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January 20th, 2019, 12:22 PM   #1
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Math Focus: Complex Analysis
Laplace transform dilemma

I was explicitly asked to find the Laplace transform of the following function:

$\displaystyle f(t)=\frac{\sin(\omega t)}{\cos^2(\omega t) +1}$

There's no way I can solve it manually with my current abilities. I've tried to run it on Mathematica but it can't find a solution either.

By the comparison test for improper integrals, I have concluded that the Laplace transform of $\displaystyle f(t)$ does actually exist. What do you think?

Thank you in advance
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Last edited by skipjack; January 20th, 2019 at 02:41 PM. Reason: symbolism error
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January 20th, 2019, 01:53 PM   #2
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Since $|f(t)|\le 1$ for all $t$, Laplace transform exists.
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January 20th, 2019, 02:19 PM   #3
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Use integration by parts, $\displaystyle u=e^{-wt}$ and $\displaystyle dv=-\frac{d\cos (wt)}{1+\cos^2 (wt)}$ .

Last edited by skipjack; January 20th, 2019 at 02:42 PM.
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January 31st, 2019, 11:22 AM   #4
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I tried this way but then I do not know how to calculate the laplace transform
of L[arctan(cos(ωt))] : do you have any suggestions about it?

L[sin(ωt)/(1+cos²(ωt)] =
∫_0,∞ e^(-st)[sin(ωt)/(1+cos²(ωt)] dt



∫e^(-st)[sin(ωt)/(1+cos²(ωt)] dt =
(1/ω)∙∫e^(-st)[d(-cos(ωt)) / (1+cos²(ωt)] =
- (1/ω)∙∫e^(-st)[d(cos(ωt)) / (1+cos²(ωt)] =
- (1/ω)∙∫e^(-st) d [ arctan(cos(ωt)) ] =

- (1/ω)∙e^(-st) arctan(cos(ωt)) +(1/ω)∙∫arctan(cos(ωt)) d e^(-st) =
- (1/ω)∙e^(-st) arctan(cos(ωt)) -(s/ω)∙∫arctan(cos(ωt)) e^(-st) dt


L[sin(ωt)/(1+cos²(ωt)] =
(1/ω)∙(π/4) - (s/ω)∙L[arctan(cos(ωt))]

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January 31st, 2019, 12:17 PM   #5
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Rog are you absynthe ? However .
The way I used integration by parts leads to the easiest possible form of integral.
Once there is no standard antiderivative, then series and approximations are included.
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Last edited by skipjack; January 31st, 2019 at 04:25 PM.
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January 31st, 2019, 12:36 PM   #6
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Quote:
Originally Posted by idontknow View Post
Rog are you absynthe ? However .
The way I used integration by parts leads to the easiest possible form of integral.
Once there is no standard antiderivative, then series and approximations are included.
I'm not Absynthe, but I'm also interested in solving the problem.

Regarding the resolution with the series, can you give me some further suggestion?

Best regards

Last edited by skipjack; January 31st, 2019 at 04:26 PM.
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