The Laplace transform of the exponential function e to the power at is 1/(s-a). In this article, we will learn how to prove this Laplace transform formula of exponential functions.
Laplace Transform of eat Formula
The formula of the Laplace transform of eat is 1/(s-a). Mathematically, we write it as
L{eat} = 1/(s-a).
eat Laplace Transform
The Laplace transform of eat is equal to 1 divided by (s-a). That is,
L{eat} = 1/(s-a).
Proof:
Recall, the definition of the Laplace transform of a function f(t).
L{f(t)} = $\int_0^\infty$ e-st f(t) dt
Put f(t)=eat.
∴ L{eat} = $\int_0^\infty$ e-st eat dt
= $\int_0^\infty$ e-st+at dt
= $\int_0^\infty$ e-(s-a)t dt
= limT→∞ $\left[\dfrac{e^{-(s-a)t}}{-(s-a)}\right]_0^T$
= limT→∞ $\left(\dfrac{e^{-(s-a)T}}{-(s-a)}-\dfrac{1}{-(s-a)}\right)$
= 0 – $\dfrac{1}{-(s-a)}$
= $\dfrac{1}{s-a}$
So, the Laplace transform of e^at is 1/(s-a).
Now, replacing a with -a, we obtain the Laplace transform of e^-at which is 1/(s+a).
Summary: Denoting the Laplace transform of f(t) by L{f(t)}, we list the Laplace transform formulas for the exponential functions in the below table:
1 | L{eat} | 1/(s-a) |
2 | L{e-at} | 1/(s+a) |
3 | L{et} | 1/(s-1) |
4 | L{e-t} | 1/(s+1) |
Read Also:
Concept of Laplace Transform: Definition, Table, Formulas, Properties & Examples
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.
Question: Find the Laplace transform of eat+e-at.
Solution:
By the linearity property of Laplace transform, we have
L{eat+e-at} = L{eat} + L{e-at}
= 1/(s-a) + 1/(s+a) from the above table
= (s+a+s-a)/(s-a)(s+a)
= 2s/(s2-a2)
So the Laplace transform of the sum of eat and e-at is 2s/(s2-a2).
FAQs
Answer: The Laplace transform of eat is 1/(s-a).
Answer: The Laplace transform of et is 1/(s-1).
Answer: The Laplace transform of e-at is 1/(s+a).
Answer: The Laplace transform of e-t is 1/(s+1).