{
  "id": "math-laplace-transform",
  "name": "Comprehensive Laplace Transform Table Dataset",
  "category": "math",
  "source": null,
  "updated": "2025-12-28",
  "columns": [
    {
      "key": "category",
      "label": "Category",
      "type": "string"
    },
    {
      "key": "f_t",
      "label": "f(t) (Time Domain)",
      "type": "string"
    },
    {
      "key": "F_s",
      "label": "F(s) (s-Domain)",
      "type": "string"
    },
    {
      "key": "condition",
      "label": "Condition",
      "type": "string"
    },
    {
      "key": "description",
      "label": "Description",
      "type": "string"
    }
  ],
  "rows": [
    {
      "category": "Basic",
      "f_t": "1",
      "F_s": "1/s",
      "condition": "s > 0",
      "description": "Unit step (constant)"
    },
    {
      "category": "Basic",
      "f_t": "t",
      "F_s": "1/s²",
      "condition": "s > 0",
      "description": "Ramp function"
    },
    {
      "category": "Basic",
      "f_t": "tⁿ",
      "F_s": "n!/s^(n+1)",
      "condition": "s > 0, n ≥ 0",
      "description": "Power function"
    },
    {
      "category": "Basic",
      "f_t": "t²",
      "F_s": "2/s³",
      "condition": "s > 0",
      "description": "Quadratic"
    },
    {
      "category": "Basic",
      "f_t": "t³",
      "F_s": "6/s⁴",
      "condition": "s > 0",
      "description": "Cubic"
    },
    {
      "category": "Basic",
      "f_t": "√t",
      "F_s": "√π/(2s^(3/2))",
      "condition": "s > 0",
      "description": "Square root"
    },
    {
      "category": "Exponential",
      "f_t": "e^(at)",
      "F_s": "1/(s-a)",
      "condition": "s > a",
      "description": "Exponential"
    },
    {
      "category": "Exponential",
      "f_t": "t·e^(at)",
      "F_s": "1/(s-a)²",
      "condition": "s > a",
      "description": "Exponential ramp"
    },
    {
      "category": "Exponential",
      "f_t": "tⁿ·e^(at)",
      "F_s": "n!/(s-a)^(n+1)",
      "condition": "s > a",
      "description": "Exponential power"
    },
    {
      "category": "Exponential",
      "f_t": "1 - e^(-at)",
      "F_s": "a/(s(s+a))",
      "condition": "s > 0",
      "description": "Exponential decay from 1"
    },
    {
      "category": "Trigonometric",
      "f_t": "sin(ωt)",
      "F_s": "ω/(s²+ω²)",
      "condition": "s > 0",
      "description": "Sine function"
    },
    {
      "category": "Trigonometric",
      "f_t": "cos(ωt)",
      "F_s": "s/(s²+ω²)",
      "condition": "s > 0",
      "description": "Cosine function"
    },
    {
      "category": "Trigonometric",
      "f_t": "tan(ωt)",
      "F_s": "Complex",
      "condition": "-",
      "description": "Tangent (no simple form)"
    },
    {
      "category": "Trigonometric",
      "f_t": "t·sin(ωt)",
      "F_s": "2ωs/(s²+ω²)²",
      "condition": "s > 0",
      "description": "Ramp sine"
    },
    {
      "category": "Trigonometric",
      "f_t": "t·cos(ωt)",
      "F_s": "(s²-ω²)/(s²+ω²)²",
      "condition": "s > 0",
      "description": "Ramp cosine"
    },
    {
      "category": "Trigonometric",
      "f_t": "sin²(ωt)",
      "F_s": "2ω²/(s(s²+4ω²))",
      "condition": "s > 0",
      "description": "Sine squared"
    },
    {
      "category": "Trigonometric",
      "f_t": "cos²(ωt)",
      "F_s": "(s²+2ω²)/(s(s²+4ω²))",
      "condition": "s > 0",
      "description": "Cosine squared"
    },
    {
      "category": "Damped",
      "f_t": "e^(-at)·sin(ωt)",
      "F_s": "ω/((s+a)²+ω²)",
      "condition": "s > -a",
      "description": "Damped sine"
    },
    {
      "category": "Damped",
      "f_t": "e^(-at)·cos(ωt)",
      "F_s": "(s+a)/((s+a)²+ω²)",
      "condition": "s > -a",
      "description": "Damped cosine"
    },
    {
      "category": "Hyperbolic",
      "f_t": "sinh(at)",
      "F_s": "a/(s²-a²)",
      "condition": "s > |a|",
      "description": "Hyperbolic sine"
    },
    {
      "category": "Hyperbolic",
      "f_t": "cosh(at)",
      "F_s": "s/(s²-a²)",
      "condition": "s > |a|",
      "description": "Hyperbolic cosine"
    },
    {
      "category": "Special",
      "f_t": "δ(t)",
      "F_s": "1",
      "condition": "all s",
      "description": "Dirac delta (impulse)"
    },
    {
      "category": "Special",
      "f_t": "δ(t-a)",
      "F_s": "e^(-as)",
      "condition": "all s",
      "description": "Delayed impulse"
    },
