Laplace Transform Calculator

What is the Laplace Transform?

The Laplace Transform is a powerful mathematical tool used primarily in engineering and physics to simplify the analysis of linear time-invariant (LTI) systems. It converts a function of time, f(t), which often represents a signal or system response, into a function of complex frequency, F(s).

Mathematically, the Laplace Transform of a function f(t), defined for t ≥ 0, is given by the integral:

L{f(t)}(s) = F(s) = ∫0∞ e-st f(t) dt

Here, s is a complex variable, s = σ + jω, where σ represents exponential decay or growth, and ω represents angular frequency. The key benefit is that this transformation converts complex operations like differentiation and integration in the time domain (t) into simpler algebraic operations (multiplication and division) in the complex frequency domain (s). This makes solving linear ordinary differential equations significantly easier.

How This Calculator Works

This online Laplace Transform calculator determines the transform F(s) for a given function f(t). It works by recognizing common function patterns and applying known Laplace Transform pairs.

1. Input: You enter your time-domain function f(t) into the input field.
2. Parsing: The calculator attempts to match your input against a predefined list of common functions using pattern recognition (regular expressions). Spaces are generally ignored.
3. Transformation: If a recognized pattern is found, the tool applies the corresponding standard Laplace Transform formula.
4. Output: The resulting Laplace Transform F(s) is displayed in the result area. If the input function is not recognized or is in an unsupported format, an error message will be shown.

Currently, this calculator supports the following basic functions (where 'a' and 'n' are constants, and 'n' must be a non-negative integer):

• Constant: c → c/s
• Power of t: t^n → n! / s^(n+1) (e.g., t, t^2, t^3)
• Exponential: exp(a*t) or e^(a*t) → 1 / (s - a) (e.g., exp(t), e^(-2*t))
• Sine: sin(a*t) → a / (s^2 + a^2) (e.g., sin(t), sin(5t))
• Cosine: cos(a*t) → s / (s^2 + a^2) (e.g., cos(t), cos(0.5*t))

Note: This tool focuses on basic, single-term functions. It does not currently support combinations (e.g., t*sin(t)), sums (e.g., t + 5), or more complex functions like Dirac delta or Heaviside step functions directly, although some might be representable (e.g., a constant c is like c*u(t) where u(t) is the Heaviside function).

Frequently Asked Questions (FAQs)

Q1: What is 's' in the Laplace Transform F(s)?

s is a complex variable, often written as s = σ + jω. It represents the complex frequency. The real part σ (sigma) relates to exponential decay or growth of the signal, and the imaginary part ω (omega) relates to its angular frequency (oscillation).

Q2: Why are Laplace Transforms useful?

They are incredibly useful in engineering (especially electrical and control systems) and physics because they transform linear ordinary differential equations (ODEs) with constant coefficients into algebraic equations. Solving algebraic equations is much simpler than solving ODEs directly. Once the solution is found in the 's' domain (F(s)), the Inverse Laplace Transform can be used to convert it back to the time domain (f(t)).

Q3: What input formats does this calculator accept?

Please use standard mathematical notation for the supported functions listed in the "How This Calculator Works" section. Examples include: 5, t, t^3, exp(2*t), e^(-t), sin(4*t), cos(t). Ensure you use * for multiplication where necessary (e.g., 2*t in exponents or arguments). For t^n, n must be a whole number (0, 1, 2, ...).

Q4: Can this calculator handle functions like `t * sin(t)` or `f(t) + g(t)`?

No, not currently. This calculator is designed for basic, single-term functions. It does not parse or apply rules for multiplication of functions (like the frequency convolution theorem) or the linearity property (transforming sums term-by-term). For more complex functions, you might need more advanced software like WolframAlpha, MATLAB, or Symbolab, or consult Laplace Transform tables for relevant theorems.

Q5: Is the result F(s) exact?

Yes, for the functions it recognizes, this calculator applies the exact analytical formula for the Laplace Transform. It performs symbolic manipulation based on predefined rules, not numerical approximation.

Q6: What does the factorial symbol `n!` mean in the `t^n` transform?

The factorial `n!` (read as "n factorial") is the product of all positive integers up to `n`. For example, 3! = 3 * 2 * 1 = 6, and 5! = 5 * 4 * 3 * 2 * 1 = 120. By definition, 0! = 1. This is used in the formula L{t^n} = n! / s^(n+1).