From d955e6d349847d85ad23aabfc75d34639b5600f8 Mon Sep 17 00:00:00 2001 From: unlockable Date: Wed, 7 Jun 2023 13:50:48 +0800 Subject: [PATCH] =?UTF-8?q?=E7=AC=AC=E5=8D=81=E5=85=AD=E5=91=A8=E3=80=82?= =?UTF-8?q?=E7=BB=93=E8=AF=BE=E3=80=82?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 20含参变量积分.tex | 119 +++++++++++++++++++++++++++++++++++++++++++-- 高等微积分.tex | 1 + 2 files changed, 117 insertions(+), 3 deletions(-) diff --git a/20含参变量积分.tex b/20含参变量积分.tex index 570713b..bf585f1 100644 --- a/20含参变量积分.tex +++ b/20含参变量积分.tex @@ -43,14 +43,14 @@ \end{theorem} \begin{proof} - 令$J(t, \alpha, \beta) = \dint_\alpha^\beta g(t, x) \dif x$，由$g(t, x), \deriv{g_t}(t, x)$的连续性， + 令$J(t, \alpha, \beta) = \dint_\alpha^\beta g(t, x) \dif x$，由$g(t, x), \dfrac{\partial g}{\partial t}(t, x)$的连续性， \[\deriv{J_t} = \int_\alpha^\beta \dfrac{\partial g}{\partial t}(t, x) \dif x, \deriv{J_\alpha} = - g(t, \alpha), \deriv{J_\beta} = g(t, \beta)\] 均在$(t, \alpha, \beta) \in D = [a, b] \times [c, d] \times [c, d]$上连续。因此$J(t, \alpha, \beta)$在$D$上可微，复合函数 \[f(t) = \int_{\alpha(t)}^{\beta(t)} g(t, x) \dif x = J(t, \alpha(t), \beta(t))\] 在$t \in [a, b]$上可微，且 \begin{align*} \deriv{f}(t) & = \deriv{J_t} + \deriv{J_\alpha} \cdot \deriv{\alpha}(t) + \deriv{J_\beta} \cdot \deriv{\beta}(t)\\ - & = \int_{\alpha(t)}^{\beta(t)} \dfrac{\partial g}{\partial t}(t, x) \dif x + g(t, \beta(t)) \deriv{\beta}(t) - g(t, \alpha(t)) \deriv{\alpha}(t)\eqper + & = \int_{\alpha(t)}^{\beta(t)} \dfrac{\partial g}{\partial t}(t, x) \dif x + g(t, \beta(t)) \cdot \deriv{\beta}(t) - g(t, \alpha(t)) \cdot \deriv{\alpha}(t)\eqper \end{align*} \end{proof} @@ -113,4 +113,117 @@ 那么积分 \[\int_a^{+\infty} f(t, x) g(t, x) \dif x\] 在$\Omega$上一致收敛。 -\end{theorem} \ No newline at end of file +\end{theorem} + +\section{含参变量反常积分的性质} +\begin{theorem} + 如果函数$f(t, x)$在$[\alpha, \beta] \times [a, +\infty)$上连续，而且积分 + \[I(t) = \int_a^{+\infty} f(t, x) \dif x\] + 关于$t \in [\alpha, \beta]$一致收敛，则$I(t) \in C[\alpha, \beta]$。 +\end{theorem} + +\begin{theorem} + 设函数$f(t, x), \dfrac{\partial f}{\partial t}(t, x) \in C([\alpha, \beta] \times [a, +\infty])$，且对任意的$t \in [\alpha, \beta]$， + \[I(t) = \int_a^{+\infty} f(t, x) \dif x\] + 收敛，同时 + \[\int_a^{+\infty} \dfrac{\partial f}{\partial t}(t, x) \dif x\] + 关于$t \in [\alpha, \beta]$一致收敛，那么$I(t) \in C^1[\alpha, \beta]$且 + \[\deriv{I}(t) = \frac{\dif}{\dif t} \int_a^{+\infty} f(t, x) \dif x = \int_a^{+\infty} \frac{\partial f}{\partial t}(t, x) \dif x\eqper\] +\end{theorem} + +\begin{theorem} + 设函数$f(x, y) \in C([a, +\infty) \times [\alpha, \beta])$，含参积分 + \[I(t) = \int_a^{+\infty} f(x, y) \dif x\] + 关于$y \in [\alpha, \beta]$一致收敛，则$I(y)$在$[\alpha, \beta]$上可积，且 + \[\int_\alpha^\beta I(y) \dif y = \int_a^{+\infty} \left(\int_\alpha^\beta f(x, y) \dif y\right)\dif x\] + 即 + \[\int_\alpha^\beta \dif y \int_a^{+\infty} f(x, y) \dif x = \int_a^{+\infty} \dif x \int_\alpha^\beta f(x, y) \dif y\eqper\] +\end{theorem} + +\begin{theorem} + 如果函数$f(x, y)$满足下列条件： + \begin{enumerate} + \item $f$在$[a, +\infty) \times [\alpha, +\infty)$上连续； + \item 