{
  "id": "inverse-laplace-transform",
  "version": "7848b966f35c",
  "status": "published",
  "name": "Inverse Laplace Transform Calculator",
  "question": "What is the inverse Laplace of F(s)?",
  "summary": "Finds f(t) from its Laplace transform F(s), by partial fractions and term by term, checked by numeric integration.",
  "category": "math",
  "subcategory": "calculus",
  "url": "https://www.acalculator.org/math/inverse-laplace-transform-calculator",
  "markdown": "https://www.acalculator.org/math/inverse-laplace-transform-calculator.md",
  "kind": "function",
  "method": "F(s) is split into partial fractions; a computer algebra system inverts each one, and each result is checked: its Laplace transform, by numeric integration, must equal the fraction at three values of s.",
  "assumptions": [
    "f(t) is found 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": "Transform F(s)",
        "description": "A function of s, typed like (3s + 8)/(s^2 + 2s + 5) or 1/(s^2 - 1).",
        "type": "string",
        "maxLength": 200
      }
    }
  },
  "outputs": {
    "f": {
      "label": "f(t) =",
      "description": "The function whose Laplace transform is F(s).",
      "format": "math"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "F": "(3s + 2)/(s^2 - 3s + 2)"
    },
    "outputs": {
      "f": "8 e^(2t) - 5 e^t"
    },
    "text": "The inverse Laplace transform of (3s + 2)/(s^2 - 3s + 2) is f(t) = 8 e^(2t) - 5 e^t."
  },
  "examples": [
    {
      "given": {
        "F": "(3s + 2)/(s^2 - 3s + 2)"
      },
      "expect": {
        "f": "8 e^(2t) - 5 e^t"
      },
      "source": "Trench, Elementary Differential Equations (2013), section 8.2 The Inverse Laplace Transform, Example 8.2.4. https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/08%3A_Laplace_Transforms/8.02%3A_The_Inverse_Laplace_Transform"
    },
    {
      "given": {
        "F": "1/(s^2 - 1)"
      },
      "expect": {
        "f": "e^t/2 - 1/(2 e^t)"
      },
      "source": "Trench, Elementary Differential Equations (2013), section 8.2 The Inverse Laplace Transform, Example 8.2.1(a) (sinh t)"
    },
    {
      "given": {
        "F": "(s^2 - 5s + 7)/(s + 2)^3"
      },
      "expect": {
        "f": "1/e^(2t) - 9t/e^(2t) + 21t^2/(2 e^(2t))"
      },
      "source": "Trench, Elementary Differential Equations (2013), section 8.2 The Inverse Laplace Transform, Example 8.2.7 (e^(−2t)(1 − 9t + 21t²/2))"
    }
  ],
  "sources": [
    "William F. Trench, Elementary Differential Equations with Boundary Value Problems (2013), section 8.2 The Inverse Laplace Transform (LibreTexts): https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/08%3A_Laplace_Transforms/8.02%3A_The_Inverse_Laplace_Transform"
  ],
  "related": [
    "laplace-transform",
    "integral",
    "improper-integral",
    "series"
  ],
  "changelog": []
}