Turn on thread page Beta
    • Thread Starter
    Offline

    0
    ReputationRep:
    Ok, so the goal is to find the Laplace transform of f(t)=cos(at) where a is a real constant.

    L=\displaystyle\int^\infty_0 e^{-st}cos(at)\ dt

    Working it out using integration by parts twice I arrive at
    \displaystyle\lim_{t\to \infty} \frac{e^{-st}cos(at)}{s}-\frac{a}{s}[\frac{e^{-st}sin(at)}{s}+\frac{a}{s}\displ  aystyle\int^t_0 e^{-st}cos(at) dt]

    At this point I would normally substitute the remaining integral for what's on the other side of the original equation but here there is nothing. Is there some trick I'm not seeing or should I just give up and use a table of integrals?
    Offline

    15
    ReputationRep:
    What's happened to the t=0 limit when you did your first integration by parts?
    Offline

    0
    ReputationRep:
    Cos (at) = the real part of e^iat
    Offline

    19
    ReputationRep:
    (Original post by D-Day)
    Ok, so the goal is to find the Laplace transform of f(t)=cos(at) where a is a real constant.

    L=\displaystyle\int^\infty_0 e^{-st}cos(at)\ dt

    Working it out using integration by parts twice I arrive at
    \displaystyle\lim_{t\to \infty} \frac{e^{-st}cos(at)}{s}-\frac{a}{s}[\frac{e^{-st}sin(at)}{s}+\frac{a}{s}\displ  aystyle\int^t_0 e^{-st}cos(at) dt]

    At this point I would normally substitute the remaining integral for what's on the other side of the original equation but here there is nothing. Is there some trick I'm not seeing or should I just give up and use a table of integrals?
    If you let your integral equal to I, then you'll have I = "some stuff with an I in". You then solve for I.

    In this case you'll have:

    I = 1/s - (a^2/s^2) * I.

    Sorry your solutions may or may not be correct (I've quoted you and it has gone into latex code) but as long as you check your integration by parts to make sure you've included all your limits etc you'll end up with:

    I = s/(s^2 + a^2) where I is the Laplace of cos(at).

    If you can't get that I'll check your integration by parts with mine.
    Offline

    12
    ReputationRep:
    Another way to integrate this is to notice that it's the real part of e^st*e^iat.
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: July 30, 2009

871

students online now

800,000+

Exam discussions

Find your exam discussion here

Poll
Should predicted grades be removed from the uni application process
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.