Inverse Laplace Transform
of
(1/s) tanh^-1 (s)

done most of it...

Question

Inverse Laplace Transform
of
(1/s) tanh^-1 (s)

done most of it...

hartnn · Answer

attachment

hartnn · Answer

what next ?

hartnn · Answer

@SithsAndGiggles 
@myininaya

hartnn · Answer

i am thinking convolution with 
u(t)
is that correct ?

hartnn · Answer

@dumbcow 
@ganeshie8 
@campbell_st

anonymous · Answer

I'm not sure I follow the derivation of the inverse transform for $\tanh^{-1}s$, but supposing that's right, your last step can use this formula:
$$\mathcal{L}^{-1}\left\{\frac{1}{s}F(s)\right\}=\int_0^t f(v)~dv$$
where $f(t)=\dfrac{\sinh t}{t}$, according to your work.

hartnn · Answer

thats convolution

hartnn · Answer

but ok,
so how would i start with that integral ?

anonymous · Answer

I don't think you'd be able to find an elementary result for it... http://mathworld.wolfram.com/Shi.html

hartnn · Answer

;-;

any other approach
forget what i did

anonymous · Answer

Hmm, not sure just yet. WA doesn't seem to agree wholeheartedly with your result either, I'm afraid (past $s>1$)... http://www.wolframalpha.com/input/?i=LaplaceTransform%5BSinh%5Bt%5D%2Ft%2Ct%2Cs%5D

anonymous · Answer

That might just be an artifact from the logarithms you used though.

hartnn · Answer

@satellite73

hartnn · Answer

i highly doubt people are getting notifications for tags

anonymous · Answer

Well, if it's any consolation, your answer is right. I don't think you're expected to be able evaluate the integral.

anonymous · Answer

Unless it's part of an IVP, in which case you'll need to approximate that integral...

hartnn · Answer

so just leave the answer in integral form ? :O

anonymous · Answer

I would, yes. Unless you were formally introduced to that special function $\text{Shi}(t)$... (hehe)

hartnn · Answer

lol

hartnn · Answer

not introduced to that shi(t) yet
so i should better assume, the question is incorrect
and its 1/2 instead of 1/s :P

anonymous · Answer

This site has some info, but it's hard to read... http://ddmf.msr-inria.inria.fr/1.9.1/ddmf?service=P0&rendering=MathML&mac=&sf_id=sf_Shi

anonymous · Answer

Sigh, it's hard to look up any useful info with such an unoriginal name for the function...

anonymous · Answer

Here you have the table of contents of a book that might be helpful. Too bad the book is nowhere to be found >.< http://www.mathtable.com/gr/gr5_toc/ which can be purchased for about $100 used, or if you're smart you can find a free PDF of it online :) http://www.lepp.cornell.edu/~ib38/tmp/reading/Table_of_Integrals_Series_and_Products_Tablicy_Integralov_Summ_Rjadov_I_Proizvedennij_Engl._2.pdf

hartnn · Answer

would take the easier way out and say the question has a typing mistake :P
thanks!

anonymous · Answer

No problem!

hartnn · Answer

$100 ? amazon gives it for $14 http://www.amazon.com/Table-Integrals-Series-Products-Gradshteyn/dp/0122947606

anonymous · Answer

According to that textbook, the result is some linear combination of complex functions. Oh well, let's hope it's a typo.

Ah but that's for the 5th ed, that PDF has the 7th.