LaPlace Transforms

Integral Transforms

  • K(s,t)
    A function known as the kernel of the transform
  • α,β
    The limits of the transform
  • F(s)
    The integral transform of f(t)
  • F s = α β K s t f t d t

    The Laplace Transform

    L f t = F s = 0 e - s t f t d t

    The laplace transform of a function only exists only if there exist numbers M and C such that: f t =< M e C t For all values of t > 0.

    Important Transforms
    f(t) F(s)
    δ t
    (Dirac Delta Function)
    u t
    (Heviside Step Function)
    1 s
    tn n ! s n + 1
    e-at 1s+a
    sin(ωt) ωs2+ω2
    cos(ωt) ss2+ω2
    Operational Transforms
    f(t) F(s)
    C f(t) (multiplication by a constant) C F(s)
    f(t)+g(t) F(t)+G(t)
    d f t d x s F s - f 0
    d 2 f t d x 2 s 2 F s - s f 0 - d f 0 d x
    d 3 f t d x 3 s 3 F s - s 2 f 0 - s d f 0 d x - d 2 f 0 d x 2
    f(t-a)u(t-a), a>0 (time shift) e -as F s
    f(at), a>0 (time scale) 1a F sa

    Integral Transforms

    The Laplace Transform is an integral transform. It is useful for solving differential equations with non-continuous forcing functions such as modeling the response of a resonator after applying some pulse. s can be a complex value.

    Shifts in the time domain

    Shifts in the frequency domain

    Solving Differential Equations with Laplace Transforms

    Transforming a differential equation into the s domain by means of the laplace transform makes its solution an algebraic problem. Then the solution must be transformed back into the t domain.

    Heviside Step Function

    Dirac Delta Function