Comprehensive Laplace Transform Table Dataset
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.
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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.
Showing 39 of 39
| Category | f(t) (Time Domain) | F(s) (s-Domain) | Condition | Description |
|---|---|---|---|---|
| Basic | 1 | 1/s | s > 0 | Unit step (constant) |
| Basic | t | 1/s² | s > 0 | Ramp function |
| Basic | tⁿ | n!/s^(n+1) | s > 0, n ≥ 0 | Power function |
| Basic | t² | 2/s³ | s > 0 | Quadratic |
| Basic | t³ | 6/s⁴ | s > 0 | Cubic |
| Basic | √t | √π/(2s^(3/2)) | s > 0 | Square root |
| Exponential | e^(at) | 1/(s-a) | s > a | Exponential |
| Exponential | t·e^(at) | 1/(s-a)² | s > a | Exponential ramp |
| Exponential | tⁿ·e^(at) | n!/(s-a)^(n+1) | s > a | Exponential power |
| Exponential | 1 - e^(-at) | a/(s(s+a)) | s > 0 | Exponential decay from 1 |
| Trigonometric | sin(ωt) | ω/(s²+ω²) | s > 0 | Sine function |
| Trigonometric | cos(ωt) | s/(s²+ω²) | s > 0 | Cosine function |
| Trigonometric | tan(ωt) | Complex | - | Tangent (no simple form) |
| Trigonometric | t·sin(ωt) | 2ωs/(s²+ω²)² | s > 0 | Ramp sine |
| Trigonometric | t·cos(ωt) | (s²-ω²)/(s²+ω²)² | s > 0 | Ramp cosine |
| Trigonometric | sin²(ωt) | 2ω²/(s(s²+4ω²)) | s > 0 | Sine squared |
| Trigonometric | cos²(ωt) | (s²+2ω²)/(s(s²+4ω²)) | s > 0 | Cosine squared |
| Damped | e^(-at)·sin(ωt) | ω/((s+a)²+ω²) | s > -a | Damped sine |
| Damped | e^(-at)·cos(ωt) | (s+a)/((s+a)²+ω²) | s > -a | Damped cosine |
| Hyperbolic | sinh(at) | a/(s²-a²) | s > |a| | Hyperbolic sine |
| Hyperbolic | cosh(at) | s/(s²-a²) | s > |a| | Hyperbolic cosine |
| Special | δ(t) | 1 | all s | Dirac delta (impulse) |
| Special | δ(t-a) | e^(-as) | all s | Delayed impulse |
| Special | u(t) | 1/s | s > 0 | Unit step function |
| Special | u(t-a) | e^(-as)/s | s > 0 | Delayed step |
| Bessel | J₀(at) | 1/√(s²+a²) | s > 0 | Bessel function J₀ |
| Bessel | J₁(at) | (√(s²+a²)-s)/(a√(s²+a²)) | s > 0 | Bessel function J₁ |
| Logarithmic | ln(t) | -(ln(s)+γ)/s | s > 0 | Natural log (γ = Euler) |
| Logarithmic | (1-e^(-t))/t | ln((s+1)/s) | s > 0 | Logarithmic form |
| Error Function | erf(√t) | 1/(s√(s+1)) | s > 0 | Error function |
| Error Function | erfc(√t) | 1/s - 1/(s√(s+1)) | s > 0 | Complementary error |
| Property | f'(t) | s·F(s) - f(0) | - | First derivative |
| Property | f''(t) | s²·F(s) - s·f(0) - f'(0) | - | Second derivative |
| Property | ∫f(τ)dτ | F(s)/s | - | Integration |
| Property | f(t-a)·u(t-a) | e^(-as)·F(s) | a > 0 | Time shift |
| Property | e^(at)·f(t) | F(s-a) | - | Frequency shift |
| Property | t·f(t) | -dF(s)/ds | - | Multiplication by t |
| Property | f(at) | (1/a)·F(s/a) | a > 0 | Time scaling |
| Property | f(t)*g(t) | F(s)·G(s) | - | Convolution |
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.