对任意的$\beta > \alpha$，积分 + \[\int_a^{+\infty} f(x, y) \dif x\] + 在$y \in [\alpha, \beta]$上一致收敛； + 对任意的$b > a$，积分 + \[\int_\alpha^{+\infty} f(x, y) \dif y\] + 在$x \in [a, b]$上一致收敛； + \item 积分 + \[\int_a^{+\infty} \left(\int_\alpha^{+\infty} \abs{f(x, u)} \dif u\right)\dif x, \quad \int_\alpha^{+\infty} \left(\int_a^{+\infty} \abs{f(x, u)} \dif x\right)\dif u\] + 中至少有一个存在， + \end{enumerate} + 那么积分 + \[\int_a^{+\infty} \left(\int_\alpha^{+\infty} f(x, u) \dif u\right)\dif x, \quad \int_\alpha^{+\infty} \left(\int_a^{+\infty} f(x, u) \dif x\right)\dif u\] + 都存在且相等，即 + \[\int_a^{+\infty} \left(\int_\alpha^{+\infty} f(x, u) \dif u\right)\dif x = \int_\alpha^{+\infty} \left(\int_a^{+\infty} f(x, u) \dif x\right)\dif u\eqper\] +\end{theorem} + +\section[Γ函数与B函数]{$\Gamma$函数与$\Beta$函数} +含参变量的广义积分 +\begin{align*} + \Gamma(s) = \int_0^{+\infty} t^{s - 1} e^{-t} \dif t, \quad (s > 0)\\ + \Beta(p, q) = \int_0^1 t^{p - 1} (1 - t)^{q - 1} \dif t, \quad (p > 0 , q > 0) +\end{align*} +分别称为$\Gamma$函数和$\Beta$函数。 + +\subsection[Γ函数]{$\Gamma$函数} +\begin{theorem} + $\Gamma(s)$在$(0, +\infty)$上连续，且有各阶连续导数。 +\end{theorem} + +\begin{theorem} + $\Gamma$函数具有下面三条性质： + \begin{enumerate} + \item 对任意$s > 0$，$\Gamma(s) > 0$，且$\Gamma(1) = 1$； + \item $\Gamma(s + 1) = s \Gamma(s)$对任意$s > 0$成立； + \item $\ln \Gamma(s)$是$(0, +\infty)$上的凸函数。 + \end{enumerate} +\end{theorem} + +$\Gamma$函数可以看作是阶乘函数的推广： +\[\Gamma(n + 1) = n!, n = 1, 2, 3, \dots\] + +\begin{theorem}[Bohr-Mollerup] + 如果$(0, +\infty)$上的函数$f$满足下面三个条件： + \begin{enumerate} + \item 对任意$x > 0$，$f(x) > 0$且$f(1) = 1$； + \item $f(x + 1) = xf(x)$对任意$x > 0$成立； + \item $\ln f$是$(0, +\infty)$上的凸函数， + \end{enumerate} + 那么$f(x) = \Gamma(x)$对任意$x > 0$成立。 +\end{theorem} + +\begin{theorem} + 对任意$x > 0$，有 + \[\Gamma(x) = \tolim{n}{\infty} \frac{n^x n!}{x(x + 1) \dots (x + n)}\eqper\] +\end{theorem} + +\begin{theorem}[$\Gamma$函数的余元公式] + 对任意的$x \in (0, 1)$，有 + \[\Gamma(x) \Gamma(1 - x) = \frac{\pi}{\sin \pi x}\eqper\] +\end{theorem} + +\subsection[B函数]{$\Beta$函数} +\begin{theorem} + $\Beta$函数有下面的性质： + \begin{enumerate} + \item $\Beta(p, q) > 0$，且$\Beta(1, q) = \dfrac{1}{q}$； + \item $\Beta(p + 1, q) = \dfrac{x}{p + q} \Beta(p, q)$； + \item 给定$q > 0$，$\ln \Beta(p, q)$关于$x$在$(0, + \infty)$是凸函数。 + \end{enumerate} +\end{theorem} + +\begin{theorem} + 对任意$p > 0$，$q > 0$，有 + \[\Beta(p, q) = \frac{\Gamma(p) \Gamma(q)}{\Gamma(p + q)}\eqper\] +\end{theorem} + +\begin{corollary} + 对任意的$p > 0, q > 0$，$\Beta$函数有如下性质： + \begin{enumerate} + \item $\Beta(p, q) = \Beta(q, p)$； + \item $\Beta(p + 1, q + 1) = \dfrac{pq}{(p + q + 1)(p + q)}B(p, q)$。 + \end{enumerate} +\end{corollary} \ No newline at end of file diff --git a/高等微积分.tex b/高等微积分.tex index 54d6059..6edf9e4 100644 --- a/高等微积分.tex +++ b/高等微积分.tex @@ -63,6 +63,7 @@ \newcommand{\closure}[1]{\overline{#1}} \newcommand{\ndreal}{\ensuremath{\realnum^n}} \newcommand{\boldf}{\ensuremath{\bvec{f}}} +\newcommand{\Beta}{{\rm B}} \renewcommand{\parallel}{\mathrel{/\mskip-2.5mu/}} \renewcommand{\gradient}{\nabla} \renewcommand{\divergence}{\nabla \cdot}