Compute Laplace transforms and inverses for common functions. Visualize time-domain and s-domain representations side by side.
A function written in t can be converted into its Laplace-domain expression, and an expression in s can be converted back to a common time-domain form. The calculator presents the symbolic transform together with time-domain and s-domain views so you can relate the formula to both representations. It is intended for standard transform pairs, including forms used in signals and control work. The usual source of confusion is treating the displayed algebraic expression as the whole answer: convergence conditions and the chosen one-sided or bilateral convention can matter, especially for inverse transforms.
A concrete input and expected output from the current implementation.
Input
t (with forward transform selected)
Expected output
1/s^2
The forward transform is the integral from 0 to infinity of t times e^(-st). Evaluating that integral gives 1/s^2, with convergence for Re(s) > 0.
In forward mode, the mathematical relationship is F(s) = integral from 0 to infinity of f(t)e^(-st) dt, provided the integral converges. Powers, exponentials, and sine or cosine functions produce their standard expressions in s. The paired displays show the entered time function and its transform together, making the symbolic mapping easier to inspect.
In inverse mode, the input is an expression in s and the result is a time-domain function whose forward transform matches it under the calculator's convention. For rational expressions, poles and repeated factors are reflected in exponential, oscillatory, or polynomial time terms. Algebraically different-looking answers may still represent the same function after simplification.
Not by itself. It can supply the forward and inverse transforms used in a Laplace-domain solution, but you still have to form the transformed equation, include the correct initial values, solve for the unknown expression in s, and request the inverse. A plotted transform pair does not verify that the original equation or its conditions were entered correctly.
Yes, if the desired response is the inverse transform of the expression you enter. A transfer function alone normally represents a system, not its response to every possible input. For a particular zero-state response, first multiply the transfer function by the input's Laplace transform, then invert the resulting s-domain expression.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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