Laplace transform calculator
Enter your values. The answer and the full worked solution update as you type.
Result
Find the Laplace transform of a function made of common terms: powers of t, exponentials, sines and cosines (with a phase), sinh and cosh, shifted step functions u(t − c) and impulses δ(t − c). The calculator transforms each term with the standard table and the shift theorems, shows which entries it used, and gives F(s) with its region of convergence.
Step-by-step solution
Transform each term
Table entries used
Add the transforms (linearity)
How to find a Laplace transform with the table
- Split f(t) into a sum of terms; the Laplace transform is linear.
- Use the basic pairs: L{tⁿ} = n!/sⁿ⁺¹, L{sin ωt} = ω/(s² + ω²) and L{cos ωt} = s/(s² + ω²).
- Multiplying by e^(at) shifts s: L{e^(at) f(t)} = F(s − a).
- A delayed term g(t)·u(t − c) transforms to e^(−cs)·L{g(t + c)}.
- Add the transforms of the terms; F(s) exists for Re(s) larger than the largest exponent a.
Questions and answers
How do I type f(t)?
For example 3t^2 e^(-2t) + 5sin(4t) - 2e^(-t)cos(3t) + 4u(t-1). Use ^ for powers, e^(at) or exp(at) for exponentials, u(t-c) for a step and δ(t-c) or delta(t-c) for an impulse. A phase is allowed, as in sin(2t + 1).
What is the difference between t·u(t − 1) and (t − 1)·u(t − 1)?
The second is a delayed ramp and transforms to e^(−s)/s². The first is t·u(t − 1) = ((t − 1) + 1)·u(t − 1), which transforms to e^(−s)(1/s² + 1/s). The calculator handles both by writing g(t + c).
How are t·sin(ωt) and t·cos(ωt) transformed?
Through L{tⁿ e^(jωt)} = n!/(s − jω)ⁿ⁺¹: the imaginary part gives the sine and the real part the cosine, for example L{t sin ωt} = 2ωs/(s² + ω²)².
What is the region of convergence?
The values of s for which the transform integral converges: Re(s) larger than the largest growth rate in f(t). For e^(3t) that is Re(s) > 3.
Can I enter a product of two sines?
Not directly. Rewrite it first with a product-to-sum identity, for example sin(at)·cos(bt) = ½[sin((a + b)t) + sin((a − b)t)].
Practise it yourself
Practise Laplace transforms
A calculator gives the answer. In the exam you need the method: practise it with new problems, hints and worked solutions.