671 lines
32 KiB
TeX
671 lines
32 KiB
TeX
\chapter{函数的积分}
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\section{积分的概念}
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问题来源：求曲边梯形的面积。方法：将所求区域分割为小长方形、将面积叠加来逼近。
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\begin{definition}
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设$f: [a,b] \to \realnum$，如果$\exists A \in \realnum$使得$\forall \varepsilon > 0, \exists \delta > 0$，对于区间$[a,b]$上任意有限分割$T: a = x_0 < x_1 < x_2 < \cdots < x_n = b$，记分割宽度$\Vert T \Vert = \max \limits_{i = 1, 2, \cdots, n} \Delta x_i, \Delta x_i = x_i - x_{i-1}, i = 1, 2, \cdots, n$。只要$\Vert T \Vert < \delta$，都有
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\[\left| \sum_{i = 1}^n f(c_i) \Delta x_i = A \right| < \varepsilon\]
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其中$c_i \in [x_{i-1}, x_i]$任取，$i = 1, 2, \cdots, n$。
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这时称$f$在$[a,b]$上Riemann可积，记为$f \in R[a,b]$，并记
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\[\tolim{\Vert T \Vert}{0} \sum_{i=1}^n f(c_i) \Delta x_i = A = \int_a^b f(x) \dif x\]
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称为$f$在$[a,b]$上的积分值，$a$称为积分下限，$b$称为积分上限。
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\end{definition}
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\begin{proposition}[积分的线性性质]
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设$f, g \in R[a, b]$，$\alpha, \beta \in \realnum$，则
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\[\alpha f + \beta g \in R[a,b]\]
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且
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\[\int_a^b [\alpha f(x) + \beta g(x)] \dif x = \alpha \int_a^b f(x) \dif x + \beta \int_a^b g(x) \dif x\eqper\]
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\end{proposition}
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\begin{proof}
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任取$[a,b]$上有限分割$T: a = x_0 < x_1 < x_2 < \cdots < x_n = b$。令$\Delta x_i = x_i - x_{i-1}$，$\forall c_i \in [x_{i-1}, x_i], i = 1, 2, \cdots, n$，考虑相应的和式
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\[\sum_{i=1}^n [\alpha f(c_i) + \beta g(c_i)] \Delta x_i = \alpha \sum_{i=1}^n f(c_i) \Delta x_i + \beta \sum_{i=1}^n g(c_i) \Delta x_i\]
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设$\Vert T \Vert \max \limits_{i = 1, 2, \cdots, n} \Delta x_i$。令$\Vert T \Vert \to 0$。等式右侧有极限$\alpha \dint_a^b f(c_i) \Delta x_i + \beta \dint_a^b g(c_i) \Delta x_i$。因此左侧有相同的极限，依照定义$\alpha f + \beta g \in R[a,b]$且
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\[\int_a^b [\alpha f(x) + \beta g(x)] \dif x = \alpha \int_a^b f(x) \dif x + \beta \int_a^b g(x) \dif x\eqper\]
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\end{proof}
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\begin{proposition}[积分的保号性质]
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设$f \in R[a,b]$，且$f \geq 0$，则
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\[\int_a^b f(x) \dif x \geq 0 \eqper\]
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\end{proposition}
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\begin{corollary}[积分的保序性质]
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设$f, g \in R[a,b]$，且$f \geq g$，则
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\[\int_a^b f(x) \dif x \geq \int_a^b g(x) \dif x \eqper\]
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\end{corollary}
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\begin{corollary}
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设$\abs{f}, f \in R[a,b]$，则
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\[\left| \int_a^b f(x) \dif x \right| \leq \int_a^b \vert f(x) \vert \dif x \eqper\]
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\end{corollary}
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\begin{proposition}[积分的区间可加性质]
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设$a < c < b$，则
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\[f \in R[a,b] \Leftrightarrow f \in R[a,c] \text{且} f \in R[c,b]\]
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这时有
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\[\int_a^b f(x) \dif x = \int_a^c f(x) \dif x + \int_c^b f(x) \dif x\eqper\]
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\end{proposition}
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我们规定：
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\[\int_a^a f(x) \dif x = 0,\quad \int_b^a f(x) \dif x = -\int_a^b f(x) \dif x\eqper\]
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\begin{corollary}
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\[\int_a^b f(x) \dif x + \int_b^a f(x) \dif x = \int_a^a f(x) \dif x\eqper\]
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\end{corollary}
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\begin{corollary}
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\[\int_a^c f(x) \dif x = \int_a^b f(x) \dif x + \int_b^c f(x) \dif x\eqper\]
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\end{corollary}
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\section{定积分的计算}
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\begin{enumerate}
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\item 利用定义：为简化计算通常取均匀分割
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\[T: \Delta x = \frac{b-a}{n}, x_i = a + i\Delta x, i = 0, 1, \cdots, n\]
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则可以取$c_i = x_i$，于是有
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\[\int_a^b f(x) \dif x = \tolim{n}{\infty} \sum_{i=1}^n f(x_i) \Delta x\]
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\item 借助几何定义：$f(x)$在$x$轴上方围成的面积$-$ $f(x)$在$x$轴下方围成的面积。
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\end{enumerate}
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\begin{example}
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$f_1(x) \equiv c, x \in [a,b]$，求$\int_a^b f(x) \dif x$。
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\end{example}
