Laplace Transform: Definition, Table, Formulas, Properties

The Laplace transform is a method of transforming a time variable function into a complex variable function. In recent years, the theory of Laplace transform has been an essential part of solving many problems arising in engineering. This helps to directly find out the solution of differential equations with boundary values. In this post, we will learn about Laplace transforms with their formulas, properties, applications, and many more.

Laplace Transform Definition

Let f(t) be a function of the variable t, defined for t≥0. Then the Laplace transform of f(t), denoted by L{f(t)}, is given by the following integral formula:

L{f(t)} = $\int_0^\infty$ f(t)e^-stdt,

provided that the integral converges. Note that the Laplace transform of f(t) is a function of a complex variable s.

For example, the Laplace transform of 1 is 1/s.

Laplace transform notation: The Laplace transform of f(t) is denoted by L{f(t)} or F(s).

Condition for the Existence of Laplace Transform

The Laplace transform of a function f(t) exists, that is, $\int_0^\infty$ f(t)e^-stdt exists if the following two conditions hold true:

1. f(t) is continuous.
2. lim_t→∞ {e^-at f(t)} is finite for s>a.

Note that the above two conditions are sufficient but not necessary. For example, L{1/√t} exists; although the function 1/√t is infinite at t=0.

Laplace Transform Table

We list the Laplace transforms of all the elementary functions in the table below.

No.	f(t)	L{f(t)}=F(s)
1	1	1/s
2	t	1/s2
3	t2	2/s3
4	t3	6/s4
5	tn (n=0,1,…)	$\dfrac{n!}{s^{n+1}}$
6	tn (n≠0,1,…)	$\dfrac{\Gamma(n+1)}{s^{n+1}}$
7	√t	$\dfrac{\sqrt{\pi}}{2s^{3/2}}$
8	et	$\dfrac{1}{s-1}$
9	eat	$\dfrac{1}{s-a}$
10	e-at	$\dfrac{1}{s+a}$
11	sin t	$\dfrac{1}{s^2+1}$
12	sin at	$\dfrac{a}{s^2+a^2}$
13	cos t	$\dfrac{s}{s^2+1}$
14	cos at	$\dfrac{s}{s^2+a^2}$
15	sinh t	$\dfrac{1}{s^2-1}$
16	sinh at	$\dfrac{a}{s^2-a^2}$
17	cosh t	$\dfrac{s}{s^2-1}$
18	cosh at	$\dfrac{s}{s^2-a^2}$

Properties of Laplace Transform

(I). Linearity Property:

Let f(t) and g(t) be two functions. Then for two constants a and b, we have

L{af(t) + bg(t)} = aL{f(t)} + bL{g(t)}.

In particular, the Laplace transform of the sum of two functions is equal to the sum of their Laplace transforms.

Example 1: Find the Laplace transform of sint cos2t.

Solution:

We have sint cos2t = 1/2 [2 cos2t sint]

= 1/2 [sin3t – sint]

∴ L{sint cos2t} = 1/2 L{sin3t – sint}

= 1/2 [L{sin3t} – L{sint}]

= 1/2 [3/(s²+9) – 1/(s²+1)]

= (2s²-6)/2(s²+1)(s²+9)

= (s²-3)/(s²+1)(s²+9)

(II). First Shifting Property:

If L{f(t)} = F(s), then L{e^at f(t)} = F(s-a).

As an application of the above first shifting property, we list the Laplace transforms of the following useful functions in the table below:

No.	f(t)	L{f(t)} = F(s)
1	eat	$\dfrac{1}{s-a}$
2	teat	$\dfrac{1}{(s-a)^{2}}$
3	t2eat	$\dfrac{2}{(s-a)^{3}}$
4	tn eat	$\dfrac{n!}{(s-a)^{n+1}}$
5	et sin t	$\dfrac{1}{(s-1)^2+1}$
6	eat sin t	$\dfrac{1}{(s-a)^2+1}$
7	eat sin bt	$\dfrac{b}{(s-a)^2+b^2}$
8	et cos t	$\dfrac{s-1}{(s-1)^2+1}$
9	eat cos t	$\dfrac{s-a}{(s-a)^2+1}$
10	eat cos bt	$\dfrac{s-a}{(s-a)^2+b^2}$
10	eat sinh bt	$\dfrac{b}{(s-a)^2-b^2}$
12	eat cosh bt	$\dfrac{s-a}{(s-a)^2-b^2}$

(III). Change of Scale Property:

If L{f(t)} = F(s), then L{f(at)} = 1/a F(s/a).

