \text{ 则 }\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\text{ 发散 } \\ = 1, & \text{ 则 }\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\text{ 敛散性不定 } \end{array}\right.\)
5. 两个重要级数的敛散性
等比级数: \(\mathop{\sum }\limits_{{n = 0}}^{\infty }a{r}^{n}\left( {a \neq 0}\right)\) 当 \(\left| r\right| < 1\) 时收敛; 当 \(\left| r\right| \geq 1\) 时发散
\(p\) 一级数: \(\displaystyle\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{1}{{n}^{p}}\) 当 \(p > 1\) 时收敛,当 \(p \leq 1\) 时发散
6. 正项级数 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\) 判断敛散性的一般步骤
(1)考查 \({u}_{n} \rightarrow 0\) ,若 \(\mathop{\lim }\limits_{{n \rightarrow \infty }}{u}_{n} \neq 0\) 则级数发散
(2)若 \({u}_{n} \rightarrow 0\) ,用比值法或根值法判定级数敛散性
(3)若比值法或根值判别法均无效, 则用比较判别法
(4)若上述方法都行不通时,考虑 \({S}_{n}\) 是否有极限
从上述步骤可知, 比值法或根值法是比较重要的判别法, 也是比较易掌握的判别法
3. 任意项级数的审敛法
1. 交错级数的莱布尼兹判别法
若 \({u}_{n} > 0,{u}_{n} \geq {u}_{n + 1},\mathop{\lim }\limits_{{n \rightarrow \infty }}{u}_{n} = 0\) ,则交错级数 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{\left( -1\right) }^{n - 1}{u}_{n}\) 收敛,其和 \(S < {u}_{1}\)
2. 任意项级数 绝对收敛与条件收敛
若 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\) 为任意项级数,且 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }\left| {u}_{n}\right|\) 收敛,则 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\) 收敛,并称 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\) 为绝对收敛; 若 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\) 收敛,而 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }\left| {u}_{n}\right|\) 发散,则称 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\) 为条件收敛
3. 判定任意项级数 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\) 的敛散性的主要方法
若 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }\left| {u}_{n}\right|\) 收敛,则 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\) 绝对收敛; 若 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }\left| {u}_{n}\right|\) 发散,则 \(\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\) 敛散性判别主要利用“莱布尼兹判别法”或 \({u}_{n} \rightarrow 0\) 或求 \({S}_{n}\)
4. 幂级数
1. 函数项级数的一般概念
(1)函数项级数的定义 设给定一个定义在区间 \(\left\lbrack {a,b}\right\rbrack\) 上的函数列\[{u}_{1}\left( x\right) ,{u}_{2}\left( x\right) ,\cdots ,{u}_{n}\left( x\right) ,\cdots \]则式子\[{u}_{1}\left( x\right) + {u}_{2}\left( x\right) + \cdots + {u}_{n}\left( x\right) + \cdots \qquad ①\]叫做函数项级数
(2)函数项级数的收敛域 对于区间 \(\left\lbrack {a,b}\right\rbrack\) 上的每一个值 \({x}_{0}\) ,级数①成为常数项级数\[{u}_{1}\left( {x}_{0}\right) + {u}_{2}\left( {x}_{0}\right) + \cdots + {u}_{n}\left( {x}_{0}\right) + \cdots \qquad ②\]如果②收敛,则称 \({x}_{0}\) 是级数①的收敛点; 如果②发散,则称 \({x}_{0}\) 是级数①的发散点,①所有收敛点的全体称为函数项级数①的收敛域
(3)函数项级数的和函数 对于收敛域内的任一点 \(x\) ,级数① 都有一个确定的和\[S\left( x\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\left( x\right)\]\(S\left( x\right)\) 是定义在收敛域上的函数,称为级数①的和函数.