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\begin{proof}[解]
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取分割$\Delta x = \dfrac{b-a}{n}$，$x_i = a + i \Delta x$。则分割宽度$\Vert T \Vert = \dfrac{b-a}{n}$。因此$\Vert T \Vert \to 0 \Leftrightarrow n \to \infty$。考察分割时函数的和式
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\[\sum_{i=1}^n f(x_i) \Delta x = \sum_{i=1}^n c \Delta x = c\sum_{i=1}^n \Delta x = c(b-a)\]
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因此
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\[\int_a^b f(x) \dif x = \tolim{n}{\infty} \sum_{i=1}^n f(x_i) \Delta x = c(b-a)\]
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即
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\[\int_a^b c \dif x = c(b-a) \eqper \qedhere\]
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\end{proof}
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\begin{corollary}[积分的估值性质]
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设$f \in R[a,b]$且$m \leq f(x) \leq M$，则
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\[m(b-a) \leq \int_a^b f(x) \dif x \leq M(b-a)\eqper\]
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\end{corollary}
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\begin{theorem}[Newton-Leibniz公式]
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设函数$f$在$[a,b]$上可积，且在$(a,b)$上有原函数$F$，如果$F$在$(a,b)$上连续，那么必有
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\[\int_a^b f(x) \dif x = F(b) - F(a)\eqper\]
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其中$F(b) - F(a)$也记为$\eval{F(x)}_a^b$。
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\end{theorem}
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上面的定理中的式子也可以写成：
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\[\int_a^b \deriv{F}(x) \dif x = \int_a^b \dif F(x) = \eval{F(x)}_a^b\eqper\]
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\begin{proof}
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取$[a,b]$上均匀分割$T: a = x_0 < x_1 < x_2 < \cdots < x_n = b$，其中$x_i = a + i \Delta x, i = 0,1, \cdots, n, \Delta x = \dfrac{b - a}{n}$。这时有
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\[F(b) - F(a) = \sum_{i=1}^n [F(x_i) - F(x_{i-1})]\]
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在每个区间上应用Lagrange中值定理，得
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\[F(x_i) - F(x_{i-1}) = \deriv{F}(c_i) \Delta x, c_i \in (x_{i-1}, x_i)\]
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已知$\deriv{F} = f \in R[a,b]$，因此
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\[\tolim{n}{\infty} \sum_{i=1}^n f(c_i) \Delta x = \int_a^b f(x) \dif x\]
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这说明
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\[F(b) - F(a) = \int_a^b f(x) \dif x\eqper \qedhere\]
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\end{proof}
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因此可以将求可积函数$f$的积分问题转化为求$f$的原函数的问题。
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\section{可积函数的性质}
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目的：寻找可积函数的特性，寻找函数可积的判别条件。
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\begin{proposition}[可积函数的有界性]
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若$f \in R[a,b]$，则$f$在$[a,b]$上有界。
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\end{proposition}
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\begin{proof}
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令$\int_a^b f(x) \dif x = A$，根据积分的定义，存在均匀分割
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\[T: x_i = a + i\Delta x, i = 0, 1, \cdots, n, \Delta x = \frac{b - a}{n}\]
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使得
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\[\left|\sum_{i=1}^n f(c_i) \Delta x - A\right| < 1, c_i \in [x_{i-1}, x_i]\text{任取}, i = 1, 2, \cdots, n\]
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由此导出
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\[\left|\sum_{i=1}^n f(c_i) \Delta x \right| < \left|\sum_{i=1}^n f(c_i) \Delta x - A\right| + \abs{A} < 1 + \abs{A}\]
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因此
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\[\left|\sum_{i=1}^n f(c_i) \right| \leq \frac{1 + \abs{A}}{\Delta x}\]
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进一步有
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\[\vert f(c_i) \vert \leq \left| \sum_{i=1}^n f(c_i) \right| + \left| \sum_{i=2}^n f(c_i)\right| \leq \frac{1 + \abs{A}}{\Delta x} + \left| \sum_{i=2}^n f(x_i) \right|\]
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即
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\[\vert f(x) \vert \leq \frac{1 + \abs{A}}{\Delta x} + \left|\sum_{i\neq 1}^n f(x_i)\right| = M_1, \forall x \in [x_0, x_1]\]
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类似地可以得到
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\begin{align*}
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\vert f(x) \vert \leq \frac{1 + \abs{A}}{\Delta x} + \left|\sum_{i\neq 2}^n f(x_i)\right| & = M_2, \forall x \in [x_1, x_2]\\
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\vdots & \\
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\vert f(x) \vert \leq \frac{1 + \abs{A}}{\Delta x} + \left|\sum_{i\neq n}^n f(x_i)\right| & = M_n, \forall x \in [x_{n-1}, x_n]
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\end{align*}
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将这$n$个式子综合起来就可以得到
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\[\vert f(x) \vert \leq \max \{M_1, \cdots, M_n\}, \forall x \in [a,b] \eqper \qedhere\]
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\end{proof}
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\begin{remark}
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有界是可积的必要条件，但不是充分条件。$f$在$[a,b]$上有界未必可积，例如Dirichlet函数$D(x)$。
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\end{remark}
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\begin{proposition}[连续函数的可积性]
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$C[a,b] \subset R[a,b]$。
