# Math Help - laplace

1. ## laplace

how you get the laplace of 1/tsin(wt)
pls show me the solution

2. ## Re: laplace

Originally Posted by jomex42
how you get the laplace of 1/tsin(wt)
pls show me the solution
I will leave the details of why to you.

Given $f(w,t)=\frac{1}{t}\sin(wt) \implies \frac{\partial f}{\partial w}=\cos(wt)$

$\mathcal{L}\{ \frac{\partial f}{\partial x}\}=\frac{\partial }{\partial w}\mathcal{L}\{ f(w,t)\}$

$\frac{s}{s^2+w^2}=\frac{\partial }{\partial w}\mathcal{L}\{ f(w,t)\}$

Now integrate both sides with respect to w to solve for the Laplace transform you want

$\int \frac{s}{s^2+w^2}dw=\int \frac{\partial }{\partial w}\mathcal{L}\{ f(w,t)\}dw$

$\tan^{-1}\left( \frac{w}{s}\right)=\mathcal{L}\{ f(w,t)\}$

3. ## Re: laplace

could you show me the solution of laplace of 1/tsinwt using the formula
intergral of limits 0 to infinity e^st(ft)dt
then you will get the arctanw/s
plss the exact solution my teacher want it

4. ## Re: laplace

can you solve this and you will get the result of
arctan(α/s)

5. ## Re: laplace

Originally Posted by jomex42

can you solve this and you will get the result of
arctan(α/s)
Another method is to note that the integral converges uniformly so limits and integrals can be interchanged.

Note that

$\sin(z)=\sum_{n=0}^{\infty}\frac{(-1)^nz^{2n+1}}{(2n+1)!} \quad \tan^{-1}(z)=\sum_{n=0}^{\infty}\frac{(-1)^nz^{2n+1}}{(2n+1)}$

This gives that

$\int_{0}^{\infty}\frac{e^{-st}}{t}\left( \sum_{n=0}^{\infty}\frac{(-1)^n(\alpha t)^{2n+1}}{(2n+1)!}\right)dt = \sum_{n=0}^{\infty}\frac{(-1)^n\alpha^{2n+1}}{(2n+1)!}\int_{0}^{\infty} e^{-st} t^{2n}dt = \sum_{n=0}^{\infty}\frac{(-1)^n\alpha^{2n+1}}{(2n+1)!} \mathcal{L}\{t^{2n}\}$

Now integrate or use the definition of the Laplace transform to get

$\sum_{n=0}^{\infty}\frac{(-1)^n\alpha^{2n+1}}{(2n+1)!}\frac{(2n)!}{s^{2n+1}}= \sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)}\left( \frac{\alpha}{s}\right)^{2n+1}=\tan^{-1}\left( \frac{\alpha}{s} \right)$

6. ## Re: laplace

What do you called that big letter e in your equation
what method did you use?

Because im not familiar in that equation i have not encounter that kind of function could you explain it

7. ## Re: laplace

what method did you use so i can explain it to my teacher how it been solved

8. ## Re: laplace

how the t in your first function gone in the next function

and the summation get out of the integral sign

What did you do in the first function to comeup with the second function