2. 幂级数及其收敛域
(1)幂级数的定义 形如\[{a}_{0} + {a}_{1}\left( {x - {x}_{0}}\right) + {a}_{2}{\left( x - {x}_{0}\right) }^{2} + \cdots + {a}_{n}{\left( x - {x}_{0}\right) }^{n} + \cdots\]或 \({a}_{0} + {a}_{1}x + {a}_{2}{x}^{2} + \cdots + {a}_{n}{x}^{n} + \cdots\)的级数称为幂级数
(2) 阿贝尔 (Abel) 引理 若 \({x}_{0}\) 是幂级数 \(\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}\) 的收敛点,则对于一切满足 \(\left| x\right| < \left| {x}_{0}\right|\) 的点 \(x\) ,幂级数都绝对收敛; 若 \({x}_{0}\) 是幂级数 \(\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}\) 的发散点,则对一切满足 \(\left| x\right| > \left| {x}_{0}\right|\) 的点 \(x\) ,幂级数都发散
(3) 幂级数的收敛半径 对任一幂级数 \(\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}\) ,必存在一个非负数 \(R\left( {R\text{ 可为无穷大 }}\right)\) ,使得对一切 \(\left| x\right| < R\) 的点 \(x\) (当 \(R = 0\) 时, \(x = 0\) ),幂级数都收敛; 而对一切 \(\left| x\right| > R\) 的点 \(x\) ,幂级数都发散. \(R\) 称为幂级数的收敛半径, \(R\) 的求法如下:
设幂级数 \(\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}\) ,若 \(\mathop{\lim }\limits_{{n \rightarrow \infty }}\left| \frac{{a}_{n + 1}}{{a}_{n}}\right| = \rho\) ,则幂级数的收敛半径\[R = \left\{ \begin{array}{ll} \frac{1}{\rho }, & 0 < \rho < + \infty \\ + \infty , & \rho = 0 \\ 0, & \rho = + \infty \end{array}\right.\](4)幂级数的收敛域 在幂级数 \(\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}\) 的收敛区间(-R, R)上,加上收敛区间端点中的收敛点,就得到幂级数 \(\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}\) 的收敛域. 若 \(R\) 是不为零的有限数,则其收敛域为以下四种情形之一:\[\left( {-R,R}\right) ,\left\lbrack {-R,R}\right\rbrack ,\;\left( {-R,R}\right) ,\;\lbrack - R,R)\]
3. 幂级数的性质
若幂级数 \(\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}\) 的收敛半径为 \(R\) ,则有
(1)和函数 \(S\left( x\right)\) 在(-R, R)内是连续的. 若 \(\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}\) 在端点 \(x = R\) (或 \(x = - R\) ) 处收敛,则和函数在点 \(x = R\) 左连续 (或在点 \(x = - R\) 右连续)
(2)幂级数可以逐项微分,即\[{S}^{\prime }\left( x\right) = {\left( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}\right) }^{\prime } = \mathop{\sum }\limits_{{n = 0}}^{\infty }{\left( {a}_{n}{x}^{n}\right) }^{\prime } = \mathop{\sum }\limits_{{n = 0}}^{\infty }n{a}_{n}{x}^{n - 1},\;x \in \left( {-R,R}\right)\]若逐项微分后得到的幂级数 \(\mathop{\sum }\limits_{{n = 0}}^{\infty }n{a}_{n}{x}^{n - 1}\) 在端点 \(x = R\) (或 \(x = - R\) )处收敛,则逐项微分以前的幂级数在点 \(x = R\) (或 \(x = - R\) ) 也收敛
(3)幂级数可以逐项积分,即\[{\int }_{0}^{x}S\left( x\right) \mathrm{d}x = {\int }_{0}^{x}\left( {\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}}\right) \mathrm{d}x = \mathop{\sum }\limits_{{n = 0}}^{\infty }{\int }_{0}^{x}{a}_{n}{x}^{n}\mathrm{\;d}x = \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{a}_{n}}{n + 1}{x}^{n + 1},\;x \in \left( {-R,R}\right)\]若幂级数 \(\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}\) 在端点 \(x = R\) (或 \(x = - R\) ) 处收敛,则积分上限 \(x\) 可取为 \(x = R\) (或 \(x =\) \(- R)\)
