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如何平滑图形 $\chi(t)=\sen(\pi t)\exp\left(\dfrac{-1}{\sen^2(\pi t)}\right)$

tag-icon • tag-icon tikz-pgf graphics plot asymptote integral
如何平滑图形 $\chi(t)=\sen(\pi t)\exp\left(\dfrac{-1}{\sen^2(\pi t)}\right)$

我正在绘制这两个函数$$\chi(t)=\sin(\pi t)\exp\left(\dfrac{-1}{\sin^2(\pi t)}\right)$$和的图形$\rho(t)=\sqrt{\frac{1}{A}\int_0^{t+1}\chi(x)dx}$，其中$A=\int_0^1 \chi(x)dx$。

在此处输入图片描述

我只成功绘制了第一个图形，但我没有得到任何平滑的东西，我知道该函数$\chi$没有在整数中定义，但它可以通过$\chi(x)=0$在 x 为整数时放置来修复（我不知道如何绘制所有函数）。

\documentclass[border=5mm]{standalone}
\usepackage{tikz}
\begin{document}

\begin{tikzpicture}
\draw[-stealth] (-2.5,0) -- (2.5,0) node[below]{$t$};;
\foreach \X in {-2,-1,1,2} {\draw (\X,0.1) -- (\X,-0.1) node[below]{$\X$}; };
\draw[-stealth] (0,-1) -- (0,1) node[left]{$\chi(t)$};

\draw[smooth,domain = -2:2, color=blue] plot(\x,{exp(-1/(sin(deg(\x*pi)))^2)*sin(deg(\x*pi))});
\end{tikzpicture}
\end{document}

阴谋： 在此处输入图片描述

正如您在代码中看到的，我尝试绘制它，错误来自我上面提到的不连续点。我希望它绘制时没有错误，而且更平滑，例如与 geogebra 相比。

在此处输入图片描述

关于第二个函数，我不知道如何绘制积分图，我看到了一些相关的问题，但在编译时我没有得到任何结果。如果有错误，我很抱歉，我在 LaTeX 中绘图方面还是个新手。

更新：我已经得到了$chi(t)$函数，但是代码很长，为了不扭曲原始问题，我将把我的解决方案放在答案中。请记住，即使我没有得到图表$\rpho(t)$，如果我做错了，我很抱歉。

答案1

尝试一下渐近线。

编译于http://asymptote.ualberta.ca/。

请注意，这test=64对我的电脑来说是一个幸运数字。

import graph;
size(300,200,false);

real chi(real t)
{
  real a=sin(pi*t);
  return (a != 0) ? a*exp(-1/a^2) : 0;
}
real rho(real t)
{
  return sqrt(simpson(chi,0,t+1)/simpson(chi,0,1));
}

int test=64;
guide g1=graph(chi,0,3.5,300,Spline);
guide g2=graph(rho,0,3.5,test,Spline);
draw(Label("$\chi(t)$",Relative(.4),black),g1,red+0.7bp);
draw(Label("$\rho(t)$",Relative(.12),LeftSide,black),g2,blue+1bp);

xaxis(Label("$x$",position=EndPoint, align=NE), xmin=-0.3, xmax=3.5,
      Ticks(scale(.85)*Label(), NoZero, endlabel=false, Step=0.5,
            end=false, Size=.8mm, pTick=black),
      Arrow);
yaxis(Label("$y$",position=EndPoint, align=NE), ymin=-0.4, ymax=1.2,
      Ticks(scale(.85)*Label(), NoZero, endlabel=false, Step=0.2,
            end=false, Size=.8mm, pTick=black),
      Arrow);

在此处输入图片描述

答案2

这是我获得的功能$chi(t)$： 在此处输入图片描述

\documentclass[border=1mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.15}
\usepackage{mathrsfs}
\usetikzlibrary{arrows}
\pagestyle{empty}
\begin{document}
\definecolor{qqqqff}{rgb}{0,0,1}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1cm,y=1cm]
\begin{axis}[
x=1cm,y=2cm,
axis lines=middle,
xmin=-2.592785508994096,
xmax=2.548506587329168,
ymin=-0.6,
ymax=0.6,
xtick={0,1,...,0},
ytick={0,0.5,...,0},]
\clip(-2.592785508994096,-0.4025540845460146) rectangle (2.548506587329168,0.4257312538518619);
\draw[color=qqqqff] (-2.001536917916929,0) -- (-1.9886836876761207,0);
\draw[color=qqqqff] (-1.9886836876761207,0) -- (-1.9758304574353125,0);
\draw[color=qqqqff] (-1.9758304574353125,0) -- (-1.9629772271945043,0);
\draw[color=qqqqff] (-1.9629772271945043,0) -- (-1.950123996953696,0);
\draw[color=qqqqff] (-1.950123996953696,0) -- (-1.937270766712888,0);
\draw[color=qqqqff] (-1.937270766712888,0) -- (-1.9244175364720797,0);
\draw[color=qqqqff] (-1.9244175364720797,0) -- (-1.9115643062312715,0);
\draw[color=qqqqff] (-1.9115643062312715,0) -- (-1.8987110759904633,0);
\draw[color=qqqqff] (-1.8987110759904633,0) -- (-1.885857845749655,0);
\draw[color=qqqqff] (-1.885857845749655,0) -- (-1.873004615508847,0);
\draw[color=qqqqff] (-1.873004615508847,0) -- (-1.8601513852680387,0.0016915696259532037);
\draw[color=qqqqff] (-1.8601513852680387,0.0016915696259532037) -- (-1.8472981550272305,0.004221067810271749);
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\end{tikzpicture}
\end{document}

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