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\end{proposition}
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\begin{corollary}[积分中值定理]
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设$f,g \in C[a,b]$，且$g(x)$不变号，则$\exists c \in [a,b]$使
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\[\int_a^b f(x)g(x) \dif x = f(c) \int_a^b g(x) \dif x\eqper\]
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特例：设$f \in C[a,b]$，则$\exists c_0 \in [a,b]$使得
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\[f(c_0) = \frac{1}{b-a} \int_a^b f(x) \dif x\]
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这称为$f$在区间$[a,b]$上的平均值。
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\end{corollary}
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\begin{proof}
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不妨设$g(x) \geq 0$，由保号性$\int_a^b g(x) \dif x \geq 0$。同时已知$f \in C[a,b]$，根据最值性质，存在$m = \min \limits_{a \leq x \leq b} f(x) , M = \max \limits_{a \leq x \leq b} f(x)$。这导出
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\[mg(x) \leq f(x)g(x) \leq Mg(x), x \in [a,b]\]
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应用积分的保序性与线性性质有
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\[m \int_a^b g(x) \dif x \leq \int_a^b f(x)g(x) \dif x \leq M \int_a^b g(x) \dif x\]
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不妨设$\dint_a^b g(x) \dif x > 0$，那么上式导出
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\[m \leq \frac{\dint_a^b f(x)g(x) \dif x}{\dint_a^b g(x) \dif x} \leq M\]
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由连续函数介质性质$\exists c \in [a,b]$使得
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\[f(c) = \frac{\dint_a^b f(x)g(x) \dif x}{\dint_a^b g(x) \dif x} \eqper \qedhere\]
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\end{proof}
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\section{微积分基本定理}
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变上限积分：设$f \in R[a,b]$，则$\forall x \in [a,b], f \in R[a,x]$。定义
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\[F(x) = \int_a^x f(t) \dif t\]
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由此得到函数$F: [a,b] \to \realnum$。
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记号约定：
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\[C^n (a,b) = \{f \in C(a,b) \mid f^{(n)} \in C(a,b)\}\\
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C^n [a,b] = \{f \in C[a,b] \mid f^{(n)} \in C[a,b]\}\]
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\begin{theorem}
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设$f$在$[a,b]$上可积，那么带变动上限的积分$F(x) = \int_a^x f(t) \dif t$在$[a,b]$上连续。
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\end{theorem}
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\begin{proof}
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任取$x, x + \Delta x \in [a,b], \Delta x > 0$
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\begin{align*}
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F(x + \Delta x) - F(x) & = \int_a^{x+\Delta x} f(t) \dif t - \int_a^x f(t) \dif t\\
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& = \int_a^{x + \Delta x} f(t) \dif t + \int_x^a f(t) \dif t \\
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& = \int_x^{x + \Delta x} f(t) \dif t
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\end{align*}
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已知可积函数有界，再由估值不等式得到
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\[\vert F(x + \Delta x) - F(x) \vert \leq \int_x^{x + \Delta x} \vert f(t) \vert \dif t \leq M \Delta x\]
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因此
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\[\tolim{\Delta x}{0} \vert F(x + \Delta x) - F(x) \vert = 0\]
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因此$F(x)$连续。
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\end{proof}
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\begin{theorem}
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设函数$f$在$[a,b]$上可积且连续，那么$F$可导且
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\[\deriv{F}(x) = f(x)\eqper\]
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\end{theorem}
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\begin{proof}
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已知
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\[F(x + \Delta x) - F(x) = \int_x^{x + \Delta x} f(t) \dif t\]
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若$f \in C[a,b]$，则可应用积分中值定理：$\exists c \in (x, x + \Delta x)$满足
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\[F(x + \Delta x) - F(x) = \int_x^{x + \Delta x} f(t) \dif t = f(c) \Delta x\]
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即
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\[\frac{F(x + \Delta x) - F(x)}{\Delta x} = f(c) \quad (\Delta x \neq 0)\]
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令$\Delta x \to 0$，则$c \to x$，进而$f(c) \to f(x)$。
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\end{proof}
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结论：设$f \in R[a,b]$，则$F(x) = \int_a^x f(t) \dif t \in C[a,b]$。又若$f \in C[a,b]$，则$F \in C^1 [a,b]$且$\deriv{F}(x) = f(x)$，$\forall x \in [a,b]$。这就是下面的
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\begin{theorem}[微积分基本定理]
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设$f \in C[a,b]$，则
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\[\frac{\dif}{\dif x} \int_a^x f(t) \dif t = f(x), \forall x \in [a,b]\]
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或写成
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\[\dif \int_a^x f(t) \dif t = f(x) \dif x, \forall x \in [a,b] \eqper\]
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\end{theorem}
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\begin{theorem}[Newton-Leibniz公式]
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设$F \in C^1 [a,b]$，则
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\[\int_a^b \deriv{F}(x) \dif x = \eval{F(x)}_a^b\]
