Laplace Transform Calculator – Basic Functions

The Laplace transform swaps time-domain calculus for s-domain algebra. Integrating e⁻ˢᵗ against f(t) from 0 to infinity converts the library: constants pass through as L{1} = 1/s; exponentials shift the pole, L{e²ᵗ} = 1/(s−2); powers fall factorial-wise, L{t} = 1/s² and L{t²} = 2/s³; sines and cosines produce quadratic denominators, L{sin t} = 1/(s²+1) and L{cos t} = s/(s²+1). The calculator evaluates the integral pattern for each supported form and demonstrates the operational rules that make the transform useful: differentiation becomes multiplication, L{f′} = sF(s) − f(0), so an ODE turns into an algebra equation. Shifting: L{eᵃᵗf(t)} = F(s−a), and delay: L{f(t−c)} = e⁻ˢᶜF(s). Convergence needs s larger than the growth rate — e²ᵗ requires s > 2.

1. 1

   Choose a function form

   1, tⁿ, eᵃᵗ, sin(bt), cos(bt), or composites

2. 2

   Read the transform pair

   F(s) is returned with the integral substitution

3. 3

   Apply the rules

   Shift, delay, and derivative rules are demonstrated

4. 4

   Check convergence

   The valid s-range (s > a for exponentials) is stated

• 1

  Derivatives map to polynomials in s: L{f′} = sF(s) − f(0)

• 2

  L{eᵃᵗf} = F(s−a) — exponential factors shift the s-domain

• 3

  Convergence requires s above the function’s growth exponent

• Dropping the initial-condition term f(0) when transforming derivatives

• Using L{eᵃᵗ} = 1/(s+a); the pole sits at s = a, giving 1/(s−a)

• Forgetting the factorial: L{t²} = 2/s³, not 1/s³

• Ignoring the s > a convergence requirement for exponentials