Laplace Transform of t^n: Formula, Proof

The function tⁿ is the n^th power of t. The Laplace transform of t^n is equal to n!/s^{n+1}. In this article, we will learn how to find the Laplace transform of real powers of t.

Laplace Transform of tⁿ Formula

The formula of the Laplace transform of tⁿ, denoted by L{tⁿ}, is given below:

L{tⁿ} = n!/sⁿ⁺¹.

What is the Laplace Transform of tⁿ?

Answer: The Laplace transform of tⁿ is n!/sⁿ⁺¹.

Proof:

We will find the Laplace transform of tⁿ by the definition of Laplace transform. By definition, the Laplace transform of f(t) is given the following integral formula:

L{f(t)} = $\int_0^\infty$ f(t) e^-st dt.

Step 1: Put f(t) = tⁿ.

So the Laplace transform of tⁿ by definition is equal to

L{tⁿ} = $\int_0^\infty$ tⁿ e^-st dt. …(I)

Step 2: To use the theory of the Gamma function Γ(x) = $\int_0^\infty$ t^x-1 e^-t dx, let us assume that

z=st

∴ dz=s dt ⇒ dt = dz/s. Also, t=z/s.

t	z
0	0
∞	∞

Step 3: Therefore, from (I) we get that

L{tⁿ} = $\int_0^\infty \Big(\dfrac{z}{s} \Big)^n e^{-z} \dfrac{dz}{s}$

= (1/sⁿ⁺¹) $\int_0^\infty z^{n+1-1} e^{-z} dz$

= (1/sⁿ⁺¹) $\Gamma(n+1)$, by the definition of the Gamma function.

= (1/sⁿ⁺¹) × n! as we know that Γ(n) = (n-1)!

= $\dfrac{n!}{s^{n+1}}$.

Thus, the Laplace transform of tⁿ is n!/sⁿ⁺¹.

Find the Laplace transform of t^n. Summary: The Laplace transform of t^n is n!/sn+1.

Important Notes:

• tⁿ Laplace transform: L{tⁿ} = n!/sⁿ⁺¹.
• t² Laplace transform: L{t²} = 2/s³.
• t³ Laplace transform: L{t³} = 6/s⁴.
• t⁴ Laplace transform: L{t⁴} = 24/s⁵.

Read Also:

Concept of Laplace Transform: Definition, Table, Formulas, Properties & Examples

Laplace Transform of eat: The Laplace transform of eat is 1/(s-a).

Laplace transform of sin(at): The Laplace transform of sin(at) is a/(s2+a2).

Laplace transform of cos(at): The Laplace transform of cos(at) is s/(s2+a2).

Laplace transform of constant: The Laplace transform of c is c/s.

Inverse Laplace transform of constant: The inverse Laplace transform of c is cδ(t), where δ(t) is the Dirac delta function.

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