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Laplace Transform Calculator

Laplace Transform Calculator

Laplace Transform Calculator

Laplace Transform Calculator

Common Laplace Transforms

Time Domain Function Laplace Transform
1 \(\frac{1}{s}\)
t \(\frac{1}{s^2}\)
t^n \(\frac{n!}{s^{n+1}}\)
e^(at) \(\frac{1}{s-a}\)
sin(ωt) \(\frac{ω}{s^2 + ω^2}\)
cos(ωt) \(\frac{s}{s^2 + ω^2}\)

How to Use the Laplace Transform Calculator

  1. Access the Calculator
    • Open the HTML file in a modern web browser
    • The calculator will load with an input field and reference table
  2. Enter Your Function
    • In the input field labeled “Enter Time-Domain Function”:
    • Use ‘t’ as your variable (other variables won’t work)
    • Follow these formats:
      • Constants: 5
      • Polynomials: t^3 or 2t^2
      • Exponentials: e^(-2t) or 4e^(5t)
      • Sine functions: 3sin(4t)
      • Cosine functions: 2cos(6t)
  3. Calculate the Transform
    • Click the “Calculate” button
    • Wait a moment for the calculation and rendering
  4. View Results
    • The result will appear in the blue box below the button
    • Both original function and Laplace transform will be displayed
    • Mathematical notation will render automatically using MathJax
  5. Use the Reference Table
    • Check common transforms in the reference section
    • Use the table to verify results or find standard transforms

Valid Input Examples:

plaintext

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5                 (constant)
t                 (linear term)
t^4               (t to the 4th power)
3e^(-2t)          (exponential decay)
2e^(5t)           (exponential growth)
4sin(3t)          (sine function)
5cos(2t)          (cosine function)

Error Handling:

  • If you see a red error message:
    1. Check your syntax
    2. Ensure you’re using only supported function types
    3. Verify you’re using ‘t’ as the variable
    4. Remove any spaces in the input

Mobile Usage:

  • On smartphones:
    • Input field automatically adjusts width
    • Calculate button becomes full-width
    • Table scrolls horizontally if needed
    • Mathematical equations scale to screen size

Supported Operations:

  • Constant coefficients (e.g., 5, 3.14)
  • Integer and decimal exponents
  • Combined coefficients and exponents
  • Negative exponents for decay functions
  • Positive exponents for growth functions

Limitations:

  • Does not support:
    • Complex numbers
    • Fractional exponents
    • Composite functions
    • Differential equations
    • Partial fractions
    • Unit step functions (Heaviside)
    • Dirac delta functions

Tips for Best Results:

  1. Start with simple functions first
  2. Use parentheses for exponential arguments
  3. Include explicit multiplication signs (e.g., 3*e^(-2t))
  4. Check the reference table for formatting examples
  5. Refresh the page if equations don’t render properly

Troubleshooting:

  • If equations don’t display properly:
    1. Check internet connection (MathJax requires online access)
    2. Wait a few seconds for rendering
    3. Avoid special characters except those in the examples
    4. Ensure JavaScript is enabled in your browser

The calculator provides instant Laplace transform solutions for basic functions commonly encountered in engineering and mathematics, while maintaining clear visual feedback and error messages for unsupported inputs.

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