4. 幂级数的运算
设 \(\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n} = f\left( x\right)\) 的收敛半径为 \({R}_{1},\mathop{\sum }\limits_{{n = 0}}^{\infty }{b}_{n}{x}^{n} = h\left( x\right)\) 的收敛半径为 \({R}_{2}\) ,则对于这两个幂级数可以进行下列四则运算:\[\left( {\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}}\right) \pm \left( {\mathop{\sum }\limits_{{n = 0}}^{\infty }{b}_{n}{x}^{n}}\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }\left( {{a}_{n} \pm {b}_{n}}\right) {x}^{n} = f\left( x\right) \pm h\left( x\right)\]收敛半径 \(R = \min \left\{ {{R}_{1},{R}_{2}}\right\}\)\[\left( {\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}}\right) \left( {\mathop{\sum }\limits_{{n = 0}}^{\infty }{b}_{n}{x}^{n}}\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }\left( {{a}_{0}{b}_{n} + {a}_{1}{b}_{n - 1} + \cdots + {a}_{n}{b}_{0}}\right) {x}^{n} = f\left( x\right) \vdot h\left( x\right)\]收敛半径 \(R = \min \left\{ {{R}_{1},{R}_{2}}\right\}\)\[\frac{\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}}{\mathop{\sum }\limits_{{n = 0}}^{\infty }{b}_{n}{x}^{n}} = \mathop{\sum }\limits_{{n = 0}}^{\infty }{c}_{n}{x}^{n}\;\left( {\text{ 其中 }{b}_{0} \neq 0}\right)\]系数 \({c}_{n}\) 可由幂级数的乘法 \(\left( {\mathop{\sum }\limits_{{n = 0}}^{\infty }{b}_{n}{x}^{n}}\right) \left( {\mathop{\sum }\limits_{{n = 0}}^{\infty }{c}_{n}{x}^{n}}\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}\) ,并比较同次幂的系数得到. 相除后得到的幂级数 \(\mathop{\sum }\limits_{{n = 0}}^{\infty }{c}_{n}{x}^{n}\) 的收敛区间可能比原来两个幂级数的收敛区间小得多
5. 函数展开成幂级数
1. 泰勒 (Taylor) 级数
当 \(f\left( x\right)\) 在 \({x}_{0}\) 的某邻域内存在任意阶导数时,幂级数\[\mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{f}^{\left( n\right) }\left( {x}_{0}\right) }{n!}{\left( x - {x}_{0}\right) }^{n}\]\[= f\left( {x}_{0}\right) + \frac{{f}^{\prime }\left( {x}_{0}\right) }{1!}\left( {x - {x}_{0}}\right) + \frac{{f}^{\prime \prime }\left( {x}_{0}\right) }{2!}{\left( x - {x}_{0}\right) }^{2} + \cdots + \frac{{f}^{\left( n\right) }\left( {x}_{0}\right) }{n!}{\left( x - {x}_{0}\right) }^{n} + \cdots\]称为 \(f\left( x\right)\) 在点 \({x}_{0}\) 处的泰勒级数
当 \({x}_{0} = 0\) 时,泰勒级数为\[\mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{f}^{\left( n\right) }\left( 0\right) }{n!}{x}^{n} = f\left( 0\right) + {f}^{\prime }\left( 0\right) x + \frac{{f}^{\prime \prime }\left( 0\right) }{2!}{x}^{2} + \cdots + \frac{{f}^{\left( n\right) }\left( 0\right) }{n!}{x}^{n} + \cdots\]称为麦克劳林 (Maclaurin) 级数
