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Inverse Laplace Calculator

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Inverse Laplace Transform Calculator

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💡 The inverse Laplace transform converts a function F(s) in the s-domain to a function f(t) in the time domain.

Enter a Laplace domain function to compute its inverse transform

Compute inverse Laplace transforms from F(s) to f(t) with our calculator. Perfect for engineering students solving differential equations. Shows solutions, poles, and plots.

What is an Inverse Laplace Calculator?

An Inverse Laplace Calculator computes the original time-domain function f(t) from its Laplace transform F(s). Essential for engineering and physics students solving differential equations and analyzing dynamic systems.

The Inverse Laplace Transform Formula

f(t) = (1/2πi) ∫ F(s)e^(st) ds

Common inverse pairs:

  • 1/s → 1
  • 1/(s-a) → e^(at)
  • ω/(s²+ω²) → sin(ωt)
  • s/(s²+ω²) → cos(ωt)

How to Use the Calculator

  1. Enter your Laplace function F(s) (e.g., “1/(s^2+4)”)
  2. Specify time variable (default: t)
  3. Set time range for plotting
  4. Select calculation method
  5. View results: time function, poles, and interactive plot

Key Terms Explained

  • Laplace Transform: Converts time-domain to complex frequency-domain
  • Poles: Values of s where F(s) becomes infinite
  • Residues: Coefficients used in inverse calculation
  • Time Domain: Original function with time variable (t)

Calculation Methods

  1. Partial Fractions: Breaks F(s) into simpler fractions
  2. Residue Theorem: Complex analysis method
  3. Table Lookup: Matches with known transform pairs

Important Notes

  1. Works best for rational functions (polynomial fractions)
  2. Some functions require advanced methods
  3. Always check assumptions (e.g., t > 0)
  4. Results may differ for discontinuous functions

Frequently Asked Questions

Q: Why can’t it compute my function?
A: The calculator handles common forms - try rewriting as partial fractions.

Q: Are initial conditions considered?
A: No - this computes the general inverse transform.

Q: Can I use variables other than ‘s’ and ‘t’?
A: Yes, but ‘s’ is standard for Laplace domain.

Q: How accurate is the plot?
A: Very accurate for continuous functions - may approximate discontinuities.

Q: What does “No inverse found” mean?
A: Your input may not be a valid Laplace transform or is too complex.

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