How to find the Laplace transform of |sin(t)|?

AimettiA8J

AimettiA8J

Answered question

2022-11-22

How to find the Laplace transform of | sin ( t ) | ?

Answer & Explanation

Lukas Arias

Lukas Arias

Beginner2022-11-23Added 6 answers

L s ( | sin t | ) = 0 | sin ( t ) | e s t d t = n = 0 π n π ( n + 1 ) | sin ( t ) | e s t d t = n = 0 e s π n 0 π sin ( t ) e s t d t = 1 1 exp ( π s ) 0 π sin ( t ) e s t d t
The latter integrate is easy to evaluate as follows:
0 π sin ( t ) e s t d t = 0 π e i t s t d t = ( exp ( i π π s ) 1 i s ) = 1 + exp ( π s ) 1 + s 2
Thus
L s ( | sin t | ) = 1 + exp ( π s ) 1 + s 2 1 1 exp ( π s )
Jewel Hall

Jewel Hall

Beginner2022-11-24Added 3 answers

Here is a short derivation. First note that | sin t | has a period of π. Then
L ( | sin t | )
= 1 1 e π s 0 π e s t | sin t | d t
= 1 1 e π s 0 π e s t sin t d t
= 1 1 e π s ( e s t s sin t + cos t s 2 + 1 | 0 π )
= 1 1 e π s ( e π s s 2 + 1 + 1 s 2 + 1 )
= 1 + e π s 1 e π s 1 s 2 + 1

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Differential Equations

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?