Laplace Transform of cos(at) * cos(bt)

In summary, the speaker is discussing how to find the Laplace transform of cos(at) * cos(bt) and mentions converting the cosines to exponentials and using Euler's formula. They are also confused about finding the Laplace transform of jsin(t(-a-b)) and whether the problem is meant as convolution or multiplication.
  • #1
seang
184
0
I have to find the laplace transform of cos(at) * cos(bt) and express it as a ratio of two polynomials. I converted both of the cosines into exponentials, and took the laplace transform of those. I think I'm getting confused on the complex side of things.

I get a few things like e^(jt(a+b)), and e^(-jt(a+b)) so I use Euler's formula and get some cosines and jsines. So how do you find the laplace transform of say jsin(t(-a-b)). Shouldn't it just be j*s / (s^2 + (-a-b)^2)?

Except this does not yield the correct answer. Maybe I'm converting all of the transforms to one fraction incorrectly...?
 
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  • #2
Do you mean to find the Laplace transform of [itex]\cos (at) * \cos (bt)[/itex] ? Where the [itex]*[/itex] denotes convolution? Or did you meant it as multiplication?
 
  • #3
Corneo said:
Do you mean to find the Laplace transform of [itex]\cos (at) * \cos (bt)[/itex] ? Where the [itex]*[/itex] denotes convolution? Or did you meant it as multiplication?

I assume he meant it as multiplication, since he is doing all of that work. I thought the same thing myself. If it were convolution it sure would make the problem a lot easier. :p
 

Related to Laplace Transform of cos(at) * cos(bt)

1. What is the Laplace Transform of cos(at) * cos(bt)?

The Laplace Transform of cos(at) * cos(bt) is given by the following formula:

L[cos(at) * cos(bt)] = s/(s^2 + a^2) * s/(s^2 + b^2)

where s is the Laplace variable and a and b are constants.

2. How is the Laplace Transform of cos(at) * cos(bt) derived?

The Laplace Transform of cos(at) * cos(bt) is derived using the properties of the Laplace Transform and the trigonometric identity cos(a+b) = cos(a) * cos(b) - sin(a) * sin(b). By using the property of linearity and the Laplace Transform of sin(at), the final result is obtained.

3. What is the significance of Laplace Transform of cos(at) * cos(bt) in scientific applications?

The Laplace Transform of cos(at) * cos(bt) is commonly used in engineering and physics to analyze and solve differential equations. It is also used in signal processing and control systems to model and manipulate signals and systems.

4. How does the Laplace Transform of cos(at) * cos(bt) relate to the Fourier Transform?

The Laplace Transform of cos(at) * cos(bt) is closely related to the Fourier Transform. In fact, the Fourier Transform can be derived from the Laplace Transform by setting s = jw, where j is the imaginary unit and w is the frequency. This means that the Laplace Transform provides a more general and powerful mathematical tool for analyzing signals and systems.

5. Are there any alternative forms of the Laplace Transform of cos(at) * cos(bt)?

Yes, there are alternative forms of the Laplace Transform of cos(at) * cos(bt) that can be found in tables or online calculators. These include using the complex exponential form of cos(at) and cos(bt) and expressing the result in terms of sines and cosines. However, the formula given in question 1 is the most commonly used form.

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