定积分的应用。

07函数的积分.tex

@@ -37,7 +37,7 @@
 \end{corollary}
 
 \begin{corollary}
-    设$\vert f \vert, f \in R[a,b]$，则
+    设$\abs{f}, f \in R[a,b]$，则
     \[\left| \int_a^b f(x) \dif x \right| \leq \int_a^b \vert f(x) \vert \dif x \eqper\]
 \end{corollary}
 
@@ -121,18 +121,18 @@
     使得
     \[\left|\sum_{i=1}^n f(c_i) \Delta x - A\right| < 1, c_i \in [x_{i-1}, x_i]\text{任取}, i = 1, 2, \cdots, n\]
     由此导出
-    \[\left|\sum_{i=1}^n f(c_i) \Delta x \right| < \left|\sum_{i=1}^n f(c_i) \Delta x - A\right| + \vert A \vert  < 1 + \vert A \vert\]
+    \[\left|\sum_{i=1}^n f(c_i) \Delta x \right| < \left|\sum_{i=1}^n f(c_i) \Delta x - A\right| + \abs{A}  < 1 + \abs{A}\]
     因此
-    \[\left|\sum_{i=1}^n f(c_i) \right| \leq \frac{1 + \vert A \vert}{\Delta x}\]
+    \[\left|\sum_{i=1}^n f(c_i) \right| \leq \frac{1 + \abs{A}}{\Delta x}\]
     进一步有
-    \[\vert f(c_i) \vert \leq \left| \sum_{i=1}^n f(c_i) \right| + \left| \sum_{i=2}^n f(c_i)\right| \leq \frac{1 + \vert A \vert}{\Delta x} + \left| \sum_{i=2}^n f(x_i) \right|\]
+    \[\vert f(c_i) \vert \leq \left| \sum_{i=1}^n f(c_i) \right| + \left| \sum_{i=2}^n f(c_i)\right| \leq \frac{1 + \abs{A}}{\Delta x} + \left| \sum_{i=2}^n f(x_i) \right|\]
     即
-    \[\vert f(x) \vert \leq \frac{1 + \vert A \vert}{\Delta x} + \left|\sum_{i\neq 1}^n f(x_i)\right| = M_1, \forall x \in [x_0, x_1]\]
+    \[\vert f(x) \vert \leq \frac{1 + \abs{A}}{\Delta x} + \left|\sum_{i\neq 1}^n f(x_i)\right| = M_1, \forall x \in [x_0, x_1]\]
     类似地可以得到
     \begin{align*}
-        \vert f(x) \vert \leq \frac{1 + \vert A \vert}{\Delta x} + \left|\sum_{i\neq 2}^n f(x_i)\right| & = M_2, \forall x \in [x_1, x_2]\\
+        \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]\\
         \vdots & \\
-        \vert f(x) \vert \leq \frac{1 + \vert A \vert}{\Delta x} + \left|\sum_{i\neq n}^n f(x_i)\right| & = M_n, \forall x \in [x_{n-1}, x_n]
+        \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]
     \end{align*}
     将这$n$个式子综合起来就可以得到
     \[\vert f(x) \vert \leq \max \{M_1, \cdots, M_n\}, \forall x \in [a,b] \eqper \qedhere\]
@@ -521,7 +521,7 @@ C^n [a,b] = \{f \in C[a,b] \mid f^{(n)} \in C[a,b]\}\]
 \begin{corollary}
     容易由Lebesgue定理得到以下结论：
     \begin{enumerate}
-        \item 若$f \in R[a,b]$，则$\vert f \vert \in R[a,b]$；
+        \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$\varphi \circ f \in R[a,b]$。
@@ -609,7 +609,7 @@ C^n [a,b] = \{f \in C[a,b] \mid f^{(n)} \in C[a,b]\}\]
 \end{definition}
 
 \begin{lemma}[Riemann-Lebesgue]
-    $f$在$[a,b]$上可积或广义绝对可积（$f$与$\vert f \vert$均在$[a,b]$上广义可积），则
+    $f$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}
 
@@ -646,4 +646,4 @@ C^n [a,b] = \{f \in C[a,b] \mid f^{(n)} \in C[a,b]\}\]
         \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}
+\end{proof}