From 0d2798bf6c1b05aae5e5958e6f918477697ff5f4 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Wed, 28 Sep 2022 22:21:32 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...or-algebra-and-space-analytic-geometry.pdf | Bin 37861 -> 37911 bytes ...or-algebra-and-space-analytic-geometry.tex | 4 +- .../8-infinite-series/infinite-series.pdf | Bin 84907 -> 91560 bytes .../8-infinite-series/infinite-series.tex | 50 +++++++++++++++--- ...or-algebra-and-space-analytic-geometry.pdf | Bin 115999 -> 116927 bytes ...or-algebra-and-space-analytic-geometry.tex | 5 +- .../8-infinite-series/infinite-series.pdf | Bin 175861 -> 174851 bytes .../8-infinite-series/infinite-series.tex | 12 +---- .../differential-equation.pdf | Bin 129632 -> 129954 bytes .../differential-equation.tex | 8 ++- 10 files changed, 58 insertions(+), 21 deletions(-) diff --git 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$\dfrac{1}{x+1}=\dfrac{1}{(x-1)+2}=\dfrac{1}{2}\dfrac{1}{1+\frac{x-1}{2}}=\dfrac 所以其幂级数就是其加和,收敛区间为$(-2,4)\cap(-1,3)=(-1,3)$。 +\subsubsection{先导后积} + +\textbf{例题:}求函数$f(x)=\arctan x$在$x=0$处的幂级数展开。 + +解:$f'(x)=(\arctan x)'=\dfrac{1}{1+x^2}=\dfrac{1}{1-(-x^2)}=\sum\limits_{n=0}^\infty(-1)^nx^{2n}$,$\vert-x^2\vert<1$。 + +已经求得求导后的函数的幂级数展开,所以求原函数的幂级数展开只需要积分,利用先导后积公式:$f(x)=f(0)+\int_0^xf'(t)\,\textrm{d}t=\int_0^x\sum\limits_{n=0}^\infty(-1)^nt^{2n}\,\textrm{d}t=\sum\limits_{n=0}^\infty(-1)^n\dfrac{t^{2n+1}}{2n+1}\bigg|_0^x=\sum\limits_{n=0}^\infty(-1)^n\dfrac{x^{2n+1}}{2n+1}$。 + +求导的级数要求$\vert x\vert<1$,代入$x=\pm1$到最后结果得到两个交错级数,所以收敛域其实为$[-1,1]$(可以不写)。 + \subsection{级数求和} 即对展开式进行逆运算,根据幂级数展开式反推原幂级数。 可以利用展开式求和函数,但是很多展开式的通项都不是公式中的,就需要对通项进行变形。 +在求和之前要先计算收敛半径和收敛域。 + +无论是哪个方法都要求求导和积分后系数$n+a$与幂次$n$相等,所以求导或积分的目的就是为了让他们相同,从而能被看成一个整体。 + \subsubsection{先导后积} -$n$在分母上,先导后积。使用变限积分:$\int_{x_0}^xS'(t)\,\textrm{d}t=S(x)-S(x_0)$,即$S(x)=S(x_0)+\int_{x_0}^xS'(t)\,\textrm{d}t$。一般选择$x_0$为展开点。 +$\sum\dfrac{x^{f(n)}}{P(n)}$:$n$在分母上,先导后积。使用变限积分:$\int_{x_0}^xS'(t)\,\textrm{d}t=S(x)-S(x_0)$,即$S(x)=S(x_0)+\int_{x_0}^xS'(t)\,\textrm{d}t$。一般选择$x_0$为展开点。 -\textbf{例题:}求级数$\sum\limits_{n=1}^\infty\dfrac{x^n}{n}$的和函数。 +主要公式:$\sum\limits_{n=1}^\infty\dfrac{(-1)^{n-1}}{n}x^n=\ln(1+x)$($(-1,1]$);$\sum\limits_{n=1}^\infty\dfrac{x^n}{n}=-\ln(1-x)$($[-1,1)$)。 -解:已知$\sum\limits_{n=0}^\infty x^n=\dfrac{1}{1-x}$,而这里求和是$\dfrac{x^n}{n}$,所以需要对其进行转换。 +目的是让$P(n)=f(n)$。 -对$\dfrac{x^n}{n}$求导就得到了$x^{n-1}$消去了分母的$n$,所以使用先导后积的方法。 +% \textbf{例题:}求级数$\sum\limits_{n=1}^\infty\dfrac{x^n}{n}$的和函数。 -记$S(x)=\sum\limits_{n=1}^\infty\dfrac{x^n}{n}$,则$x^n=(x-0)^n$,取$x_0=0$。 +% 解:已知$\sum\limits_{n=0}^\infty x^n=\dfrac{1}{1-x}$,而这里求和是$\dfrac{x^n}{n}$,所以需要对其进行转换。 -$\therefore S(x)=S(0)+\displaystyle{\int_0^x\left(\sum\limits_{n=1}^\infty\dfrac{t^n}{n}\right)_t'\,\textrm{d}t}=0+\int_0^x(\sum\limits_{n=1}^\infty t^{n-1})\,\textrm{d}t=\displaystyle{\int_0^x\dfrac{1}{1-t}\textrm{d}t}=-\ln(1-x)$。收敛域为$[-1,1)$。 +% 对$\dfrac{x^n}{n}$求导就得到了$x^{n-1}$消去了分母的$n$,所以使用先导后积的方法。 + +% 记$S(x)=\sum\limits_{n=1}^\infty\dfrac{x^n}{n}$,则$x^n=(x-0)^n$,取$x_0=0$。 + +% $\therefore S(x)=S(0)+\displaystyle{\int_0^x\left(\sum\limits_{n=1}^\infty\dfrac{t^n}{n}\right)_t'\,\textrm{d}t}=0+\int_0^x(\sum\limits_{n=1}^\infty t^{n-1})\,\textrm{d}t=\displaystyle{\int_0^x\dfrac{1}{1-t}\textrm{d}t}=-\ln(1-x)$。收敛域为$[-1,1)$。 + +\textbf{例题:}求级数$\sum\limits_{n=0}^\infty\dfrac{(-1)^n}{2n+1}x^{2n}$的和函数。 + +解:首先$\lim\limits_{n\to\infty}\left\vert\dfrac{a_{n+1}}{a_n}\right\vert=1$,$R=1$。 + +当$x=\pm1$时,$x^2n=1$,所以原式$=\sum\limits_{n=0}^\infty\dfrac{(-1)^n}{2n+1}$,为交错级数,由莱布尼茨判别法可知极限为0且单调递减,从而该级数收敛。从而收敛域为$[-1,1]$。 + +令$S(x)=\sum\limits_{n=0}^\infty\dfrac{(-1)^n}{2n+1}x^{2n}$。易得$x=0$时$S(x)=1$。 + +当$x\neq0$时,$S(x)=\dfrac{1}{x}\sum\limits_{n=0}^\infty\dfrac{(-1)^n}{2n+1}x^{2n+1}=\dfrac{1}{x}\sum\limits_{n=0}^\infty\int_0^x(-1)^nx^{2n}\,\textrm{d}x\\=\dfrac{1}{x}\int_0^x[\sum\limits_{n=0}^\infty(-1)^nx^{2n}]\textrm{d}x$。 + +所以$(-1)^nx^{2n}$为一个几何级数,所以$q=\dfrac{(-1)^{n+1}x^{2n+2}}{(-1)^nx^{2n}}=-x^2$。 + +从而$=\displaystyle{\dfrac{1}{x}\int_0^x\dfrac{1}{1+x^2}\textrm{d}x=\dfrac{\arctan x}{x}}$。 \subsubsection{先积后导} -$n$在分子上,先积后导。$(\int S(x)\,\textrm{d}x)'=S(x)$。 +$\sum P(n)x^{f(n)}$:$n$在分子上,先积后导。$(\int S(x)\,\textrm{d}x)'=S(x)$。 + +主要公式:$\sum\limits_{n=0}^\infty(-1)^nx^n=\dfrac{1}{1+x}$($(-1,1)$);$\sum\limits_{n=0}^\infty x^n=\dfrac{1}{1-x}$($(-1,1)$)。 + +目的是让$P(n)=f^{(n)}(n)\cdots f'(n)$。 \textbf{例题:}求级数$\sum\limits_{n=1}^\infty nx^n$的和函数。 