Laplace Transform Calculator

How to Use This Calculator

1. Select f(t) from the dropdown (1, t, t², e^(at), sin(at), cos(at), δ(t)).
2. Enter the parameter a (used for e^(at), sin(at), cos(at)).
3. Enter an s value to evaluate F(s) numerically.
4. The Linearity tab combines two transforms with coefficients.
5. The Professional tab shows derivative and integral transform properties.

Formula

Example

Frequently Asked Questions

• What is the Laplace transform? ▼
  The Laplace transform converts a function f(t) into F(s) = ∫₀^∞ f(t)e^(−st) dt. It transforms differential equations into algebraic equations, making them easier to solve.
• What is L{e^(at)}? ▼
  L{e^(at)} = 1/(s−a), valid for Re(s) > a. For example L{e^(2t)} = 1/(s−2) with region of convergence Re(s) > 2.
• What is the derivative property of the Laplace transform? ▼
  L{f'(t)} = sF(s) − f(0). This converts differentiation in the t-domain to multiplication by s in the s-domain, which is why Laplace transforms simplify ODE solving.
• What is the region of convergence (ROC)? ▼
  The ROC is the set of s values for which the Laplace integral converges. For e^(at) it is Re(s) > a. For bounded functions like sin(at) it is Re(s) > 0.
• What is the convolution theorem? ▼
  L{f*g} = F(s)·G(s). Convolution in the time domain equals multiplication in the s-domain. The inverse is used to find responses of linear systems.

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