2. 函数的幂级数展开式
函数展为幂级数有直接方法与间接方法两种 (以下主要讨论函数 \(f\left( x\right)\) 在点 \(x = 0\) 处展为幂级数的问题)
直接法 用直接法将函数展开为 \(x\) 的幂级数的步骤是
(1)求出 \(f\left( x\right)\) 在 \(x = 0\) 处各阶导数值 \({f}^{\left( n\right) }\left( 0\right) ,n = 0,1,2,\cdots\)
(2)写出幂级数\[\mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{f}^{\left( n\right) }\left( 0\right) }{n!}{x}^{n} = f\left( 0\right) + {f}^{\prime }\left( 0\right) x + \frac{{f}^{\prime \prime }\left( 0\right) }{2!}{x}^{2} + \cdots + \frac{{f}^{\left( n\right) }\left( 0\right) }{n!}{x}^{n} + \cdots\]并求出收敛半径 \(R\)
(3)在收敛区间(-R, R)内考察泰勒级数余项 \({R}_{n}\left( x\right)\) 的极限\[\mathop{\lim }\limits_{{n \rightarrow \infty }}{R}_{n}\left( x\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{{f}^{\left( n + 1\right) }\left( \xi \right) }{\left( {n + 1}\right) !}{x}^{n + 1}\;\left( {\xi \text{ 介于 }0\text{ 与 }x\text{ 之间 }}\right)\]是否为零,如果为零,则第 (2) 步写出的幂级数就是 \(f\left( x\right)\) 的幂级数展开式
间接法 这种方法是利用已知的函数展开式, 经过适当的四则运算、复合步骤以及逐项微分、逐项积分等把所给函数展为幂级数. 常用的函数展开式见需熟记的重要公式——泰勒公式及常用泰勒展开。
6. 傅立叶级数
1. 函数的傅立叶 (Fourier) 级数
设函数 \(f\left( x\right)\) 在区间 \(\left\lbrack {-\pi ,\pi }\right\rbrack\) (或 \(\left\lbrack {0,{2\pi }}\right\rbrack\) ) 上可积,则称\[{a}_{n} = \frac{1}{\pi }{\int }_{-\pi }^{\pi }f\left( x\right) \cos {nx}\mathrm{\;d}x,\;n = 0,1,2,\cdots\]\[{b}_{n} = \frac{1}{\pi }{\int }_{-\pi }^{\pi }f\left( x\right) \sin {nx}\mathrm{\;d}x,\;n = 1,2,\cdots\]为函数 \(f\left( x\right)\) 的傅立叶系数. 由上述 \({a}_{n},{b}_{n}\) 所形成的三角级数\[\frac{{a}_{0}}{2} + \mathop{\sum }\limits_{{n = 1}}^{\infty }\left( {{a}_{n}\cos {nx} + {b}_{n}\sin {nx}}\right)\]称为函数 \(f\left( x\right)\) 的傅立叶级数
2. 狄立克莱 (Dirichlet) 定理
设函数 \(f\left( x\right)\) 在区间 \(\left\lbrack {-\pi ,\pi }\right\rbrack\) (或 \(\left\lbrack {0,{2\pi }}\right\rbrack\) ) 上满足条件:
(1)连续或只有有限个第一类间断点
(2)至多只有有限个极值点
则 \(f\left( x\right)\) 的傅立叶级数\[\frac{{a}_{0}}{2} + \mathop{\sum }\limits_{{n = 1}}^{\infty }\left( {{a}_{n}\cos {nx} + {b}_{n}\sin {nx}}\right)\]在区间 \(\left\lbrack {-\pi ,\pi }\right\rbrack\) (或 \(\left\lbrack {0,{2\pi }}\right\rbrack\) ) 上收敛,并且,若其和函数为 \(S\left( x\right)\) ,则有:
(1)在 \(f\left( x\right)\) 的连续点处, \(S\left( x\right) = f\left( x\right)\)
(2)在 \(f\left( x\right)\) 的间断点 \(x\) 处, \(\displaystyle S\left( x\right) = \frac{f\left( {x - 0}\right) + f\left( {x + 0}\right) }{2}\)
(3)在端点 \(x = \pm \pi\) 处, \(\displaystyle S\left( x\right) = \frac{f\left( {-\pi + 0}\right) + f\left( {\pi - 0}\right) }{2}\)