diff --git a/advanced-math/knowledge/5-vector-algebra-and-space-analytic-geometry/vector-algebra-and-space-analytic-geometry.pdf b/advanced-math/knowledge/5-vector-algebra-and-space-analytic-geometry/vector-algebra-and-space-analytic-geometry.pdf index 4f0dab25fde16f46762b6328d8bd7ea9e2b47c59..75911444e3f7dec783b49f1b456f77185b1513f9 100644 GIT binary patch delta 96512 zcmZUaV|Sok&}Ng4I(EmlZ6_VuHafO-$F{AGZQHidv27>w%&htF&YHE(hdRHYcI~<< 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a/advanced-math/knowledge/5-vector-algebra-and-space-analytic-geometry/vector-algebra-and-space-analytic-geometry.tex +++ b/advanced-math/knowledge/5-vector-algebra-and-space-analytic-geometry/vector-algebra-and-space-analytic-geometry.tex @@ -75,7 +75,8 @@ a_x & a_y & a_z \\ b_x & b_y & b_z \end{array}\right\vert$,其中$\vert a\times b\vert=\vert a\vert\vert b\vert\sin\theta$,用右手规则确定方向(转向角不超过$\pi$),其中$\theta$为$\vec{a},\vec{b}$夹角。 - \item $a//b\Leftrightarrow\theta=0$或$\pi\Leftrightarrow\dfrac{a_x}{b_x}=\dfrac{a_y}{b_y}=\dfrac{a_z}{b_z}$。 + \item $\vec{a}//\vec{b}\Leftrightarrow\theta=0\Leftrightarrow\vec{a}\times\vec{b}=0$或$\pi\Leftrightarrow\dfrac{a_x}{b_x}=\dfrac{a_y}{b_y}=\dfrac{a_z}{b_z}$。 + \item $\vec{a}\times\vec{a}=0$。 \end{itemize} 向量积的计算也可以如此理解,将两个向量上下摞在一起,然后右边再复制一份: @@ -93,6 +94,8 @@ $\left[\begin{array}{cccccc} b_x & b_y & b_z \\ c_x & c_y & c_z \\ \end{array}\right\vert$。 + \item 交换两行不改变值:$a\cdot(b\times c)=b\cdot(c\times a)=c\cdot(a\times b)$。 + \item 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\subsection{概念} -\textcolor{violet}{\textbf{定义:}}形如$x^2\dfrac{\textrm{d}^2y}{\textrm{d}x^2}+px\dfrac{\textrm{d}y}{\textrm{d}x}+qy=f(x)$的方程称为\textbf{欧拉方程},其中$pq$为常数,$f(x)$为已知函数。 +\textcolor{violet}{\textbf{定义:}}形如$x^2y''+pxy'+qy=f(x)$的方程称为\textbf{欧拉方程},其中$pq$为常数,$f(x)$为已知函数。 + +注意与一般的二阶常系数非齐次线性微分方程$y''+py'+qy=f(x)$对比,其二阶导和一阶导前面多了$x$的多项式。 + +% $x^2\dfrac{\textrm{d}^2y}{\textrm{d}x^2}+px\dfrac{\textrm{d}y}{\textrm{d}x}+qy=f(x)$ \subsection{解法} -使用换元法。 +使用换元法$x=e^t$将欧拉方程换为二阶常系数非齐次线性方程。 当$x>0$时,令$x=e^t$,则$t=\ln x$,$\dfrac{\textrm{d}t}{\textrm{d}x}=\dfrac{1}{x}$,$\dfrac{\textrm{d}y}{\textrm{d}x}=\dfrac{\textrm{d}y}{\textrm{d}t}\dfrac{\textrm{d}t}{\textrm{d}x}=\dfrac{1}{x}\dfrac{\textrm{d}y}{\textrm{d}t}$,$\dfrac{\textrm{d}^2y}{\textrm{d}x^2}=\dfrac{\textrm{d}}{\textrm{d}x}\left(\dfrac{1}{x}\dfrac{\textrm{d}y}{\textrm{d}t}\right)=-\dfrac{1}{x^2}\dfrac{\textrm{d}y}{\textrm{d}t}+\dfrac{1}{x}\dfrac{\textrm{d}}{\textrm{d}x}\left(\dfrac{\textrm{d}y}{\textrm{d}t}\right)=-\dfrac{1}{x^2}\dfrac{\textrm{d}y}{\textrm{d}t}+\dfrac{1}{x^2}\dfrac{\textrm{d}^2y}{\textrm{d}t^2}$,方程化为$\dfrac{\textrm{d}^2y}{\textrm{d}t^2}+(p-1)\dfrac{\textrm{d}y}{\textrm{d}t}+qy=f(e^t)$,解出结果,组后用$t=\ln x$回代。