Laplace Transform Calculator
The Laplace transform turns a function of time into a function of frequency, converting differential equations into algebra. This calculator evaluates the transform of the versatile standard family f(t) = C times t to the power n times e to the power a t, which covers constants, polynomials, exponentials, and their products. Using the exact transform pair C times n factorial over (s minus a) to the (n plus 1), it returns the transform value at your chosen s, the factorial used, and the region of convergence so you know the result is valid.
Laplace transform formula
f(t) = C * t^n * e^(a t)
Laplace of f(t) at s = C * n! / (s - a)^(n + 1)
valid for s > a
n must be a non-negative integer
n! = 1 * 2 * 3 * ... * n
The transform of this standard family is a closed-form rational expression. The factorial of the power appears in the numerator, and the denominator is the shifted variable raised to one more than the power.
Notes on the Laplace transform
- The power n must be a non-negative integer for the factorial to be defined.
- The transform converges only when s is greater than the rate a.
- With n = 0 and a = 0 the result is C divided by s.
- With a = 0 the result is the power transform C times n factorial over s to the (n+1).
- The transform linearizes differentiation, turning derivatives into multiplication by s.
Laplace transform: frequently asked questions
What is the Laplace transform?
The Laplace transform converts a function of time f(t) into a function of a complex frequency variable s by integrating f(t) times e to the minus s t from zero to infinity. It turns differential equations into algebraic ones, simplifying their solution.
What function does this calculator transform?
It handles the standard family f(t) = C times t to the power n times e to the power a t, where C is a constant, n is a non-negative integer power, and a is a real growth rate. This single family covers constants, powers, exponentials, and their products.
What is the formula used?
The Laplace transform of C times t^n times e^(at) is C times n factorial divided by (s minus a) raised to the power (n plus 1). This is an exact, standard transform pair, valid for s greater than a.
What is the region of convergence?
The integral converges only when the real part of s exceeds a, the exponential growth rate. For real inputs that means s must be greater than a; otherwise the transform diverges and this calculator returns n/a.
What are common special cases?
With n = 0 and a = 0 you get the transform of a constant C, which is C over s. With n = 0 you get C over (s minus a), the exponential transform. With a = 0 you get C times n factorial over s to the (n plus 1), the power transform.
Official sources
- NIST Digital Library of Mathematical Functions: Integral transforms.
- NIST Digital Library of Mathematical Functions: Gamma and factorial functions.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.