{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Function Plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from smpl import plot\n",
    "import numpy as np\n",
    "import smpl\n",
    "import math\n",
    "smpl.__version__"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## without uncertainties"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\dot x = 1- \\exp(- x^2)$\n",
    "\n",
    "Fixed point $x = 0$ and \n",
    "\n",
    "$\\ddot x = -2x \\exp(-x^2) \\implies \\ddot x(x = 0)=0$\n",
    "\n",
    "only metastable for $x\\lt0$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plot.function( lambda x : 1- np.exp(-x**2), xaxis=\"$x$\", yaxis=\"$\\\\dot x$\",xmin=-10, xmax=10 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\dot x = \\ln x$\n",
    "\n",
    "Fixed point $x = 1$\n",
    "\n",
    "$$\\ddot x = \\frac{1}{x} \\implies \\ddot x(x=1) = 1 > 0$$\n",
    "\n",
    "$\\implies$  unstable"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plot.function( lambda x : np.log(x), xaxis=\"$x$\", yaxis=\"$\\\\dot x$\",xmin=0.1, xmax=5 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\dot x = -\\tan x$\n",
    "Fixed points for $x=0$ or $x=\\pm n\\pi$ with $n\\in \\mathbb{N}$\n",
    "\n",
    "$$\\ddot x = -\\frac{1}{\\cos^2(x)}$$\n",
    "\n",
    "\n",
    "$$\\ddot x(x=0) = -1 \\lt 0$$\n",
    "\n",
    "$$\\ddot x(x=n \\pi) = -1 \\lt 0$$\n",
    "\n",
    "$\\implies$ stable"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plot.function( lambda x : -np.tan(x), xaxis=\"$x$\", yaxis=\"$\\\\dot x$\",xmin=0.1, xmax=5,steps=100 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## with uncertainties"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import uncertainties as unc\n",
    "a = unc.ufloat(1,0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plot.function(lambda x : 1- a*np.exp(-x**2), xaxis=\"$x$\", yaxis=\"$\\\\dot x$\",xmin=-1, xmax=1,sigmas=1 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Complex"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from smpl.stat import fft\n",
    "y = np.sin(np.arange(256))\n",
    "print(len(fft(y)))\n",
    "plot.data(*fft(y),label=\"FFT\",fmt=\"-\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from smpl.stat import fft\n",
    "plot.data(*fft(np.sin(np.arange(256))),*fft(np.sin(1/np.pi*np.arange(100))),label=\"FFT\",fmt=\"-\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## without xmin and xmax\n",
    "xmin and xmax will have to be guessed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from smpl import plot\n",
    "plot.function(lambda x: x**2,)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from smpl import plot\n",
    "import numpy as np\n",
    "def f(x):\n",
    "    return np.exp(x)\n",
    "plot.function(f,label=\"exp\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from smpl import plot\n",
    "from smpl import functions as f\n",
    "def gauss(x):\n",
    "    \"\"\"Gauss\"\"\"\n",
    "    return f.gauss(x,0,1,3,0)\n",
    "plot.function(gauss)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gauss(x):\n",
    "    return np.arctan(x)\n",
    "plot.function(gauss)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gauss(x):\n",
    "    return np.tan(x)\n",
    "plot.function(gauss)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gauss(x):\n",
    "    return np.log(x)\n",
    "plot.function(gauss)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gauss(x):\n",
    "    return x**3+5*x**2-2\n",
    "plot.function(gauss)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gauss(x):\n",
    "    return x**0.5\n",
    "plot.function(gauss)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Guessing the interesting regions of a function can't always be correct/satisfactory, especially in numerical unstable regions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "c=299792458#m/s\n",
    "h=4.13566769692*10**-15#eVs\n",
    "kb=8.617333262*10**-5#eV/K\n",
    "T=273\n",
    "def Strahlungsgesetz(x):\n",
    "    return 8*np.pi/c**3*h*x**3/(np.exp((h*x)/(kb*T))-1)\n",
    "plot.function(Strahlungsgesetz,xaxis=\"$x$\", yaxis=\"$\\\\dot x$\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plot.function(Strahlungsgesetz,xaxis=\"$x$\", yaxis=\"$\\\\dot x$\",xmin=1e-7-2e-2,xmax=1e-7+2e-2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plot.function(Strahlungsgesetz,xaxis=\"$x$\", yaxis=\"$\\\\dot x$\",xmin=1,xmax=0.3e15)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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