    {
      "category": "Special",
      "f_t": "u(t)",
      "F_s": "1/s",
      "condition": "s > 0",
      "description": "Unit step function"
    },
    {
      "category": "Special",
      "f_t": "u(t-a)",
      "F_s": "e^(-as)/s",
      "condition": "s > 0",
      "description": "Delayed step"
    },
    {
      "category": "Bessel",
      "f_t": "J₀(at)",
      "F_s": "1/√(s²+a²)",
      "condition": "s > 0",
      "description": "Bessel function J₀"
    },
    {
      "category": "Bessel",
      "f_t": "J₁(at)",
      "F_s": "(√(s²+a²)-s)/(a√(s²+a²))",
      "condition": "s > 0",
      "description": "Bessel function J₁"
    },
    {
      "category": "Logarithmic",
      "f_t": "ln(t)",
      "F_s": "-(ln(s)+γ)/s",
      "condition": "s > 0",
      "description": "Natural log (γ = Euler)"
    },
    {
      "category": "Logarithmic",
      "f_t": "(1-e^(-t))/t",
      "F_s": "ln((s+1)/s)",
      "condition": "s > 0",
      "description": "Logarithmic form"
    },
    {
      "category": "Error Function",
      "f_t": "erf(√t)",
      "F_s": "1/(s√(s+1))",
      "condition": "s > 0",
      "description": "Error function"
    },
    {
      "category": "Error Function",
      "f_t": "erfc(√t)",
      "F_s": "1/s - 1/(s√(s+1))",
      "condition": "s > 0",
      "description": "Complementary error"
    },
    {
      "category": "Property",
      "f_t": "f'(t)",
      "F_s": "s·F(s) - f(0)",
      "condition": "-",
      "description": "First derivative"
    },
    {
      "category": "Property",
      "f_t": "f''(t)",
      "F_s": "s²·F(s) - s·f(0) - f'(0)",
      "condition": "-",
      "description": "Second derivative"
    },
    {
      "category": "Property",
      "f_t": "∫f(τ)dτ",
      "F_s": "F(s)/s",
      "condition": "-",
      "description": "Integration"
    },
    {
      "category": "Property",
      "f_t": "f(t-a)·u(t-a)",
      "F_s": "e^(-as)·F(s)",
      "condition": "a > 0",
      "description": "Time shift"
    },
    {
      "category": "Property",
      "f_t": "e^(at)·f(t)",
      "F_s": "F(s-a)",
      "condition": "-",
      "description": "Frequency shift"
    },
    {
      "category": "Property",
      "f_t": "t·f(t)",
      "F_s": "-dF(s)/ds",
      "condition": "-",
      "description": "Multiplication by t"
    },
    {
      "category": "Property",
      "f_t": "f(at)",
      "F_s": "(1/a)·F(s/a)",
      "condition": "a > 0",
      "description": "Time scaling"
    },
    {
      "category": "Property",
      "f_t": "f(t)*g(t)",
      "F_s": "F(s)·G(s)",
      "condition": "-",
      "description": "Convolution"
    }
  ],
  "meta": {
    "seo_title": "Laplace Transform Table - Free Download (Excel/CSV/PDF)",
    "seo_description": "Get a free download of 39 Laplace Transform entries. Essential for engineers & students. Available in Excel, CSV, and PDF formats for easy use.",
    "title": "Comprehensive Laplace Transform Table Dataset",
    "description": "This dataset provides a comprehensive table of 39 Laplace Transform pairs, mapping functions from the time domain (f(t)) to the s-domain (F(s)). It includes categories, conditions, and descriptions.",
    "key_takeaways": [
      "Access 39 essential Laplace Transform pairs for quick reference.",
      "Explore common functions and their s-domain equivalents.",
      "Download the complete table in multiple convenient formats.",
      "Leverage this data for engineering, physics, and mathematics studies."
    ],
    "keywords": [
      "laplace transform table",
      "laplace transform formulas excel",
      "time domain s-domain csv",
      "f(t) F(s) dataset",
      "engineering mathematics pdf",
      "laplace transform free download"
    ],
    "use_cases": [
      "Import the **CSV** file into your Python scripts or simulation software to automate calculations involving Laplace Transforms.",
      "Use the **Excel** file to filter and sort specific transform pairs, create custom reference sheets, or integrate into engineering reports.",
      "Print the **PDF** version for quick offline reference during exams, lectures, or practical problem-solving sessions.",
      "Integrate this dataset into educational platforms or e-learning tools to provide students with a structured learning resource."
    ]
  },
  "record_count": 39,
  "title": "Comprehensive Laplace Transform Table Dataset"
}