{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sp\n",
    "\n",
    "s, t = sp.symbols('s, t')\n",
    "z = sp.symbols('z', real = True, positive = True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\left(\\cos{\\left(3 t \\right)} - 1.0\\right) \\theta\\left(t\\right)$"
      ],
      "text/plain": [
       "-(cos(3*t) - 1.0)*Heaviside(t)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expression = 1/s* 9/(s**2+0*s+9)\n",
    "\n",
    "a=(sp.inverse_laplace_transform(expression, s, t))\n",
    "a=sp.N(a,5)\n",
    "a=sp.simplify(a)\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0.17082 e^{- \\frac{9 t}{2} - \\frac{3 \\sqrt{5} t}{2}} \\theta\\left(t\\right) - 1.1708 e^{- \\frac{9 t}{2} + \\frac{3 \\sqrt{5} t}{2}} \\theta\\left(t\\right) + 1.0 \\theta\\left(t\\right)$"
      ],
      "text/plain": [
       "0.17082*exp(-9*t/2 - 3*sqrt(5)*t/2)*Heaviside(t) - 1.1708*exp(-9*t/2 + 3*sqrt(5)*t/2)*Heaviside(t) + 1.0*Heaviside(t)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expression = 1/s* 9/(s**2+9*s+9)\n",
    "\n",
    "a=(sp.inverse_laplace_transform(expression, s, t))\n",
    "a=sp.N(a,5)\n",
    "a=sp.simplify(a)\n",
    "a=sp.powsimp(a)\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1.0 \\theta\\left(t\\right) - 1.0 e^{- 3 t} \\left(e^{t}\\right)^{1.5} \\cos{\\left(\\frac{3 \\sqrt{3} t}{2} \\right)} \\theta\\left(t\\right) - 0.6 e^{- \\frac{3 t}{2}} \\sin{\\left(\\frac{3 \\sqrt{3} t}{2} \\right)} \\theta\\left(t\\right)$"
      ],
      "text/plain": [
       "1.0*Heaviside(t) - 1.0*exp(-3*t)*exp(t)**1.5*cos(3*sqrt(3)*t/2)*Heaviside(t) - 0.6*exp(-3*t/2)*sin(3*sqrt(3)*t/2)*Heaviside(t)"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expression = 1/s* 9/(s**2+3*s+9)\n",
    "\n",
    "a=(sp.inverse_laplace_transform(expression, s, t))\n",
    "a=sp.N(a,2)\n",
    "a=sp.simplify(a)\n",
    "a=sp.powsimp(a)\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
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   "nbconvert_exporter": "python",
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