{
  "$schema": "https://siliconverify.com/schemas/tool-v1.json",
  "slug": "laplace",
  "title": "Inverse Laplace Solver — F(s) → f(t)",
  "category": "signal-processing",
  "category_label": "Signal processing",
  "urls": {
    "interactive": "https://siliconverify.com/laplace.html",
    "static_sheet": "https://siliconverify.com/ref-laplace.html",
    "api_doc": "https://siliconverify.com/api/laplace.json",
    "blog_index": "https://siliconverify.com/llms-signalprocessing.txt"
  },
  "description": "The Inverse Laplace Transform Solver takes F(s) as a rational function (numerator and denominator polynomials) and returns f(t) for t ≥ 0. Supports simple poles, repeated poles, complex conjugate pairs, and Dirac delta terms. Use it to go from transfer function H(s) to impulse response h(t), or from block diagram to time-domain step response.",
  "keywords": [
    "inverse Laplace",
    "Laplace transform",
    "partial fractions",
    "ROC",
    "F(s) to f(t)",
    "symbolic"
  ],
  "pricing": {
    "class": "workbench-tool",
    "protocol": "x402",
    "network": "base",
    "currency": "USDC",
    "price_atomic": 1000,
    "price_usd": 0.001,
    "payTo": "0x97DAA5649Fd7Dfe3e46fCd9f75516e36E750eed7",
    "static_sheet_price_atomic": 1000,
    "export_price_atomic": 5000
  },
  "content": {
    "formulas": [
      "L{f(t)} = F(s) = ∫₀^∞ f(t)·e^{−s·t} dt",
      "L⁻¹{1/(s−a)} = e^{a·t}·u(t) (simple pole)",
      "L⁻¹{n!/(s−a)^{n+1}} = tⁿ·e^{a·t}·u(t) (repeated pole, multiplicity n+1)",
      "L⁻¹{(s+a)/((s+a)² + b²)} = e^{−a·t}·cos(b·t)·u(t) (complex pair)",
      "Final value: lim t→∞ f(t) = lim s→0 s·F(s) (if poles of sF(s) are in LHP)",
      "Initial value: f(0⁺) = lim s→∞ s·F(s)"
    ],
    "citations": [
      "Oppenheim & Willsky, \"Signals and Systems,\" 2nd ed., 1997, Ch. 9.",
      "Spiegel, M. R., \"Schaum's Outline of Laplace Transforms,\" McGraw-Hill, 1965."
    ],
    "faq": [
      {
        "q": "What if F(s) has a pole at s = 0?",
        "a": "Then f(t) has a constant (u(t)) or ramp (t·u(t)) component. The partial-fraction expansion includes a 1/s or 1/s² term."
      },
      {
        "q": "Can I inverse-Laplace a transfer function with a delay e^{−s·T}?",
        "a": "Yes, but the inverse transform is f(t−T)·u(t−T). The workbench handles delays by factoring them out and time-shifting the inverse result."
      }
    ]
  },
  "accessibility": {
    "crawlable": true,
    "javascript_required": true,
    "static_alternative": "/ref-laplace.html",
    "structured_data": [
      "TechArticle",
      "SoftwareApplication",
      "FAQPage",
      "BreadcrumbList"
    ]
  },
  "x402_challenge_example": {
    "request": "curl -A 'GPTBot/1.0' -i https://siliconverify.com/laplace.html",
    "response_headers": [
      "HTTP/1.1 402 Payment Required",
      "X-Siliconverify-Resource-Class: workbench-tool",
      "X-Siliconverify-Price-Atomic: 1000",
      "X-Siliconverify-Price-USD: 0.001"
    ],
    "response_body": "{\n  \"x402Version\": 1,\n  \"accepts\": [\n    {\n      \"scheme\": \"exact\",\n      \"network\": \"base\",\n      \"maxAmountRequired\": \"1000\",\n      \"resource\": \"https://siliconverify.com/laplace.html\",\n      \"description\": \"SiliconVerify workbench tool — interactive browser-based engineering calculator. Client-side computation, no server cost.\",\n      \"mimeType\": \"text/html\",\n      \"payTo\": \"0x97DAA5649Fd7Dfe3e46fCd9f75516e36E750eed7\",\n      \"validAfter\": 1783274656,\n      \"validBefore\": 1783274956\n    }\n  ]\n}",
    "paid_request": "curl -A 'GPTBot/1.0' -H 'X-PAYMENT: 0x<txhash>' -i https://siliconverify.com/laplace.html"
  },
  "last_updated": "2026-07-05"
}