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或写成
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\[\int_a^b \dif F(x) = \eval{F(x)}_a^b\eqper\]
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\end{theorem}
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\begin{remark}
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考虑变下限积分的导数：若$f$连续，
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\[\frac{\dif}{\dif x} \int_x^c f(t) \dif t = \frac{\dif }{\dif x} \left[-\int_c^x f(t) \dif t\right] = -f(x)\eqper\]
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\end{remark}
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\begin{example}
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求$\tolim{x}{0} \dfrac{1}{x^3} \dint_0^x tf(t) \dif t$，其中$f$满足$f(0) = 0$，且$\deriv{f}(0)$存在。
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\end{example}
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\begin{proof}[解]
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考虑应用L'Hospital法则：
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\[\frac{\dif}{\dif x} \int_0^x tf(t) \dif t = x f(x)\]
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因此
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\begin{align*}
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\text{所求} & = \tolim{x}{0} \frac{1}{\deriv{\left(x^3\right)}} \deriv{\left(\int_0^x tf(t) \dif t\right)} = \tolim{x}{0} \frac{xf(x)}{3x^2}\\
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& = \frac{1}{3} \tolim{x}{0} \frac{f(x)}{x} = \frac{1}{3} \tolim{x}{0} \frac{f(x) - f(0)}{x - 0} = \frac{\deriv{f}(0)}{3} \eqper\qedhere
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\end{align*}
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\end{proof}
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\begin{example}
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设$f \in C[a,b], u(t), v(t)$可导且$a \leq u(t), v(t) \leq b$。计算导数
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\[\frac{\dif }{\dif t}\int_{v(t)}^{u(t)} f(x) \dif x\eqper\]
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\end{example}
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\begin{proof}[解]
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记
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\[F(u) = \int_a^u f(x) \dif x\]
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则$\deriv{F}(u) = f(u)$。
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因此有
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\begin{align*}
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\int_v^u f(x) \dif x & = \int_v^a f(x) \dif x + \int_a^u f(x) \dif x\\
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& = \int_a^u f(x) \dif x - \int_a^v f(x) \dif x\\
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& = F(u) - F(v)
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\intertext{应用链式法则}
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\frac{\dif }{\dif t} \int_{v(t)}^{u(t)} f(x) \dif x & = \frac{\dif}{\dif t} [F(u) - F(v)]\\
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& = \deriv{F}(u)\deriv{u} - \deriv{F}(v) \deriv{v}\\
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& = f(u(t))\deriv{u}(t) - f(v(t))\deriv{v}(t) \eqper\qedhere
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\end{align*}
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\end{proof}
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\section{分部积分与换元}
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\subsection{分部积分}
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分部积分公式
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\[\int_a^b u(x) \deriv{v}(x) \dif x = \eval{u(x) v(x)}_a^b - \int_a^b v(x) \deriv{u}(x) \dif x\]
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或者可以写成
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\[\int_a^b u(x) \dif [v(x)] = \eval{u(x) v(x)}_a^b - \int_a^b v(x) \dif [u(x)]\]
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\begin{example}
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求$I = \dint_0^\pi x^3 \sin x \dif x$。
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\end{example}
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\begin{proof}[解]
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考虑应用分部积分。取$u = x^3, v = -\cos x$。
|
||
\begin{align*}
|
||
I & = \eval{-x^3 \cos x}_0^\pi + 3 \int_0^\pi x^2 \cos x \dif x = \pi^3 + 3 \int_0^\pi x^2 \cos x \dif x\\
|
||
& = \pi^3 + 3\left(\eval{x^2 \sin x}_0^\pi - 2\int_0^\pi x \sin x \dif x\right)\\
|
||
& = \pi^3 + 3\times 2 \int_0^\pi x \dif \cos x\\
|
||
& = \pi^3 + 6 \left(\eval{x \cos x}_0^\pi - \int_0^\pi \cos x \dif x\right)\\
|
||
& = \pi^3 + 6 \left(-\pi - \eval{\sin x}_0^\pi\right)\\
|
||
& = \pi^3 - 6\pi \eqper\qedhere
|
||
\end{align*}
|
||
\end{proof}
|
||
|
||
分部积分的应用：设$f$在$[a,b]$上所需的导数均存在。那么做分部积分
|
||
\begin{equation*}
|
||
f(x) = f(a) + \int_a^x \deriv{f}(t) \dif t \tag{1} \label{积分Taylor公式1}
|
||
\end{equation*}
|
||
同时注意到
|
||
\begin{align*}
|
||
\int_a^x f(t) \dif t & = \int_a^x \deriv{f}(t) \dif (t - x)\\
|
||
& = \eval{(t - x) \deriv{f}(t)}_a^x - \int_a^x (t-x)f^{\prime \prime}(t) \dif t\\
|
||
& = (x-a) \deriv{f}(a) - \int_a^x (t-x)f^{\prime \prime} (t) \dif t \tag{2} \label{积分Taylor公式2}
|
||
\end{align*}
|
||
|
||
将\eqref{积分Taylor公式2}式带回\eqref{积分Taylor公式1}式得到
|
||
\[f(x) = f(a) + \deriv{f}(a) (x-a) + \int_a^x (x-t)f^{\prime \prime} (t) \dif t\]
|
||
|
||
类似地有
|
||
\begin{align*}
|
||
\int_a^x (x-t)f^{\prime \prime} (t) \dif t & = -\frac{1}{2} \int_a^x f^{\prime \prime} (t) \dif \left[(t-x)^2\right]\\
|
||
& = -\frac{1}{2} \left(\eval{\left((t-x)^2 f^{\prime \prime} (t)\right)}_a^x - \int_a^x (t-x)^2 f^{\prime \prime \prime} (t) \dif t\right)\\
|
||
& = \frac{1}{2} (x-a)^2 f^{\prime \prime} (a) + \frac{1}{2} \int_a^x (t-x)^2 f^{\prime \prime \prime} (t) \dif t \tag{3} \label{积分Taylor公式3}
|
||
\end{align*}
|
||
|
||
将\eqref{积分Taylor公式3}式带回\eqref{积分Taylor公式2}式中，有
|
||
\[f(x) = f(a) + \deriv{f}(a) (x-a) + \frac{1}{2} (x-a)^2 f^{\prime \prime}(a) + \frac{1}{2}\int_a^x (x-t)^2 f^{\prime \prime \prime}(t) \dif t\]
