{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "### **Grundlegende Operatoren und Funktionen**\n", "\n", "1. **Exponentiation** (Potenz)\n", "\n", " - `**`: Exponentiation\n", " ```python\n", " x**2 # x hoch 2\n", " ```\n", " - `exp(x)`: Exponentielle Funktion \\( e^x \\)\n", " ```python\n", " from sympy import exp\n", " exp(x)\n", " ```\n", "\n", "2. **Logarithmen**\n", "\n", " - `log(x)`: Natürlicher Logarithmus \\( \\ln(x) \\)\n", " ```python\n", " from sympy import log\n", " log(x)\n", " ```\n", " - `log(x, base)`: Logarithmus zur Basis `base`\n", " ```python\n", " log(x, 10) # Logarithmus zur Basis 10\n", " ```\n", "\n", "3. **Trigonometrische Funktionen**\n", "\n", " - `sin(x)`: Sinus\n", " ```python\n", " from sympy import sin\n", " sin(x)\n", " ```\n", " - `cos(x)`: Kosinus\n", " ```python\n", " from sympy import cos\n", " cos(x)\n", " ```\n", " - `tan(x)`: Tangens\n", " ```python\n", " from sympy import tan\n", " tan(x)\n", " ```\n", " - `csc(x)`: Kosekans (1/sin(x))\n", " ```python\n", " from sympy import csc\n", " csc(x)\n", " ```\n", " - `sec(x)`: Sekans (1/cos(x))\n", " ```python\n", " from sympy import sec\n", " sec(x)\n", " ```\n", " - `cot(x)`: Kotangens (1/tan(x))\n", " ```python\n", " from sympy import cot\n", " cot(x)\n", " ```\n", "\n", "4. **Hyperbolische Funktionen**\n", "\n", " - `sinh(x)`: Sinus hyperbolicus\n", " ```python\n", " from sympy import sinh\n", " sinh(x)\n", " ```\n", " - `cosh(x)`: Kosinus hyperbolicus\n", " ```python\n", " from sympy import cosh\n", " cosh(x)\n", " ```\n", " - `tanh(x)`: Tangens hyperbolicus\n", " ```python\n", " from sympy import tanh\n", " tanh(x)\n", " ```\n", "\n", "5. **Trigonometrische Inverse Funktionen**\n", " - `asin(x)`: Arkussinus (Inverser Sinus)\n", " ```python\n", " from sympy import asin\n", " asin(x)\n", " ```\n", " - `acos(x)`: Arkuskosinus (Inverser Kosinus)\n", " ```python\n", " from sympy import acos\n", " acos(x)\n", " ```\n", " - `atan(x)`: Arkustangens (Inverser Tangens)\n", " ```python\n", " from sympy import atan\n", " atan(x)\n", " ```\n", "\n", "---\n", "\n", "### **Algebraische Operationen**\n", "\n", "1. **Addition, Subtraktion, Multiplikation, Division**\n", "\n", " - `+`, `-`, `*`, `/`\n", " ```python\n", " x + y # Addition\n", " x - y # Subtraktion\n", " x * y # Multiplikation\n", " x / y # Division\n", " ```\n", "\n", "2. **Exponenten**\n", "\n", " - `**`: Potenz\n", " ```python\n", " x**2 # x hoch 2\n", " ```\n", "\n", "3. **Wurzeln**\n", " - `sqrt(x)`: Quadratwurzel\n", " ```python\n", " from sympy import sqrt\n", " sqrt(x)\n", " ```\n", "\n", "---\n", "\n", "### **Integral- und Ableitungsoperationen**\n", "\n", "1. **Ableitungen**\n", "\n", " - `diff(f, x)`: Erste Ableitung von `f` nach `x`\n", " ```python\n", " from sympy import diff\n", " diff(f, x)\n", " ```\n", "\n", "2. **Integrale**\n", " - `integrate(f, x)`: Unbestimmtes Integral von `f` nach `x`\n", " ```python\n", " from sympy import integrate\n", " integrate(f, x)\n", " ```\n", " - `integrate(f, (x, a, b))`: Bestimmtes Integral von `f` von `a` bis `b`\n", " ```python\n", " integrate(f, (x, 0, 1))\n", " ```\n", "\n", "---\n", "\n", "### **Summen, Produkte und Faktorisierung**\n", "\n", "1. **Summe**\n", "\n", " - `sum(iterable)`: Summe über eine Reihe oder Liste\n", " ```python\n", " from sympy import Sum, symbols\n", " n = symbols('n')\n", " Sum(n, (n, 1, 10))\n", " ```\n", "\n", "2. **Produkt**\n", "\n", " - `product(iterable)`: Produkt über eine Reihe oder Liste\n", " ```python\n", " from sympy import Product\n", " Product(n, (n, 1, 10))\n", " ```\n", "\n", "3. **Faktorisierung**\n", "\n", " - `factor(x)`: Faktorisieren eines Polynoms\n", " ```python\n", " from sympy import factor\n", " factor(x**2 - 4)\n", " ```\n", "\n", "4. **Entwicklung in Reihen**\n", " - `series(f, x)`: Taylor-Reihe von `f` um `x`\n", " ```python\n", " from sympy import series\n", " series(f, x)\n", " ```\n", "\n", "---\n", "\n", "### **Matrixoperationen**\n", "\n", "1. **Matrix**\n", "\n", " - `Matrix([[1, 2], [3, 4]])`: Erstellen einer Matrix\n", " ```python\n", " from sympy import Matrix\n", " A = Matrix([[1, 2], [3, 4]])\n", " ```\n", "\n", "2. **Determinante**\n", "\n", " - `det(A)`: Determinante einer Matrix\n", " ```python\n", " A.det()\n", " ```\n", "\n", "3. **Inverses einer Matrix**\n", " - `inv(A)`: Inverse der Matrix `A`\n", " ```python\n", " A.inv()\n", " ```\n", "\n", "---\n", "\n", "### **Grenzwerte und Limes**\n", "\n", "1. **Grenzwert (Limit)**\n", "\n", " - `limit(f, x, c)`: Grenzwert von `f` für `x → c`\n", " ```python\n", " from sympy import limit\n", " limit(f, x, 0)\n", " ```\n", "\n", "2. **Unendlicher Grenzwert**\n", " - `limit(f, x, oo)`: Grenzwert für `x → ∞`\n", " ```python\n", " limit(f, x, float('inf'))\n", " ```\n", "\n", "---\n", "\n", "### **Differentialgleichungen**\n", "\n", "1. **Lösen von Differentialgleichungen**\n", " - `dsolve(eq, func)`: Lösen einer gewöhnlichen Differentialgleichung\n", " ```python\n", " from sympy import Function, dsolve, Derivative\n", " f = Function('f')\n", " dsolve(Derivative(f(x), x) - f(x), f(x))\n", " ```\n", "\n", "---\n", "\n", "### **Sonstige nützliche Funktionen**\n", "\n", "1. **Ganze Zahlen (Modulararithmetik)**\n", "\n", " - `mod(x, y)`: Modulo-Operation\n", " ```python\n", " from sympy import mod\n", " mod(x, y)\n", " ```\n", "\n", "2. **Boden- und Deckenfunktion**\n", " - `floor(x)`: Abrunden auf die nächste ganze Zahl\n", " ```python\n", " from sympy import floor\n", " floor(x)\n", " ```\n", " - `ceiling(x)`: Aufrunden auf die nächste ganze Zahl\n", " ```python\n", " from sympy import ceiling\n", " ceiling(x)\n", " ```\n", "\n", "---\n", "\n", "### **Zusammenfassung der wichtigsten Funktionen**\n", "\n", "| **Funktion** | **SymPy-Befehl** | **Beschreibung** |\n", "| --------------------------- | --------------------------------------------- | -------------------------------------- |\n", "| Exponentiation | `exp(x)` | \\( e^x \\) |\n", "| Trigonometrische Funktionen | `sin(x)`, `cos(x)`, `tan(x)` | Sinus, Kosinus, Tangens |\n", "| Ableitung | `diff(f, x)` | Ableitung von `f` nach `x` |\n", "| Integral | `integrate(f, x)` | Unbestimmtes Integral von `f` nach `x` |\n", "| Summe | `Sum(expression, (variable, start, end))` | Summe über eine Reihe |\n", "| Produkt | `Product(expression, (variable, start, end))` | Produkt über eine Reihe |\n", "| Faktorisierung | `factor(f)` | Faktorisierung von `f` |\n", "| Grenzwert | `limit(f, x, c)` | Grenzwert von `f` für `x → c` |\n", "| Matrixoperationen | `Matrix([[1, 2], [3, 4]])` | Erstellen von Matrizen |\n", "| Differentialgleichung lösen | `dsolve(eq, func)` | Lösen von Differentialgleichungen |\n", "\n", "---\n", "\n", "\n", "siehe https://chatgpt.com/share/673b2049-b010-8013-a3d7-9552c03da480\n" ] } ], "metadata": { "language_info": { "name": "python" } }, "nbformat": 4, "nbformat_minor": 2 }