LaplaceTransform

The one-sided Laplace transform $F(s) = \int_0^\infty f(t) e^{-st}\,dt$ , as a rule table over the standard pairs (powers, exponentials, trig/hyperbolic, the unit step, the impulse) plus linearity and the first shifting theorem.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageLaplaceTransform!

LaplaceTransform(f, t, s)

F(s) = \int_0^\infty f(t) e^{-st}\,dt

, for f a function of t.

Typed signatures

Type	Package	Applies
`(expression, expression, expression) -> expression`	`analytic`	always

Domain: Transforms

Details

Covered: $t^n$ ( $n$ a concrete real $> -1$ ), $e^{at}$ , $\sin(at)$ , $\cos(at)$ , $\sinh(at)$ , $\cosh(at)$ , UnitStep(t - a) and DiracDelta(t - a) (sign of a determined via isPositive/isNegative, declined if unknown), sums (linearity), a constant factor, and the first shifting theorem — an Exp(a t) factor multiplying an otherwise-transformable piece shifts $s \to s - a$ in that piece's transform, covering $t^n e^{at}$ , $e^{at}\sin(bt)$ , etc.
Also covered: $|\sin at|$ and $|\cos at|$ (the full-wave rectified waves, $a\coth(\pi s/2a)/(s^2+a^2)$ and $(s+a\,\mathrm{csch}(\pi s/2a))/(s^2+a^2)$ ), $\Gamma(\nu,a/t)\to 2(as)^{\nu/2}K_\nu(2\sqrt{as})/s$ for a provably positive a, and the multivariate LaplaceTransform(f, {t1, t2}, {s1, s2}) of a function that separates into one-variable factors, or otherwise one variable at a time.
Also covered: $\sin(c\sqrt t)\to c\sqrt\pi\,e^{-c^2/4s}/(2s^{3/2})$ , $\cos(c\sqrt t)/\sqrt t\to\sqrt{\pi/s}\,e^{-c^2/4s}$ , and the Fresnel integrals $C(c\sqrt t)$ , $S(c\sqrt t)$ (closed forms in $\sqrt{a^2+s^2}$ , $a=\pi c^2/2$ ). That last one is what makes $\cos(\sqrt{xy})/\sqrt{xy}$ , which does not separate, transform to $\pi/\sqrt{p(q+1/4p)}$ .
Also covered: $t^{-1/2}(1+t)^n\,T_n\!\left(\tfrac{1-t}{1+t}\right)\to\sqrt\pi\,s^{-n-1/2}H_{2n}(\sqrt s)/4^n$ , for the same symbolic $n$ on the power and the order (ChebyshevT, HermiteH).
Declined: an opaque function f(t) (Wolfram itself only expands the derivative/second-shifting theorems symbolically, which would mean synthesizing f(0) or a self-referential LaplaceTransform[f[t],t,s] for an arbitrary f); a product of two independently-transformable pieces outside the shifting-theorem shape (e.g. $t^2 \sin(t)$ ); UnitStep/DiracDelta at $a = 0$ or an undetermined-sign a; any exponent not linear in t.

Examples

Basic #

$\mathcal{L}\{t^3\} = 6/s^4$

In

\mathrm{LaplaceTransform}(t^3, t, s)

Out

\frac{6}{s^4}

$\mathcal{L}\{e^{2t}\} = 1/(s-2)$

In

\mathrm{LaplaceTransform}(\exp(2t), t, s)

Out

\frac{1}{s-2}

$\mathcal{L}\{\sin(3t)\} = 3/(s^2+9)$

In

\mathrm{LaplaceTransform}(\sin(3t), t, s)

Out

\frac{3}{s^2+9}

In

\mathrm{LaplaceTransform}(\frac{x}{e^{y}}, \bigl\lbrack x, y\bigr\rbrack, \bigl\lbrack p, q\bigr\rbrack)

Out

\frac{1}{(q+1)p^2}

Scope #

first shifting theorem — $\mathcal{L}\{t^3 e^{2t}\} = 6/(s-2)^4$

In

\mathrm{LaplaceTransform}(\exp(2t)t^3, t, s)