|
||
不断如此重复，我们有
|
||
\begin{theorem}[带积分余项的Taylor公式]
|
||
设函数$f$在$(a,b)$上有直到$n+1$阶的连续导函数，那么对任意固定的$x_0 \in (a,b)$，我们有
|
||
\[f(x) = f(x_0) + \frac{1}{1!}\deriv{f}(x_0) (x-x_0) + \cdots + \frac{1}{n!}f^{(n)}(x_0)(x-x_0)^n + R_n(x)\]
|
||
其中
|
||
\[R_n(x) = \frac{1}{n!}\int_{x_0}^x (x-t)^n f^{(n+1)}(t) \dif t, x \in (a, b)\eqper\]
|
||
\end{theorem}
|
||
|
||
\subsection{定积分换元法}
|
||
\begin{theorem}
|
||
设$f \in C(I)$，$I$为区间，$\varphi \in C^1 [\alpha, \beta]$，且$\varphi([\alpha, \beta]) \subset I$。令$\varphi(\alpha) = a, \varphi(\beta) = b$，则
|
||
\[\int_a^b f(x) \dif x = \int_\alpha^\beta f(\varphi(t))\deriv{\varphi}(t) \dif t\eqper\]
|
||
\end{theorem}
|
||
|
||
\begin{remark}
|
||
不定积分换元公式中$x = \varphi(t)$要存在反函数，而定积分的换元$x = \varphi(t)$并不要求单调性或反函数的存在性，只要函数$\varphi$的值域落在$f$的连续区间内（不必完全在$[a,b]$内）。
|
||
\end{remark}
|
||
|
||
\begin{proof}
|
||
记$F(u) = \dint_a^u f(x) \dif x$，则$\deriv{F}(u) = f(u) \in C(I)$。已知$\varphi \in C^1 [a,b]$，且$\varphi([\alpha, \beta]) \subset I$，应用链式法则求导公式
|
||
\[\frac{\dif}{\dif t}[F(\varphi(t))] = \deriv{F}(\varphi(t))\deriv{\varphi}(t) \in C[\alpha, \beta]\]
|
||
再应用Newton-Leibniz公式
|
||
\[\int_\alpha^\beta f(\varphi(t))\deriv{\varphi}(t) \dif t = \eval{F(\varphi(t))}_\alpha^\beta = \int_{\varphi(\alpha)}^{\varphi(\beta)} f(x) \dif x = \int_a^b f(x) \dif x\eqper\qedhere\]
|
||
\end{proof}
|
||
|
||
\subsubsection{对称区间积分}
|
||
设$f \in R[-a,a]$。由区间可加性
|
||
\[\int_{-a}^a f(x) \dif x = \int_0^a f(x) \dif x + \int_{-a}^0 f(x) \dif x \tag{1} \label{对称区间积分1}\]
|
||
在区间$[-a,0]$的积分中引入换元$x = -t$，则
|
||
\[\dif x = - \dif t, x(0) = 0, x(-a) = a\]
|
||
因此
|
||
\[\int_{-a}^0 f(x) \dif x = - \int_a^0 f(-t) \dif t = \int_0^a f(-x) \dif x \tag{2} \label{对称区间积分2}\]
|
||
将\eqref{对称区间积分2}式带入\eqref{对称区间积分1}式，得到
|
||
\[\int_{-a}^a f(x) \dif x = \int_0^a f(x) \dif x + \int_0^a f(-x) \dif x\eqper\]
|
||
因此对函数$f \in [-a,a]$
|
||
\begin{enumerate}
|
||
\item 若$f$为奇函数，则$\dint_{-a}^a f(x) \dif x = 0$；
|
||
\item 若$f$为偶函数，则$\dint_{-a}^a f(x) \dif x = 2 \dint_0^a f(x) \dif x$。
|
||
\end{enumerate}
|
||
|
||
\subsubsection{周期函数积分}
|
||
设$f \in C(-\infty, + \infty)$有周期$T > 0$。由区间可加性
|
||
\[\int_a^{a+T} f(x) \dif x = \int_a^0 f(x) \dif x + \int_0^T f(x) \dif x + \int_T^{a+T} f(x) \dif x \tag{1} \label{周期函数积分1}\]
|
||
引入换元$x = t + T$，那么
|
||
\[\dif x = \dif t, x(0) = T, x(a) = a + T\]
|
||
因此
|
||
\[\int_T^{a+T} f(x) \dif x = \int_0^a f(t + T) \dif t = \int_0^a f(t) \dif t \tag{2} \label{周期函数积分2}\]
|
||
将\eqref{周期函数积分2}式带入\eqref{周期函数积分1}式中，得到
|
||
\[\int_a^{a+T} f(x) \dif x = \int_a^0 f(x) \dif x + \int_0^T f(x) \dif x + \int_0^a f(x) \dif x = \int_0^T f(x) \dif x \text{对} \forall a \in \realnum\text{成立。}\]
|
||
|
||
\section{Riemann可积函数理论}
|
||
目标：寻找函数Riemann可积的充分必要条件。
|
||
假设$f: [a,b] \to \realnum$有界，记
|
||
\[M = \sup \limits_{a \leq x \leq b} f(x), \quad m = \inf \limits_{a \leq x \leq b} f(x)\]
|
||
记$\omega = M - m$，称为$f$在$[a,b]$上的振幅。
|
||
|
||
对分割$T: a = x_0 < x_1 < \cdots < x_n = b$，记
|
||
\[\Delta x_i = x_i - x_{i-1}, i = 1, 2, \cdots, n \eqco \Vert T \Vert = \max \limits_{i=1, 2, \cdots, n} \Delta x_i\]
|
||
在每个子区间上，分别记函数的上下确界和振幅为
|
||
\[M_i = \sup \limits_{[x_{i-1}, x_i]} f\eqco m_i = \inf \limits_{[x_{i-1}, x_i]} f \eqco \omega_i = M_i - m_i\]
|
||
|
||
同时我们定义Darboux上下和：
|
||
\[\text{上和：}\overline{S}(f, T) = \sum_{i=1}^n M_i \Delta x_i \eqco \text{下和：}\underline{S}(f, T) = \sum_{i=1}^n m_i \Delta x_i\eqper\]
|
||
|
||
\begin{corollary}
|
||
\[0 \leq \overline{S}(f, T) - \underline{S}(f, T) = \sum_{i=1}^n \omega_i \Delta x_i\eqper\]
|
||
\end{corollary}
|
||
|
||
再记Riemman和为
|
||
\[S(f,T) = \sum_{i=1}^n f(c_i) \Delta x_i, c_i \in [x_{i-1}, x_i]\text{任取}, i = 1, 2, \cdots, n\]
|
||
|
||
\begin{corollary}
|
||
\[\underline{S}(f,T) \leq S(f, T) \leq \overline{S}(f, T) \eqper\]
|
||
\end{corollary}
|
||
|
||
在$x_0$与$x_1$之间多加一个分割点$t$，获得加细的分割$T^\prime: a = x_0 < t < x_1 < x_2 < \cdots < x_n = b$记
|
||
\[M_1^\prime = \sup \limits_{[x_{0}, t]}\eqco m_1^\prime = \inf \limits_{[x_0,t]} f\eqco M_2^\prime = \sup \limits_{[t,x_1]} f\eqco m_2^\prime \inf_{[t,x_1]} f\]
|
||
易得
|
||
\[M_1^\prime, M_2^\prime \leq M_1, m_1 \leq m_1^\prime, m_2^\prime\]
|
||
分割加细之后的Darboux和
|
||
\begin{align*}
|
||
\overline{S}(f, T^\prime) & = M_1^\prime (t-x_0) + M_2^\prime (x_1 - t) + \sum_{i=2}^n M_i \Delta x_i\\
|
||
& \leq M_1 (t-x_0) + M_1(x_1 - t) + \sum_{i=2}^n M_i \Delta x_i\\
|
||
& = \overline{S}(f,T)\\
|
||
\underline{S}(f, T^\prime) & = m_1^\prime (t-x_0) + m_2^\prime (x_1 - t) + \sum_{i=2}^n m_i \Delta x_i\\
|
||
& \geq m_1 (t-x_0) + m_1(x_1 - t) + \sum_{i=2}^n m_i \Delta x_i\\
|
||
& = \underline{S}(f, T)
|
||
\end{align*}
|
||
|
||
\begin{corollary}
|
||
对任意的加细分割$T^\prime$有
|
||
\[\overline{S}(f, T^\prime) \leq \overline{S}(f, T)\eqco \underline{S}(f, T^\prime) \geq \underline{S}(f, T)\eqper\]
|
||
\end{corollary}
|
||
|
||
\begin{corollary}
|
||
设$T_1, T_2$为$[a,b]$上的任意两个有限分割，则
|
||
\[m(b-a) \leq \underline{S}(f, T_1) \leq \overline{S}(f, T_2) \leq M(b-a)\eqper\]
|
||
\end{corollary}
|
||
|
||
\begin{proof}
|
||
记$T_0$是区间$[a,b]$两个端点构成的分割，再记$T_1 + T_2$为$T_1$与$T_2$合成的分割。那么有
|
||
\begin{align*}
|
||
\underline{S}(f, T_0) \leq \underline{S}(f, T_1) & \leq \underline{S}(f, T_1 + T_2)\\
|
||
& \leq \overline{S}(f, T_1 + T_2) \leq \overline{S}(f, T_2) \leq \overline{S}(f, T_0)
|
||
\end{align*}
|
||
注意$\underline{S}(f, T_0) = m(b-a), \overline{S}(f, T_0) = M(b-a)$。
|
||
\end{proof}
|
||
|
||
我们引入上下积分：
|
||
\[\text{上积分：}\overline{\int_a^b} f(x) \dif x = \inf \limits_T \overline{S}(f, T)\]
|
||
\[\text{下积分：}\underline{\int_a^b} f(x) \dif x = \sup \limits_T \underline{S}(f, T)\]
|
||
|
||
\begin{corollary}\label{Riemann可积推论4}
|
||
设$T_1, T_2$为$[a,b]$上任意两个有限分割，则
|
||
\[\underline{S}(f, T_1) \leq \underline{\int_a^b} f(x) \dif x \leq \overline{\int_a^b}f(x) \dif x \leq \overline{S}(f, T_2)\]
|
||
\end{corollary}
|
||
|
||
\begin{proof}
|
||
依照上下界的定义，第一和第三个不等号成立。假设$\beta = \sup \limits_T \underline{S}(f, T) > \alpha = \inf \limits_T \overline{S}(f, T)$，记$2 \varepsilon = \beta - \alpha > 0$。由确界的定义，
|
||
\begin{equation*}
|
||
\left.
