How Do I Prove the Laplace Transform Formula for Functions with Finite Jumps?

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SUMMARY

The Laplace transform of the function sin(t) * cos(t) can be computed using the identity \(\mathcal{L}\{f(t) * g(t)\} = \mathcal{L}\{f(t)\} \cdot \mathcal{L}\{g(t)\}\), leading to the result \(\frac{s}{{(s^2 + 1)}^2}\). To derive this, one can utilize the definition of the Laplace transform, specifically \(\mathcal{L}(\sin(t) \cos(t)) = \int_0^{\infty} e^{-st} \sin(t) \cos(t) dt\), which can be simplified using integration by parts. Additionally, the relationship \(\sin(t) \cos(t) = \frac{1}{2} \sin(2t)\) can be applied to further simplify the computation. The discussion also touches on the proof of the Laplace transform for functions with finite jumps, specifically the formula \(L\{f(t)\} = sF(s) - f(0) - f(a^+) - f(a^-) - e^{-as}\).

PREREQUISITES
  • Understanding of Laplace transforms and their properties
  • Familiarity with integration techniques, particularly integration by parts
  • Knowledge of trigonometric identities, specifically \(\sin(t) \cos(t)\)
  • Concept of functions with finite jumps and continuity
NEXT STEPS
  • Study the derivation of the Laplace transform for products of functions
  • Learn about the properties of Laplace transforms, including the shift theorem
  • Explore the application of integration by parts in Laplace transform calculations
  • Investigate the treatment of discontinuities in Laplace transforms
USEFUL FOR

Students and professionals in mathematics, engineering, and physics who are working with Laplace transforms, particularly those dealing with functions that exhibit finite jumps or require integration techniques for their analysis.

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how do i go about solving the laplace transform of sint * cost ?

i know the answer becomes 1/(s^2 + 4) but what is the method?

thanks.
 
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do you know that
\mathcal{L}\{f(t)*g(t)\} = \mathcal{L}\{f(t)\} \cdot \mathcal{L}\{g(t)\} = F(s) \cdot G(s)

and i think the answer comes out to be \frac{s}{{\left(s^2+1\right)}^2}
 
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yes i did know that but i didnt think it was as simple as that. From reading the certain rules we can apply to problems e.g the shift rule to solve L(sin2t * e^3t) do we HAVE to use the shift rule or can we separate each part and solve then multiply them together?
 
i am sorry. do you mean convolution by the asterisk? or do you mean multiplication? what i said in my last post is only correct if the asterisk means convolution.
 
i meant multiplication
 
Then use the definition
\mathcal{L}(sin(x)cos(x)}= \int_0^{\infnty}e^{-st}sin(t)cos(t)dt
which can be done by integration by parts.
 
Mybe it would help to notice that sin(t)cos(t) is equal to \frac{1}{2}sin(2t)
 
I want to ask about a proof of this transform: L{(f(t)}= sF(s) - f(0) - f(a^+) - f(a^-) - exp^(-as) inwhich f(t) is continuous except for an ordinary discontinuity (finite jump) at t=a, a>0
 
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