Method on @sym: laplace (f, t, s) ¶

Method on @sym: laplace (f) ¶

Method on @sym: laplace (f, s) ¶

Laplace transform.

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

syms f(t) s
G(s) = rewrite(laplace(f), 'Integral')
  ⇒ G(s) = (symfun)
      ∞
      ⌠
      ⎮       -s⋅t
      ⎮ f(t)⋅ℯ     dt
      ⌡
      0

Example:

syms t
f = t^2;
laplace(f)
  ⇒ (sym)
      2
      ──
       3
      s

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

syms t s z
laplace(exp(t))
  ⇒ (sym)
        1
      ─────
      s - 1
laplace(exp(s))
  ⇒ (sym)
        1
      ─────
      z - 1
laplace(exp(t), z)
  ⇒ (sym)
        1
      ─────
      z - 1

If not specified by t, the independent variable is chosen by looking for a symbol named t. If no such symbol is found, see ‘@sym/symvar’ is used, which chooses a variable close to x:

syms a y
laplace (a*exp (y))
  ⇒ (sym)
        a
      ─────
      s - 1

See also: @sym/ilaplace.

Package: symbolic