|
||
\begin{aligned}
|
||
\exists T_1 \text{使得} \underline{S}(f, T_1) \in (\beta - \varepsilon, \beta]\\
|
||
\exists T_2 \text{使得} \overline{S}(f, T_2) \in [\alpha, \alpha + \varepsilon)
|
||
\end{aligned}
|
||
\right\}
|
||
\overline{S}(f, T_2) \leq \underline{S}(f, T_1)
|
||
\end{equation*}
|
||
这与推论\ref{Riemann可积推论4}矛盾。
|
||
\end{proof}
|
||
|
||
\begin{theorem}[Darboux定理]
|
||
对$[a,b]$上的任意有界函数，有
|
||
\[\overline{\int_a^b}f(x) \dif x = \tolim{\Vert T \Vert}{0}\overline{S}(f, T) \eqco \underline{\int_a^b}f(x) \dif x = \tolim{\Vert T \Vert }{0} \underline{S}(f, T)\]
|
||
\end{theorem}
|
||
|
||
\begin{proof}
|
||
假设极限存在。由下界的定义，极限保序
|
||
\[\alpha = \overline{\int_a^b}f(x) \dif x \leq \tolim{\Vert T \Vert}{0} \overline{S}(f, T)\]
|
||
由下确界的性质，$\forall \varepsilon > 0$，$\exists T_1$使得$\alpha \leq \overline{S}(f, T_1) < \alpha + \varepsilon$。因此只要极限存在，必有
|
||
\[\tolim{\Vert T \Vert}{0} \overline{S}(f, T) = \alpha\]
|
||
\end{proof}
|
||
|
||
\begin{theorem}
|
||
设$f:[a,b] \to \realnum$有界，则以下结论等价：
|
||
\begin{enumerate}
|
||
\item $f \in R[a,b]$；
|
||
\item $\tolim{\Vert T \Vert}{0} \left[\overline{S}(f, t) - \underline{S}(f, T)\right] = \tolim{\Vert T \Vert}{0} \displaystyle\sum_{i=1}^n \omega_i\Delta x_i = 0$；
|
||
\item $\forall \varepsilon > 0$，存在分割$T$，使得$0 \leq \overline{S}(f, T) - \underline{S}(f, T) < \varepsilon$；
|
||
\item $\overline{\dint_a^b}f(x) \dif x = \underline{\dint_a^b} f(x) \dif x$。
|
||
\end{enumerate}
|
||
\end{theorem}
|
||
|
||
\begin{theorem}
|
||
设$f$是定义在$[a,b]$上的单调函数，则$f$在$[a,b]$上可积。
|
||
\end{theorem}
|
||
|
||
\begin{proof}
|
||
不妨令函数单调增，$f(b) > f(a)$。任取一个$[a,b]$上的分割$T$，由$f$的单调性
|
||
\begin{align*}
|
||
0 & \leq \sum_{i=1}^n \omega_i \Delta x_i = \sum_{i=1}^n (M_i - m_i) \Delta x_i = \sum_{i=1}^n [f(x_i) - f(x_{i-1})] \Delta x_i\\
|
||
& \leq \Vert T \Vert \sum_{i=1}^n [f(x_i) - f(x_{i-1})] = \Vert T \Vert [f(b) - f(a)]
|
||
\end{align*}
|
||
由此可得
|
||
\[\tolim{\Vert T \Vert}{0}\sum_{i=1}^n \omega_i \Delta x_i = 0\]
|
||
因此$f \in R[a,b]$。
|
||
\end{proof}
|
||
|
||
\begin{theorem}
|
||
设$f: [a,b] \to \realnum$是连续函数，则$f$在$[a,b]$上可积。
|
||
\end{theorem}
|
||
|
||
\begin{proof}
|
||
$f$在$[a,b]$上一致连续，因此$\forall \varepsilon > 0$，$\exists \delta > 0$使得$\forall x, b \in [a,b]$，只要$\vert x - y \vert < \delta$都有$\vert f(x) - f(y) \vert < \varepsilon$。
|
||
|
||
那么只要任取$[a,b]$上的一个分割$T$满足$\Vert T \Vert < \delta$，那么对于分割的每个子区间$[x_{i-1}, x_i]$都有$\Delta x_i = x_i - x_{i-1} < \delta$。因此$f$在$[x_{i-1}, x_i]$上的振幅$\omega_i = M_i - m_i \leq \varepsilon$。那么
|
||
\[0 \leq \sum_{i=1}^n \omega_i \Delta x_i \leq \varepsilon \sum_{i=1}^n \Delta x_i = \varepsilon(b-a)\]
|
||
由此可得
|
||
\[\tolim{\Vert T \Vert}{0}\sum_{i=1}^n \omega_i \Delta x_i = 0\]
|
||
因此$f \in R[a,b]$。
|
||
\end{proof}
|
||
|
||
\section{Lebesgue定理}
|
||
\begin{definition}
|
||
定义函数$f$的间断点集$D(f) = \{x_0 \in [a,b] \mid f\text{在}x_0\text{间断}\}$。
|
||
\end{definition}
|
||
\begin{definition}
|
||
设$A$为实数的集合。如果对任意给定的$\varepsilon > 0$，存在至多可数的一列开区间$\{I_n, n \in \naturalnum^\ast\}$，使得
|
||
\begin{enumerate}
|
||
\item $A \subset \displaystyle\bigcup_{i=1}^\infty I_i$；
|
||
\item $\displaystyle\sum_{i=1}^N \vert I_n \vert < \varepsilon, N = 1, 2, \cdots$（$\vert I_n \vert$表示区间$I_i$的长度）。
|
||
\end{enumerate}
|
||
则称$A$为一维零测集，简称零测集。
|
||
\end{definition}
|
||
|
||
\begin{proposition}
|
||
零测集有下列简单性质：
|
||
\begin{enumerate}
|
||
\item 至多可数个零测集的并集是零测集。
|
||
\item 设$A$为零测集，若$B \subset A$，则$B$也是零测集。
|
||
\end{enumerate}
|
||
\end{proposition}
|
||
|
||
\begin{theorem}[Lebesgue定理]
|
||