Out

\frac{6}{(s-2)^4}

$\mathcal{L}\{e^{2t}\sin(3t)\} = 3/((s-2)^2+9)$

In

\mathrm{LaplaceTransform}(\exp(2t)\sin(3t), t, s)

Out

\frac{3}{(s-2)^2+9}

In

\mathrm{LaplaceTransform}(1, t, s)

Out

\frac{1}{s}

In

\mathrm{LaplaceTransform}(3t^2+5t-7, t, s)

Out

-\frac{7}{s}+\frac{5}{s^2}+\frac{6}{s^3}

In

\mathrm{LaplaceTransform}(\exp(-(at)), t, s)

Out

\frac{1}{a+s}

In

\mathrm{Simplify}(\mathrm{LaplaceTransform}(\frac{b\sinh(at)-a\sinh(bt)}{a^2-b^2}, t, s))

Out

\mathrm{LaplaceTransform}(\frac{b\sinh(at)-a\sinh(bt)}{a^2-b^2}, t, s)

In

\mathrm{LaplaceTransform}(\frac{e^{t}-1}{t}, t, s)

Out

\ln(\frac{s}{s-1})

A Chebyshev polynomial of a Möbius argument against $t^{-1/2}$ transforms to a Hermite polynomial: $\sqrt\pi\,s^{-n-1/2} H_{2n}(\sqrt s)/4^n$

In

\mathrm{LaplaceTransform}((\mathrm{ChebyshevT}(n, \frac{1-t}{t+1})(t+1)^{n})/\sqrt{t}, t, s)

Out

\frac{\mathrm{HermiteH}(2n, \sqrt{s})\sqrt{π}s^{-n-\frac{1}{2}}}{4^{n}}

In

\mathrm{LaplaceTransform}(Γ(ν, \frac{a}{t}), t, s)

Out

\mathrm{LaplaceTransform}(Γ(ν, \frac{a}{t}), t, s)

In

\mathrm{LaplaceTransform}(\mathrm{DiracDelta}(t-1), t, s)

Out

\exp(-s)

In

\mathrm{LaplaceTransform}(\frac{x^{m}y^{n}}{Γ(m+1)Γ(n+1)}, \bigl\lbrack x, y\bigr\rbrack, \bigl\lbrack p, q\bigr\rbrack)

Out

\mathrm{LaplaceTransform}(\frac{x^{m}y^{n}}{Γ(m+1)Γ(n+1)}, \bigl\lbrack x, y\bigr\rbrack, \bigl\lbrack p, q\bigr\rbrack)

In

\mathrm{LaplaceTransform}(\frac{\cos(\sqrt{xy})}{\sqrt{xy}}, \bigl\lbrack x, y\bigr\rbrack, \bigl\lbrack p, q\bigr\rbrack)

Out

\frac{π}{\sqrt{p}\sqrt{q+\frac{1}{4p}}}

In

\mathrm{LaplaceTransform}(\frac{\sin(3t)}{t}, t, s)

Out

\arctan(\frac{3}{s})

In

\mathrm{FullSimplify}(\mathrm{LaplaceTransform}(\mathrm{RealAbs}(\sin(t)), t, s))

Out

\frac{\coth(\frac{πs}{2})}{s^2+1}

a separable function transforms variable by variable

In

\mathrm{LaplaceTransform}(x^2\exp(3y), \bigl\lbrack x, y\bigr\rbrack, \bigl\lbrack p, q\bigr\rbrack)

Out

\frac{2}{qp^3-3p^3}

In

\mathrm{Together}(\mathrm{LaplaceTransform}((1+at)\exp(at), t, s))

Out

\mathrm{LaplaceTransform}((at+1)\exp(at), t, s)

In

\mathrm{LaplaceTransform}(\ln(t), t, s)

Out

\frac{-\ln(s)-γ}{s}

In

\mathrm{LaplaceTransform}(\mathrm{FresnelC}(\sqrt{at}\sqrt{\frac{2}{π}}), t, s)

Out

\frac{\sqrt{a}\sqrt{s+\sqrt{a^2+s^2}}}{2s\sqrt{a^2+s^2}}

About

Examples	22
Checked against	Wolfram Language 17 agree4 differ1 unevaluated
Crosswalk	Wolfram Language

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/transforms.ts

See also: InverseLaplaceTransform, FourierTransform