Example 2: If the Laplace transform of (sin t)/t is tan^-1(1/s), then find L{sin(at)/t}

Solution:

Given that L{sin(t)/t} = tan^-1(1/s).

In the above change of scale property, put f(t) = sin(t)/ t. Thus, by the given condition we have F(s)=tan^-1(1/s). Then, we deduce that

L{f(at)} = 1/a F(s/a)

⇒ L{sin(at)/at} = 1/a tan^-1(1/(s/a))

⇒ 1/a L{sin(at)/t} = 1/a tan^-1(a/s)

Canceling 1/a from both sides,

L{sin(at)/t} = tan^-1(a/s)

Thus, the Laplace transform of sin(at)/t is tan^-1(a/s).

Laplace Transform of Derivative

Laplace transform derivative formula: Let f(t) be a function of t with Laplace transform L{f(t)} = F(s). Let the derivative f(t) be denoted by $f^\prime(t):=\dfrac{df}{dt}$. We assume the following:

• The function f(t) is a continuous function of exponential order s on [0, ∞), that is, we have lim_t→∞e^-stf(t) = 0.
• $f^\prime(t)$ is at least piecewise continuous on (0, ∞).

Then the Laplace transform of the derivative of f(t) exists and is given by the formula

$L\{ f^\prime(t) \} = f(0) + sF(s)$.

Laplace transform of second derivative formula: The Laplace transform of the second derivative of a function f(t) is

$L\{f^{\prime\prime}(t) \} = s^2F(s)-sf(0)-f^\prime(0)$.

Laplace transform of n-th derivative formula: Moreover, the Laplace transform of the n-th derivative of the function f(t) is given by the following formula:

L{f(n)(t)} = snF(s) – sn-1f(0) – sn-2$f^\prime(0)$ – … -f(n-1)(0),

where f⁽ⁿ⁾ denotes the n-th derivative of f(t).

Laplace Transform of Integral

Laplace transform Integral formula: Let f(t) be a function of t with Laplace transform L{f(t)} = F(s). Then the formula of the Laplace transform of integration is as follows:

$L\{\int_0^t f(u) du \}=\frac{1}{s}F(s)$.

Multiplication by tⁿ Formula

Let L{f(t)} = F(s). The Laplace transform of a function multiplied by tⁿ is given by the formula:

L{tnf(t)} = (-1)n $\frac{d^n}{ds^n}[F(s)]$,

where n is a natural number.

Division by t Formula

Let L{f(t)} = F(s). The Laplace transform of a function divided by t is given by the formula:

$L\{\frac{f(t)}{t} \} =\int_s^\infty F(s) ds$.

Laplace Transforms of Special Functions

We will now discuss the Laplace transform formulas for some special functions, for example, periodic functions, Bessel functions, and the error function.

Laplace Transform of Periodic Function

If f(t) is a periodic function with period T, that is, f(t+T)=f(t) for all t, then the Laplace transform of f(t) is given by the following formula:

L{f(t)} = $\dfrac{\int_0^T e^{-st} f(t)dt}{1-e^{-sT}}$

Laplace Transform of Bessel Function

Recall that the Bessel functions J₀(x) and J₁(x) are defined as follows:

J₀(x) = 1 – x²/2² + x⁴/(2²⋅4²) – x⁶/(2²⋅4²⋅6²) + …

and J₁(x) = $-J_0^{\prime}(x)$

The Laplace transforms of the Bessel functions are given by

• L{J₀(x)} = $\dfrac{1}{\sqrt{s^2+1}}$
• L{J₁(x)} = $1-\dfrac{s}{\sqrt{s^2+1}}$

Laplace Transform of Error Function

Recall that the Error function erf(√x) is defined as follows:

erf(√x) = 2/√π $\int_0^{\sqrt{x}} e^{-t^2} dt$

= 2/√π $\left(x^{1/2}-\dfrac{x^{3/2}}{3}+\dfrac{x^{7/2}}{5 \cdot 2!}+\cdots \right)$

The Laplace transform of the Error function erf(√x) is given by

L{erf(√x)} = $\dfrac{1}{s\sqrt{s+1}}$

FAQs on Laplace Transform