设$f: [a,b] \to \realnum$有界。那么$f \in R[a,b]$当且仅当$D(f)$是零测集。
|
||
\end{theorem}
|
||
换言之，有界函数可积的充要条件是其所有间断点总长度为0。
|
||
|
||
\begin{corollary}
|
||
容易由Lebesgue定理得到以下结论：
|
||
\begin{enumerate}
|
||
\item 若$f \in R[a,b]$，则$\abs{f} \in R[a,b]$；
|
||
\item 若$f, g \in R[a,b]$，则$fg \in R[a,b]$；
|
||
\item 若$f \in R[a,b]$且$\dfrac{1}{f}$有界，则$\dfrac{1}{f} \in R[a,b]$；
|
||
\item 设$f \in R[a,b], \varphi \in C[\alpha, \beta]$且$f([a,b]) \subset [\alpha, \beta]$，则$\varphi \circ f \in R[a,b]$。
|
||
\end{enumerate}
|
||
\end{corollary}
|
||
|
||
\begin{proposition}
|
||
令$a < c < b$，则$f \in R[a,b]$当且仅当$f \in R[a,c]$且$f \in R[c,b]$。
|
||
\end{proposition}
|
||
\begin{proof}
|
||
注意$D(f) \cap [a,c] \subset D(f)$，$D(f) \cap [c,b] \subset D(f)$，以及
|
||
\[D(f) = \left(D(f) \cap [a,c]\right)\cup\left(D(f) \cap [c,b]\right)\eqper\]
|
||
\end{proof}
|
||
|
||
\section{反常积分}
|
||
前述的积分都是研究有界函数在有限区间上的积分，如果突破这两个限制，将得到反常积分。
|
||
|
||
\subsection{无穷积分}
|
||
设$f:[a, +\infty) \to \realnum$，且$\forall A > a, f \in R[a,A]$。定义
|
||
\[\int_a^{+\infty} f(x) \dif x := \tolim{A}{+\infty} \int_a^A f(x) \dif x\]
|
||
若极限存在，则称该无穷积分收敛，否则称其发散。
|
||
|
||
类似地，我们可对$f: (-\infty, b] \to \realnum$定义
|
||
\[\int_{-\infty}^b f(x) \dif x := \tolim{B}{-\infty} \int_B^b f(x) \dif x\]
|
||
也可对$f: (-\infty, +\infty) \to \realnum$定义
|
||
\[\int_{-\infty}^{+\infty} f(x) \dif x := \int_{-\infty}^a f(x) \dif x + \int_a^{+\infty} f(x) \dif x, a \in \realnum \text{任取}\]
|
||
|
||
\begin{remark}
|
||
另外可定义Principal Value主值，其收敛性较弱：
|
||
\[\int_{-\infty}^{+\infty} f(x) \dif x := \tolim{A}{\infty} \int_{-A}^A f(x) \dif x \eqper\]
|
||
\end{remark}
|
||
|
||
\begin{example}
|
||
研究$\dint_1^{+\infty} \frac{\dif x}{x^p}$的收敛性。
|
||
\end{example}
|
||
|
||
\begin{proof}[解]
|
||
对任意的$A > 1$，
|
||
|
||
若$p \neq 1$，
|
||
\[\int_1^A \frac{\dif x}{x^p} = \frac{1}{1 - p} \eval{x^{1-p}}_1^A = \frac{A^{1-p} - 1}{1 - p} \to
|
||
\begin{cases}
|
||
\frac{1}{p-1} & p > 1\\
|
||
+\infty & p < 1
|
||
\end{cases} (A \to +\infty)\]
|
||
|
||
若$p = 1$，则
|
||
\[\int_1^A \frac{\dif x}{x} = \eval{\ln x}_1^A = \ln A \to +\infty (A \to +\infty)\]
|
||
|
||
综上，当$p > 1$时无穷积分收敛，
|
||
\[\int_1^{+\infty} \frac{\dif x}{x^p} = \frac{1}{p-1}\]
|
||
当$p \leq 1$时无穷积分$\dint_a^{+\infty} \dfrac{\dif x}{x^p}$发散到无穷。
|
||
\end{proof}
|
||
|
||
\begin{example}\label{广义积分例子}
|
||
讨论广义积分$\dint_e^{+\infty} \dfrac{1}{x(\ln x)^p} \dif x$的收敛性。
|
||
\end{example}
|
||
|
||
\begin{proof}[解]
|
||
$p = 1$时，
|
||
\[\int_e^{+\infty} \frac{1}{x(\ln x)^p} \dif x = \int_e^{+\infty} \frac{1}{\ln x} \dif \ln x = \eval{\ln \ln x}_e^{+\infty} = + \infty \eqper\]
|
||
|
||
$p \neq 1$时，
|
||
\begin{align*}
|
||
\int_e^{+\infty} \frac{1}{x(\ln x)^p \dif x} & = \int_e^{+\infty} \frac{1}{(\ln x)^p} \dif \ln x\\
|
||
& = \eval{\frac{1}{1 - p}(\ln x)^{1 - p}}_e^{+\infty}\\
|
||
& = \begin{cases}
|
||
\frac{1}{p - 1}, p > 1\\
|
||
+\infty, p < 1
|
||
\end{cases}
|
||
\end{align*}
|
||
|
||
综上，$\dint_1^{+\infty} \frac{1}{x (\ln x)^p} \dif x$当$p > 1$时收敛，当$p \leq 1$时发散。
|
||
\end{proof}
|
||
|
||
\begin{theorem}[广义Newtown-Leibniz公式]
|
||
设$f \in C[a,+\infty)$有原函数$F \in C[a,+\infty)$，则
|
||
\[\int_a^{+\infty} f(x) \dif x = \eval{F(x)}_a^{+\infty} = \tolim{x}{+\infty} F(x) - F(a)\eqper\]
|
||
相应地，有
|
||
\[\int_{-\infty}^b f(x) \dif x = \eval{F(x)}_{-\infty}^b = F(b) - \tolim{x}{-\infty}F(x)\]
|
||
\[\int_{-\infty}^{+\infty} f(x) \dif x = \eval{F(x)}_{-\infty}^{+\infty} = \tolim{x}{+\infty}F(x) - \tolim{x}{-\infty}F(x)\eqper\]
|
||
\end{theorem}
|
||
|
||
分部积分、换元、积分的线性性质、保序性、积分区间可加性仍然成立。
|
