Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by

\(F(s)=\int_{0}^{\infty} e^{-s t} f(t) \ d t\),

where we assume then s is a positive real number. For example, to find the Laplace transform oj f(t)= e?t, the following improper integral is evaluated using integration by parts:

\(F(s)=\int_{0}^{\infty} e^{-s t} e^{-t} \ d t=\int_{0}^{\infty} e^{-(s+1) t} \ d t=\frac{1}{s+1}\)

Verify the following Laplace transforms, where a is a real number.

\(f(t)=e^{a t} \quad \ \rightarrow \quad \ F(s)=\frac{1}{s-a}\)

Problem 77E

Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by where we assume s is a positive real number. For example, to find the Laplace transform of , the following improper integral is evaluated using integration by parts:

Verify the following Laplace transforms, where a is a real number.

Solution

Step 1

In this problem we have to verify the given laplace transform.

That is we have to prove : where a is a real number.

The laplace transform of a function is defined as follows.

Given a function f(t), the Laplace transform is a new function F(s) defined by where we assume s is a positive real number.