(或在 \(x = 0,{2\pi }\) 处, \(\displaystyle S\left( x\right) = \frac{f\left( {0 + 0}\right) + f\left( {{2\pi } - 0}\right) }{2}\) ),其中 \(f\left( {{x}_{0} - 0}\right) ,f\left( {{x}_{0} + 0}\right)\) 分别表示 \(f\left( x\right)\) 在 \({x}_{0}\) 处的左、右极限\[{a}_{n} = \frac{1}{\pi }{\int }_{-\pi }^{\pi }f\left( x\right) \cos {nx}\mathrm{\;d}x,\;n = 0,1,2,\cdots\]\[{b}_{n} = \frac{1}{\pi }{\int }_{-\pi }^{\pi }f\left( x\right) \sin {nx}\mathrm{\;d}x,\;n = 1,2,\cdots\]
3. 正弦级数
若 \(f\left( x\right)\) 是 \(\left( {-\pi ,\pi }\right)\) 上的奇函数,则\[f\left( x\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }{b}_{n}\sin {nx}\]其中 \({b}_{n} = \frac{2}{\pi }{\int }_{0}^{\pi }f\left( x\right) \sin {nx}\mathrm{\;d}x,\;n = 1,2,\cdots\)
4. 余弦级数
若 \(f\left( x\right)\) 是 \(\left( {-\pi ,\pi }\right)\) 上的偶函数,则\[f\left( x\right) = \frac{{a}_{0}}{2} + \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}\cos {nx}\]其中 \({a}_{n} = \frac{2}{\pi }{\int }_{0}^{\pi }f\left( x\right) \cos {nx}\mathrm{\;d}x,\;n = 0,1,2,\cdots\)
7. 一般周期函数的傅立叶级数
任意区间 \(\left\lbrack {-l,l}\right\rbrack\) 上的傅立叶级数
设函数 \(f\left( x\right)\) 在长为 \({2l}\) 的区间 \(\left\lbrack {-l,l}\right\rbrack\) (或 \(\left\lbrack {0,{2l}}\right\rbrack\) ) 上满足狄立克莱定理条件,作变量置换 \(\displaystyle t =\frac{\pi x}{l}\) ,则函数 \(\displaystyle f\left( x\right) = f\left( {\frac{l}{\pi }t}\right) = \varphi \left( t\right)\) ,而 \(\varphi \left( t\right)\) 在区间 \(\left\lbrack {-\pi ,\pi }\right\rbrack\) 上满足狄立克莱定理条件,所以 \(f\left( x\right)\) 在 \(\left\lbrack {-l,l}\right\rbrack\) 上的傅立叶级数 \(\displaystyle \frac{{a}_{0}}{2} + \mathop{\sum }\limits_{{n = 1}}^{\infty }\left( {{a}_{n}\cos \frac{n\pi }{l}x + {b}_{n}\sin \frac{n\pi }{l}x}\right)\) 收敛,并且,若其和函数为 \(S\left( x\right)\) ,则有
(1)在 \(f\left( x\right)\) 的连续点处, \(S\left( x\right) = f\left( x\right)\)
(2)在 \(f\left( x\right)\) 的间断点 \(x\) 处, \(\displaystyle S\left( x\right) = \frac{f\left( {x - 0}\right) + f\left( {x + 0}\right) }{2}\)
(3)在端点 \(x = \pm l\) 处, \(\displaystyle S\left( x\right) = \frac{f\left( {-l + 0}\right) + f\left( {l - 0}\right) }{2}\)
\(\displaystyle \left( {\text{ 或在 }x = 0,{2l}}\right.\) 处, \(\left. {S\left( x\right) = \frac{f\left( {0 + 0}\right) + f\left( {{2l} - 0}\right) }{2}}\right)\)
其中\[{a}_{n} = \frac{1}{l}{\int }_{-l}^{l}f\left( x\right) \cos \frac{n\pi x}{l}\mathrm{\;d}x,\;n = 0,1,2,\cdots\]\[{b}_{n} = \frac{1}{l}{\int }_{-l}^{l}f\left( x\right) \sin \frac{n\pi x}{l}\mathrm{\;d}x,\;n = 1,2,\cdots\]可见,任意区间 \(\left\lbrack {-l,l}\right\rbrack\) 上的傅立叶级数是区间 \(\left\lbrack {-\pi ,\pi }\right\rbrack\) 上的傅立叶级数的推广. 而区间 \(\left\lbrack {-\pi ,\pi }\right\rbrack\) 上的傅立叶级数是区间 \(\left\lbrack {-l,l}\right\rbrack\) 上的傅立叶级数的特殊情况