From 0a283cfcf433f672602a4b1ab4639d10b6f5480d Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Mon, 7 Jun 2021 00:00:24 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E6=9C=89=E7=90=86=E7=A7=AF?= =?UTF-8?q?=E5=88=86=E9=83=A8=E5=88=86?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...efinite-integral-and-definite-integral.pdf | Bin 107318 -> 129017 bytes ...efinite-integral-and-definite-integral.tex | 150 +++++++++++++++++- .../knowledge/1-determinant/determinant.pdf | Bin 0 -> 11979 bytes .../knowledge/1-determinant/determinant.tex | 32 ++++ linear-algebra/knowledge/2-matrix/matrix.pdf | Bin 0 -> 11950 bytes linear-algebra/knowledge/2-matrix/matrix.tex | 32 ++++ 6 files changed, 209 insertions(+), 5 deletions(-) create mode 100644 linear-algebra/knowledge/1-determinant/determinant.pdf create mode 100644 linear-algebra/knowledge/1-determinant/determinant.tex create mode 100644 linear-algebra/knowledge/2-matrix/matrix.pdf create mode 100644 linear-algebra/knowledge/2-matrix/matrix.tex diff --git a/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf b/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf index 92afe5ebee03e6721e9aa8f46f473e22d0e19dda..09eb227a28c42e785523b51c4e973b516e909157 100644 GIT binary patch delta 118605 zcmZsiLwKN3v#sNf)v;~cwr$(CzBuV59orptY}>YN+et_FKleFUtiR%Rvv0a#Zz7jt8KSg-Xf9oe`84*2dD z%~vjAW*TZ0R7krQrF{MT7V(qitG-`J2JJ$aRI^_n^S^S$0%FK9RVNjKi6Tw4_9+AW zez`mq_rpGhsMo6B&Pi2zfI4M|Na_AQ2rteSiJ?e&5LrSoW6sfotNp;mvi**6d@(A@aiVkHu=geCM$!zDYfyQ>nR$)(9m}D2V3es2N!`;tD6Y+Fo95)$gBfAeqxS!2fMF{w^6?*Wm*4QgKp^u{0@@gb*Ni zh?g4!_m>qK>qMgYP7j0_U|xRW z5IOb0*=Ofj+1NH1uJ@ZpnT=d>t7?-rjhd|g!h6R<$iG>8 zQF*xD{WI5wkDsb(+?SDm5P_vZeUpMKAy6zTvg~9dEh?&A0ln%z7nP#@vbAv)|2 zQ%W6reg_Pi@9gL3~W#3R|OBKuJIQnw`l_@ruDcXKUAiystY_ecLN0i7)vXP0? 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a/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex +++ b/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex @@ -68,11 +68,11 @@ 将复杂的式子转换为简单的一个因式放到$\textrm{d}$后面看作一个整体,然后利用基本积分公式计算。 -\textbf{例题:}$\displaystyle{\int\dfrac{\textrm{d}x}{x\ln x\ln\ln x}}$。 \medskip +\textbf{例题:}求$\displaystyle{\int\dfrac{\textrm{d}x}{x\ln x\ln\ln x}}$。 \medskip $=\displaystyle{\int\dfrac{\textrm{d}(\ln x)}{\ln x\ln\ln x}}=\displaystyle{\int\dfrac{\textrm{d}(\ln\ln x)}{\ln\ln x}}=\ln\vert\ln\ln x\vert+C$。\medskip -\textbf{例题:}$\displaystyle{\int\dfrac{10^{2\arccos x}}{\sqrt{1-x^2}}\,\textrm{d}x}$。 +\textbf{例题:}求$\displaystyle{\int\dfrac{10^{2\arccos x}}{\sqrt{1-x^2}}\,\textrm{d}x}$。 $=-\displaystyle{\int10^{2\arccos x}\,\textrm{d}(\arccos x)}=-\dfrac{1}{2}\displaystyle{\int10^{2\arccos x}\,\textrm{d}(2\arccos x)}=-\dfrac{10^{2\arccos x}}{2\ln10}+C$。 @@ -80,7 +80,7 @@ $=-\displaystyle{\int10^{2\arccos x}\,\textrm{d}(\arccos x)}=-\dfrac{1}{2}\displ 对于两个三角函数的乘积可以使用积化和差简单计算。 -\textbf{例题:}$\displaystyle{\int\sin2x\cos3x\,\textrm{d}x}$。 +\textbf{例题:}求$\displaystyle{\int\sin2x\cos3x\,\textrm{d}x}$。 $=\displaystyle{\int\cos3x\sin2x\,\textrm{d}x=\dfrac{1}{2}\int(\sin5x-\sin x)\,\textrm{d}x}$ @@ -88,12 +88,62 @@ $=\dfrac{1}{2}\int\sin5x\,\textrm{d}x-\dfrac{1}{2}\int\sin x\,\textrm{d}x=-\dfra \paragraph{三角拆分} \leavevmode \medskip -主要用于$\sec^2-1=\tan^2x$,当出现$\tan^2$、$\tan^3$等与$\sec x$在一起作为乘积时可以考虑拆分。 +主要用于$\sec^2-1=\tan^2x$,当出现$\tan^2$、$\tan^3$等与$\sec x$在一起作为乘积时可以考虑拆分换元。 \textbf{例题:}$\displaystyle{\int\tan^3x\sec x\,\textrm{d}x}$。 $=\displaystyle{\int(\sec^2x-1)}\tan x\sec x\,\textrm{d}x=\displaystyle{\int(\sec^2x-1)}\,\textrm{d}(\sec x)=\dfrac{1}{3}\sec^3x-\sec x+C$。 +需要利用到有理积分的高阶多项式分配与低阶多项式因式分解。 + +\paragraph{有理换元} \leavevmode \medskip + +书上这个类型属于有理函数部分,我这里移动到第一类换元中。即将无理因式设为一个变量,从而提高式子的阶数,消除无理式变为有理式。 + +\textbf{例题:}求$\displaystyle{\int\dfrac{\textrm{d}x}{1+\sqrt[3]{x+1}}}$。 + +令$u=\sqrt[3]{x+1}$,从而$x=u^3-1$,$\textrm{d}x=3u^2\,\textrm{d}u$。 + +$=\displaystyle{\int\dfrac{3u^2}{1+u}\textrm{d}u=\int\dfrac{3u^2+3u-3u-3+3}{1+u}\textrm{d}u=\int\left(3u-3+\dfrac{3}{1+u}\right)\textrm{d}u}$ + +$=\dfrac{3}{2}u^2-3u+3\ln\vert1+u\vert+C=\dfrac{3}{2}\sqrt[3]{(x+1)^2}-3\sqrt[3]{x+1}+3\ln\vert1+\sqrt[3]{x+1}\vert+C$。 + +\textbf{例题:}求$\displaystyle{\int\sqrt{\dfrac{1-x}{1+x}}\dfrac{\textrm{d}x}{x}}$。\medskip + +令$u=\sqrt{\dfrac{1-x}{1+x}}$,$x=\dfrac{1-u^2}{1+u^2}$,$\textrm{d}x=\textrm{d}\left(\dfrac{1-u^2}{1+u^2}\right)=\dfrac{-2u(1+u^2)-2u(1-u^2)}{(1+u^2)^2}$ + +$=\displaystyle{\int u\cdot\dfrac{1+u^2}{1-u^2}\cdot\dfrac{-4u}{(1+u^2)^2}\textrm{d}u=\int\dfrac{-4u^2}{(1-u)(1+u)(1+u^2)}\textrm{d}u}$ + +令$\dfrac{A}{1-u}+\dfrac{B}{1+u}+\dfrac{Cu+D}{1+u^2}=\dfrac{-4u^2}{(1-u)(1+u)(1+u^2)}$。 + +通分:$A(1+u^2+u+u^3)+B(1+u^2-u-u^3)+(Cu-Cu^3+D-Du^2)$ + +$=(A-B-C)u^3+(A+B-D)u^2+(A-B+C)u+(A+B+D)=-4u^2$ + +$\therefore A-B-C=0$,$A+B-D=-4$,$A-B+C=0$,$A+B+D=0$。 + +$\therefore A=B=-1$,$C=0$,$D=2$。 + +原式$=\displaystyle{\int\left(\dfrac{2}{1+u^2}-\dfrac{1}{1-u}-\dfrac{1}{1+u}\right)\textrm{d}u}$ + +$=2\arctan u+\ln\vert1-u\vert-\ln\vert1+u\vert+C$ + +$=2\arctan\sqrt{\dfrac{1-x}{1+x}}+\ln\left\vert\dfrac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right\vert+C$。 + +\paragraph{万能公式} \leavevmode \medskip + +同样属于有理积分的内容,但是本质还是属于三角函数的部分。 + +令$u=\tan\dfrac{x}{2}$,$\sin x=\dfrac{2u}{1+u^2}$,$\cos x=\dfrac{1-u^2}{1+u^2}$,$\textrm{d}x=\dfrac{2}{1+u^2}\textrm{d}u$。 + +\textbf{例题:}求$\displaystyle{\int\dfrac{\textrm{d}x}{3+\cos x}}$。 + +令$u=\tan\dfrac{x}{2}$,从而$\cos x=\dfrac{1-u^2}{1+u^2}$: + +$=\displaystyle{\int\dfrac{1}{3+\dfrac{1-u^2}{1+u^2}}\cdot\dfrac{2}{1+u^2}\textrm{d}u=\dfrac{1}{2+u^2}\textrm{d}u=\dfrac{1}{\sqrt{2}}\arctan\dfrac{u}{\sqrt{2}}+C}$ + +$=\dfrac{1}{\sqrt{2}}\arctan\dfrac{\tan\dfrac{x}{2}}{\sqrt{2}}+C$。 + \subsubsection{第二类换元} 使用换元法做了换元之后是要带回式子中的,也就是说要保证反函数的存在才能代入有意义。为了保证反函数的存在,因此要保证原函数的单调性,所以要有一个规定的范围来使原函数保证单调。 @@ -290,9 +340,35 @@ $=\displaystyle{\int\dfrac{x^3+9x-9x}{x^2+9}\,\textrm{d}x=\int\dfrac{x^3+9x}{x^2 $\displaystyle{=\int x\,\textrm{d}x-\dfrac{9}{2}\int\dfrac{\textrm{d}(x^2+9)}{x^2+9}}=\dfrac{x^2}{2}-\dfrac{9}{2}\ln(9+x^2)+C$。 +\subsubsection{低阶多项式分解} + +当不定积分式子形如$\displaystyle{\int\dfrac{f(x)}{g(x)}\,\textrm{d}x}$,且$f(x)$、$g(x)$都为与$x$相关的多项式,$f(x)$阶数低于$g(x)$,且$g(x)$可以因式分解为$g(x)=g_1(x)g_2(x)\cdots$时,先因式分解再进行运算。 + +因式分解时需要注意两点:一点是分解后的式子的分子最高阶要低于分母最高阶数一阶;二是当分母中出现某一因式有大于等于二的幂次时,需要把其分解为从一阶到其当前阶数的因式相加,但是阶数跟一阶因式的分子阶数一样,否则就缺一个不等式而求不出来。 + +虽然分母可以因式分解,但是整个式子不一定能因式分解,特别是某个因子的阶数高于一阶,所以若不能因式分解则可以考虑低阶多项式分配的方式。 + +\textbf{例题:}求$\displaystyle{\int\dfrac{x^2+1}{(x+1)^2(x-1)}\textrm{d}x}$。\medskip + +令$\dfrac{A}{x+1}+\dfrac{B}{(x+1)^2}+\dfrac{C}{x-1}=\dfrac{x^2+1}{(x+1)^2(x-1)}$。 \medskip + +通分:$=A(x+1)(x-1)+B(x-1)+C(x+1)^2$ + +$=A(x^2-1)+B(x-1)+C(x^2+2x+1)$ + +$=(A+C)x^2+(B+2C)x+(C-A-B)=x^2+1$。 + +从而$A+C=1$,$B+2C=0$,$C-A-B=1$。 + +所以$A=C=\dfrac{1}{2}$,$B=-1$,所以$\dfrac{x^2+1}{(x+1)^2(x-1)}=\dfrac{1}{2(x+1)}-\dfrac{1}{(x+1)^2}+\dfrac{1}{2(x-1)}$。 + +$\therefore=\displaystyle{\int\left[\dfrac{1}{2(x+1)}-\dfrac{1}{(x+1)^2}+\dfrac{1}{2(x-1)}\right]\textrm{d}x}$ \medskip + +$=\dfrac{1}{2}\ln\vert x-1\vert+\dfrac{1}{2}\ln\vert x+1\vert+\dfrac{1}{x+1}+C=\dfrac{1}{2}\ln\vert x^2-1\vert+\dfrac{1}{x+1}+C$。 + \subsubsection{低阶多项式分配} -当不定积分式子形如$\displaystyle{\int\dfrac{f(x)}{g(x)}\,\textrm{d}x}$,且$f(x)$、$g(x)$都为与$x$相关的多项式,$f(x)$阶数低于$g(x)$,且$g(x)$不能因式分解为$g(x)=g_1(x)g_2(x)\cdots$时,则可以分解式子:$\displaystyle{\int\dfrac{f(x)}{g(x)}\,\textrm{d}x=a_1\int\dfrac{\textrm{d}(f_1(x))}{g_1(x)}+a_2\int\dfrac{\textrm{d}(f_2(x))}{g_2(x)}}+\cdots$,将积分式子组合成积分结果为分式的函数,如$\ln x$、$\arcsin x$、$\arctan x$等。 +当不定积分式子形如$\displaystyle{\int\dfrac{f(x)}{g(x)}\,\textrm{d}x}$,且$f(x)$、$g(x)$都为与$x$相关的多项式,$f(x)$阶数低于$g(x)$,且$g(x)$不能因式分解为$g(x)=g_1(x)g_2(x)\cdots$时,则可以分配式子:$\displaystyle{\int\dfrac{f(x)}{g(x)}\,\textrm{d}x=a_1\int\dfrac{\textrm{d}(f_1(x))}{g_1(x)}+a_2\int\dfrac{\textrm{d}(f_2(x))}{g_2(x)}}+\cdots$,将积分式子组合成积分结果为分式的函数,如$\ln x$、$\arcsin x$、$\arctan x$等。 \textbf{例题:}求$\displaystyle{\int\dfrac{x-1}{x^2+2x+3}\textrm{d}x}$。 @@ -308,6 +384,70 @@ $-\displaystyle{\int\dfrac{1}{\left(\dfrac{x+1}{\sqrt{2}}\right)^2+1}\textrm{d}x $=\dfrac{1}{2}\ln(x^2+2x+3)-\sqrt{2}\arctan\dfrac{x+1}{\sqrt{2}}+C$。 +\subsubsection{低阶多项式因式分解与分配} + +有时候一个式子需要同时用到因式分解和分配两种方式。 + +\textbf{例题:}求$\displaystyle{\int\dfrac{x^5+x^4-8}{x^3-x}\textrm{d}x}$。 + +$=\displaystyle{\int\dfrac{x^5-x^3+x^4-x^2+x^3-x+x^2+x-8}{x^3-x}\textrm{d}x}$ + +$=\int x^2\,\textrm{d}x+\int x\,\textrm{d}x+\int\textrm{d}x+\displaystyle{\int\dfrac{x^x+x-8}{x^3-x}\textrm{d}x}=\dfrac{x^3}{3}+\dfrac{x^2}{2}+x+\displaystyle{\int\dfrac{x^2+x-8}{x^3-x}\textrm{d}x}$ + +令$\dfrac{A}{x}+\dfrac{B}{x+1}+\dfrac{C}{x-1}=\dfrac{x^2+x-8}{x^3-x}$。 + +$\therefore(A+B+C)x^2+(C-B)x-A=x^2+x-8$。$A=8$,$B=-4$,$C=-3$。 + +$=\dfrac{x^3}{3}+\dfrac{x^2}{2}+x+\displaystyle{\int\dfrac{8}{x}\textrm{d}x-\int\dfrac{4}{x+1}\textrm{d}x-\int\dfrac{3}{x-1}\textrm{d}x}$ + +$=\dfrac{x^3}{3}+\dfrac{x^2}{2}+x+8\ln\vert x\vert-4\ln\vert x+1\vert-3\ln\vert x-1\vert+C$。 + +\subsubsection{有理积分与其他积分运算} + +换元积分法可以与有理积分、分部积分共同使用。 + +\textbf{例题:}求$\displaystyle{\int\dfrac{-x^2-2}{(x^2+x+1)^2}\textrm{d}x}$。 + +首先根据因式分解:$\dfrac{-x^2-2}{(x^2+x+1)^2}=\dfrac{Ax+B}{x^2+x+1}+\dfrac{Cx+D}{(x^2+x+1)^2}$。 + +$\therefore Ax^3+(A+B)x^2+(A+B+C)x+(B+D)=-x^2-2$。 + +解得:$A=0$,$B=D=-1$,$C=1$。 + +$=\displaystyle{\int\dfrac{x-1}{(x^2+x+1)^2}\textrm{d}x-\int\dfrac{1}{x^2+x+1}\textrm{d}x}$ + +$=\displaystyle{\dfrac{1}{2}\int\dfrac{\textrm{d}(x^2+x+1)}{(x^2+x+1)^2}-\dfrac{3}{2}\int\dfrac{1}{(x^2+x+1)^2}\textrm{d}x-\int\dfrac{1}{\left(x+\dfrac{1}{2}\right)^2+\left(\dfrac{\sqrt{3}}{2}\right)^2}\textrm{d}x}$ + +$=-\dfrac{1}{2(x^2+x+1)}\displaystyle{-\dfrac{3}{2}\int\dfrac{1}{(x^2+x+1)^2}\textrm{d}x}-\dfrac{2}{\sqrt{3}}\arctan\dfrac{2x+1}{\sqrt{3}}$ \smallskip + +$\because x^2+x+1=\left(x+\dfrac{1}{2}\right)^2+\left(\dfrac{\sqrt{3}}{2}\right)^2$,$\therefore$令$u=x+\dfrac{1}{2}$,$a=\dfrac{\sqrt{3}}{2}$: + +$\displaystyle{\int\dfrac{1}{(x^2+x+1)^2}\textrm{d}x=\int\dfrac{1}{(u^2+a^2)^2}\textrm{d}u=\int\dfrac{1}{\left(a^2\left(\left(\dfrac{u}{a}\right)^2+1\right)\right)^2}}$,使用第二类换元法(三角换元):$\dfrac{u}{a}=\tan t$,$u=a\tan t$,$\textrm{d}u=a\sec^2t\,\textrm{d}t$,$t=\arctan\dfrac{u}{a}$。 + +$\therefore=\displaystyle{\int\dfrac{a\sec^2t}{a^4\sec^4t}\textrm{d}t=\dfrac{1}{a^3}\int\dfrac{\textrm{d}t}{sec^2t}}=\dfrac{1}{a^3}\int\cos^2t\,\textrm{d}t=\dfrac{1}{a^3}\int(1+\cos2t)\,\textrm{d}t$ + +$=\dfrac{1}{a^3}\int\textrm{d}t+\dfrac{1}{2a^3}\int\cos2t\,\textrm{d}(2t)=\dfrac{t}{a^3}+\dfrac{\sin2t}{2a^3}=\dfrac{\arctan\dfrac{u}{a}}{a^3}+\dfrac{\sin t\cos t}{a^3}$。 + +$\because\tan t=\dfrac{u}{a}$,$\therefore\tan^2t=\dfrac{\sin^2t}{\cos^2t}=\dfrac{1-\cos^2t}{\cos2t}=\dfrac{\sin^2t}{1-\sin^2t}=\dfrac{u^2}{a^2}$ + +$\therefore\sin t=\dfrac{u}{\sqrt{a^2+u^2}}$,$\cos t=\dfrac{a}{\sqrt{a^2+u^2}}$,$\dfrac{\sin t\cos t}{a^3}=\dfrac{u}{a^2(a^2+u^2)}$。 + +$\therefore$原式$=-\dfrac{1}{2(x^2+x+1)}-\dfrac{2}{\sqrt{3}}\arctan\dfrac{2x+1}{\sqrt{3}}\displaystyle{-\dfrac{3}{2}\int\dfrac{1}{(x^2+x+1)^2}\textrm{d}x}$ + +$=-\dfrac{1}{2(x^2+x+1)}-\dfrac{2}{\sqrt{3}}\arctan\dfrac{2x+1}{\sqrt{3}}-\dfrac{3}{2}\left(\dfrac{\arctan\dfrac{u}{a}}{a^3}+\dfrac{u}{a^2(a^2+u^2)}\right)+C$ + +$=-\dfrac{1}{2(x^2+x+1)}-\dfrac{2}{\sqrt{3}}\arctan\dfrac{2x+1}{\sqrt{3}}$ + +$-\dfrac{3}{2}\left(\dfrac{\arctan\dfrac{x+\dfrac{1}{2}}{\dfrac{\sqrt{3}}{2}}}{\left(\dfrac{\sqrt{3}}{2}\right)^3}+\dfrac{x+\dfrac{1}{2}}{\left(\dfrac{\sqrt{3}}{2}\right)^2\left(\left(\dfrac{\sqrt{3}}{2}\right)^2+\left(x+\dfrac{1}{2}\right)^2\right)}\right)+C$ + +$=-\dfrac{1}{2(x^2+x+1)}-\dfrac{2}{\sqrt{3}}\arctan\dfrac{2x+1}{\sqrt{3}}$ + +$-\dfrac{3}{2}\left(\dfrac{\arctan\dfrac{2x+1}{\sqrt{3}}}{\dfrac{3\sqrt{3}}{8}}+\dfrac{x+\dfrac{1}{2}}{\dfrac{3}{4}\left(x^2+x+1\right)}\right)+C$ + +$=-\dfrac{1}{2(x^2+x+1)}-\dfrac{2}{\sqrt{3}}\arctan\dfrac{2x+1}{\sqrt{3}}-\dfrac{4}{\sqrt{3}}\arctan\dfrac{2x+1}{\sqrt{3}}-\dfrac{2x+1}{x^2+x+1}+C$ + +$=-\dfrac{4x+3}{2(x^2+x+1)}-\dfrac{6}{\sqrt{3}}\arctan\dfrac{2x+1}{\sqrt{3}}+C$。 + \section{定积分} \subsection{变限积分} diff --git a/linear-algebra/knowledge/1-determinant/determinant.pdf b/linear-algebra/knowledge/1-determinant/determinant.pdf new file mode 100644 index 0000000000000000000000000000000000000000..14e10097b02569aacf01db1580a602495b412f31 GIT 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默认页边距为1英尺与1.25英尺 +\usepackage{indentfirst} +\setlength{\parindent}{2.45em} +% 首行缩进2个中文字符 +\usepackage{setspace} +\renewcommand{\baselinestretch}{1.5} +% 1.5倍行距 +\author{Didnelpsun} +\title{行列式} +\date{} +\begin{document} +\maketitle +\pagestyle{empty} +\thispagestyle{empty} +\tableofcontents +\thispagestyle{empty} +\newpage +\pagestyle{plain} +\setcounter{page}{1} +\section{} +\end{document} diff --git a/linear-algebra/knowledge/2-matrix/matrix.pdf b/linear-algebra/knowledge/2-matrix/matrix.pdf new file mode 100644 index 0000000000000000000000000000000000000000..08ea2e26d452539cd38b3dd3b04d62b13571c6e0 GIT binary patch literal 11950 zcmbt)1yEf}vo4+l3GN!)Hx4_v2H3%a!^Yj+A-KD{hv06(J-EBOyKJ0y{`c;!d;a>* z`|r8$t?F7eYptoCnXh_vPk%jB@?sK9tjwHJr%hZ2WfPQ@M;U zAb=mhB0{tW(&$kk`jJP=uhBklzh_)@F*@!k3HC*q75OY!NF%+5Tso9V6UQDabwuU# zY3fELo;^>J6f`~{@Q#`GWDZiuiJKx)9VBps%X;TdlyH)_ZCxrIG`&)NikO1G7jvVw97AF>m1C 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+\begin{document} +\maketitle +\pagestyle{empty} +\thispagestyle{empty} +\tableofcontents +\thispagestyle{empty} +\newpage +\pagestyle{plain} +\setcounter{page}{1} +\section{} +\end{document}