Method on @sym: ilaplace (G, s, t) ¶

Method on @sym: ilaplace (G) ¶

Method on @sym: ilaplace (G, t) ¶

Inverse Laplace transform.

The inverse Laplace transform of a function G of s is a function f of t defined by the integral below.

syms g(s) t
f(t) = rewrite(ilaplace(g), 'Integral')
  ⇒ f(t) = (symfun)
          c + ∞⋅ⅈ
             ⌠
             ⎮          s⋅t
       -ⅈ⋅   ⎮    g(s)⋅ℯ    ds
             ⌡
          c - ∞⋅ⅈ
       ────────────────────────
                 2⋅π

(This expression is usually written simply as the integral divided by 2⋅π⋅ⅈ.)

Example:

syms s t
F = 1/s^2;
ilaplace(F, s, t)
  ⇒ (sym) t⋅θ(t)

To avoid Heaviside, try:

syms t positive
ilaplace(1/s^2, s, t)
  ⇒ (sym) t

By default the output is a function of t (or x if the inverse transform happens to be with respect to t). This can be overridden by specifying t. For example:

syms s
syms t x positive
ilaplace(1/s^2)
  ⇒ (sym) t
ilaplace(1/t^2)
  ⇒ (sym) x
ilaplace(1/s^2, x)
  ⇒ (sym) x

The independent variable of the input can be specified by s; if omitted it defaults a symbol named s, or see ‘@sym/symvar’ if no such symbol is found.

See also: @sym/laplace.

Package: symbolic