||
|
||
\subsection{瑕积分}
|
||
接下来我们讨论无界函数的积分。
|
||
|
||
\begin{definition}
|
||
$f$在$[a,b)$上有定义，在$b$点附近无界。此时，称$x = b$为$f$的一个瑕点。若$\forall \delta \in (0,b-a)$，$f \in R[a,b-\delta]$，且
|
||
\[\tolim{\delta}{0^+} \int_a^{b - \delta} f(x) \dif x = I\eqco\]
|
||
则称$f$在$[a,b)$上的瑕积分收敛，称$I$为$f$在$[a,b)$上的瑕积分（值），记作
|
||
\[\int_a^b f(x) \dif x = \tolim{\delta}{0^+}\int_a^{b - \delta} f(x) \dif x\eqper\]
|
||
若该极限不存在，则称瑕积分$\int_a^b f(x) \dif x$发散。
|
||
\end{definition}
|
||
|
||
\begin{definition}
|
||
$f$在$(a,b)$上有定义，且$a,b$为瑕点。若$\exists c \in (a,b)$，满足瑕积分$\int_a^c f(x) \dif x$与$\int_c^b f(x) \dif x$均收敛，则
|
||
\[\int_a^b f(x) \dif x := \int_a^c f(x) \dif x + \int_c^b f(x) \dif x\]
|
||
此时，
|
||
\[\int_a^b f(x) \dif x = \int_a^d f(x) \dif x + \int_d^b f(x) \dif x, \forall d \in (a,b)\]
|
||
\begin{align*}
|
||
\int_a^b f(x) \dif x & = \tolim{\alpha}{a^+} \int_\alpha^c f(x) \dif x + \tolim{\beta}{b^-} \int_c^\beta f(x) \dif x\\
|
||
& = \lim \limits_{\alpha \to a^+, \beta \to b^-} \int_\alpha^\beta f(x) \dif x
|
||
\end{align*}
|
||
\end{definition}
|
||
|
||
\begin{lemma}[Riemann-Lebesgue]
|
||
$f$在$[a,b]$上可积或广义绝对可积（$f$与$\abs{f}$均在$[a,b]$上广义可积），则
|
||
\[\tolim{\lambda}{\infty} \int_a^b f(x) \cos \lambda x \dif x = 0, \tolim{\lambda}{\infty} \int_a^b f(x) \sin \lambda x \dif x = 0\eqper\]
|
||
\end{lemma}
|
||
|
||
\begin{proof}
|
||
只证第一式，第二式同理。
|
||
|
||
\emph{情况一：}设$f$在$[a,b]$上可积，则$f$在$[a,b]$上有界，设其上界为$M$，那么
|
||
\[\forall x \in [a,b], \vert f(x) \leq M\]
|
||
对任意的$\lambda > 1$，令$n = \left\lfloor \lambda \right\rfloor$。将$[a,b]$区间$n$等分，有
|
||
\[x_i = a + i\frac{b-a}{n}, i = 0, 1, 2, \cdots, n\]
|
||
设
|
||
\[\omega_i(f) = \sup \{f(\xi) - f(\eta) \mid \xi, \eta \in[x_{i-1}, x_i]\}\]
|
||
$f$在$[a,b]$上可积，则$\tolim{n}{\infty} \displaystyle\sum_{i=1}^n \omega_i(f) \Delta x_i = 0$。那么
|
||
\begin{align*}
|
||
\left|\int_a^b f(x) \cos \lambda x \dif x \right| & = \left| \sum_{i=1}^n \int_{x_{i-1}}^{x_i} f(x) \cos \lambda x \dif x \right|\\
|
||
& \leq \left| \sum_{i=1}^n \int_{x_{i-1}}^{x_i} \left(f(x) - f(x_i)\right)\cos \lambda x \dif x \right| + \left|\sum_{i=1}^n \int_{x_{i-1}}^{x_i} f(x_i) \cos \lambda x \dif x\right|\\
|
||
& \leq \sum_{i=1}^n \omega_i(f) \Delta x_i + \sum_{i=1}^n \left|f(x_i)\right|\left|\int_{x_{i-1}}^{x_i} \cos \lambda x \dif x\right|\\
|
||
& \leq \sum_{i=1}^n \omega_i(f) \Delta x_i + \frac{2Mn}{\lambda}\\
|
||
& = \sum_{i=1}^{\left\lfloor \lambda \right\rfloor} \omega_i(f) \Delta x_i + \frac{2M \left\lfloor \lambda \right\rfloor}{\lambda}
|
||
\end{align*}
|
||
当$\lambda \to +\infty$时，上式趋近于0。
|
||
|
||
\emph{情况二：}$f$在$[a,b]$上广义绝对可积。不妨设$a$为唯一的瑕点。那么$\forall \varepsilon > 0$，存在$\delta > 0$，满足$f$在$[a+\delta, b]$可积，且
|
||
\[\int_a^{a + \delta} \vert f(x) \vert \dif x < \frac{\varepsilon}{2}\]
|
||
从而
|
||
\[\left|\int_a^{a + \delta} f(x) \cos \lambda x \dif x \right| \leq \int_a^{a + \delta} \vert f(x) \vert \dif x < \frac{\varepsilon}{2}\]
|
||
因此
|
||
\[\tolim{\lambda}{+ \infty} \int_{a + \delta}^b f(x) \cos \lambda x \dif x = 0\eqper\]
|
||
|
||
于是$\exists \Lambda > 0$，当$\lambda > \Lambda$时，
|
||
\[\left|\int_{a+ \delta}^b f(x) \cos \lambda x \dif x \right| < \frac{\varepsilon}{2}\]
|
||
进一步有对任意的$\lambda > \Lambda$，
|
||
\begin{align*}
|
||
\left|\int_a^b f(x) \cos \lambda x \dif x \right| & \leq \left|\int_a^{a + \delta} f(x) \cos x \lambda x \dif x \right| + \left|\int_{a + \delta}^b f(x) \cos \lambda x \dif x \right|\\
|
||
& < \frac{\varepsilon}{2} = \frac{\varepsilon}{2} = \varepsilon\eqper \qedhere
|
||
\end{align*}
|
||
\end{proof}
|