{
  "id": "laplace-transform",
  "version": "94b386dd9de4",
  "status": "published",
  "name": "Laplace Transform Calculator",
  "question": "What is the Laplace transform of f(t)?",
  "summary": "Finds the Laplace transform F(s) of a function f(t), term by term, as partial fractions, checked by numeric integration.",
  "category": "math",
  "subcategory": "calculus",
  "url": "https://www.acalculator.org/math/laplace-transform-calculator",
  "markdown": "https://www.acalculator.org/math/laplace-transform-calculator.md",
  "kind": "function",
  "method": "F(s) = ∫ from 0 to ∞ of e^(−st) f(t) dt. A computer algebra system transforms each term; each transform is checked against numeric integration at three values of s.",
  "assumptions": [
    "f(t) is defined for t ≥ 0.",
    "An answer that fails its check is not shown."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "f": {
        "title": "Function f(t)",
        "description": "A function of t, typed like t^2 e^(3t), sin(2t) or 3t^4 - 2t + 5.",
        "type": "string",
        "maxLength": 200
      }
    }
  },
  "outputs": {
    "F": {
      "label": "F(s) =",
      "description": "The Laplace transform: the integral of e^(−st) f(t) from 0 to ∞, for s larger than the growth rate of f.",
      "format": "math"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "f": "t^3 e^(4t)"
    },
    "outputs": {
      "F": "6/(s - 4)^4"
    },
    "text": "The Laplace transform of t^3 e^(4t) is F(s) = 6/(s - 4)^4."
  },
  "examples": [
    {
      "given": {
        "f": "t^3 e^(4t)"
      },
      "expect": {
        "F": "6/(s - 4)^4"
      },
      "source": "Trench, Elementary Differential Equations (2013), section 8.1 Introduction to the Laplace Transform, Example 8.1.5. https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/08%3A_Laplace_Transforms/8.01%3A_Introduction_to_the_Laplace_Transform"
    },
    {
      "given": {
        "f": "sin(2t)"
      },
      "expect": {
        "F": "2/(4 + s^2)"
      },
      "source": "Trench, Elementary Differential Equations (2013), section 8.1 Introduction to the Laplace Transform, Example 8.1.4 (ω = 2: ω/(s² + ω²))"
    },
    {
      "given": {
        "f": "cosh(3t)"
      },
      "expect": {
        "F": "1/(2 (s - 3)) + 1/(2 (3 + s))"
      },
      "source": "Trench, Elementary Differential Equations (2013), section 8.1 Introduction to the Laplace Transform, Example 8.1.6 (b = 3: s/(s² − 9), in partial fractions)"
    }
  ],
  "sources": [
    "William F. Trench, Elementary Differential Equations with Boundary Value Problems (2013), section 8.1 Introduction to the Laplace Transform (LibreTexts): https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/08%3A_Laplace_Transforms/8.01%3A_Introduction_to_the_Laplace_Transform"
  ],
  "related": [
    "inverse-laplace-transform",
    "integral",
    "improper-integral",
    "series"
  ],
  "changelog": []
}