Inverse Laplace Transform of (1)/(e^s-1) and (s)/(e^s-1)

ymochelows 2022-09-11 Answered
Inverse Laplace Transform of 1 e s 1 and s e s 1
You can still ask an expert for help

Want to know more about Laplace transform?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Giancarlo Callahan
Answered 2022-09-12 Author has 12 answers
It's not hard to check that
0 n = 1 δ ( t n ) e s t = n = 1 e n s = 1 e s 1
and
0 n = 1 δ ( t n ) e s t = n = 1 s e n s = s e s 1
and hence from the Laplace uniqueness theorem
L 1 [ 1 e s 1 ] ( t ) = n = 1 δ ( t n )
L 1 [ s e s 1 ] ( t ) = n = 1 δ ( t n )
which is your original train of delta functions.

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-12-08
How do I find the inverse Laplace transform of 4ss2+42?
asked 2022-09-10
What is the closed-form of the following integral
I = 0 e x cos ( x ) x 2 + 1 d x
If we replaced 1 x 2 + 1 by its integral representation, 0 e x t sin ( t ) d t , , we get that
I = 0 e x cos ( x ) ( 0 e x t sin ( t ) d t ) d x
{  reverse the order of integration  }
I = 0 sin ( t ) ( 0 e x ( t + 1 ) cos ( x ) d x ) d t
I = 0 ( t + 1 ) sin ( t ) ( t + 1 ) 2 + 1 d t
{  make the change of variable  t + 1 = u }
= 1 u sin ( u 1 ) u 2 + 1 d t
= cos ( 1 ) 1 u sin ( u ) u 2 + 1 d t sin ( 1 ) 1 u cos ( u ) u 2 + 1 d t
Have anyone an idea to finish the remaining integrals ?
asked 2022-09-17
What is inverse Laplace of following function?
F ( s ) = L 1 ( Γ ( s a + b + 1 4 ) Γ ( 1 4 s a ) )
asked 2021-09-22
solve the given initial-value problem.
y3y+2y=δ(t1),y(0)=1,y(0)=0
asked 2021-12-31
Solve the following differential equations.
y4y=0
asked 2022-09-12
What is the laplace transform of the following function
f ( t ) = t 10 cosh ( t )
Is there another method that doesn't include differentiating the Laplace transform of cosh t ten times?
asked 2021-12-29
Solve the following IVP for the exact equation and find the interval of validity for the solution.
2xy9x2+(2y+x2+1) dydx=0 y(0)=2

New questions