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In the documentation, it states that

The Laplace transform of a function $f(t)$ is defined to be $\int_0^{\infty } f(t) e^{-s t} \, \mathrm{d}t$.

But why can Mathematica not get the Laplace transform of $\frac{1-\cos (t)}{t}$?

$Assumptions=s>0;

LaplaceTransform[(1-Cos[t])/t,t,s]
(* EulerGamma+LaplaceTransform[1/t,t,s]+1/2 Log[1+s^2] *)

and the integral does converge

Integrate[(1-Cos[t])/t Exp[-s t],{t,0,∞}]
(* 1/2 Log[1+1/s^2] *)

?

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Laplace transform is not defined for 1/t and that is why it did not do it. May be it used a lookup table and found that. But I do not know why it worked when doing direct integration when the cos term is there. –  Nasser Nov 20 '13 at 22:31
    
also when you do integration seperately you receive divergence message –  barznjy Nov 20 '13 at 23:26

5 Answers 5

up vote 19 down vote accepted

You can use a trick to prevent Mathematica from taking your expression apart:

LaplaceTransform[Abs[1 - Cos[t]]/t, t, s]
(* 1/2 Log[1 + 1/s^2] *)
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2  
+1, quite clever. –  Leonid Shifrin Nov 21 '13 at 0:20
1  
Usually, Abs is the last thing you want to have in your expression. With most vector norm calculations, you have to take care of it manually. Very interesting that it works here. How did you find out about it, because putting it around the whole expression doesn't work. Was it luck? ..and of course +1! –  halirutan Nov 21 '13 at 10:39
    
@halirutan It's not the last: f = {Abs, FractionalPart, Ceiling, Floor, Round, PrimePi}; GraphicsGrid[ Partition[ Plot[{#[u], D[#[x], x] /. x -> u}, {u, -1, 3}, PlotLabel -> Style[Framed[Hyperlink[#, "paclet:ref/" <> #] &@ToString@#], 16, Blue, Background -> Lighter[Yellow]]] & /@ f, 2], Frame -> All] –  belisarius Nov 21 '13 at 12:52
    
@halirutan When I realized that Mathematica tries to transform each summand separately, it was the first idea that came to my mind about how to make the expression not to be a trivially subdivisible into its summands. Of course, it can be seen that Abs is not significant in this context, but I thought that Mathematica would not try to prove it. –  Vladimir Reshetnikov Nov 21 '13 at 15:33

When Mathematica tries to pull the fraction apart, it gets

$$\mathcal{L}_t\left[\frac{1-\cos(t)}{t}\right](s)=\mathcal{L}_t\left[\frac{1}{t}\right](s) - \mathcal{L}_t\left[\frac{\cos(t)}{t}\right](s)$$

While the cosine term has a Laplace-transform, $1/t$ doesn't have a transform. That might be the reason why Mathematica cannot solve it. The problem is, that the $1/t$ term has a singularity at 0:

Limit[1/t, t -> 0, Direction -> -1]

(* ∞ *)

while the complete expression doesn't

Limit[(1 - Cos[t])/t, t -> 0, Direction -> -1]

(* 0 *)

On the other hand, calculating the back-transform works:

InverseLaplaceTransform[1/2 Log[1 + 1/s^2], s, t] // FullSimplify

(* (1 - Cos[t])/t *)

What you can do is the following. You expand your formula into a series

Series[(1 - Cos[t])/t, {t, 0, 10}] // Normal

(* t/2 - t^3/24 + t^5/720 - t^7/40320 + t^9/3628800 *)

then you use LaplaceTransform

LaplaceTransform[%, t, s]

(* 1/(10 s^10) - 1/(8 s^8) + 1/(6 s^6) - 1/(4 s^4) + 1/(2 s^2) *)

we see that this sum is pretty easy, so we write it down and let Mathematica calculate the value:

Sum[(-1)^(i/2 + 1)/(i*s^i), {i, 2, Infinity, 2}]

(* 1/2 Log[(1 + s^2)/s^2] *)
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1  
Yes, 1/t does not have a Laplace transform, but (1-Cos[t])/t has, since the integral does exist, right? –  xslittlegrass Nov 20 '13 at 22:53
1  
the sign between two transforms is minus. –  barznjy Nov 20 '13 at 23:14

Ok, I think I know why the integral worked, but not the Laplace transform.

When using the integral, there is a pole at t=0 but this is a removable singularity.

Series[1 - Cos[t], {t, 0, 6}] // Normal

Mathematica graphics

Now dividing by t

(#/t) & /@ r

Mathematica graphics

So, the t in the denominator is gone. I do not know how Mathematica actually removed this pole at t=0 in the code, but it did it when calling Integrate. It might be because it is at start of the interval? or it have done something more advanced than the above, I do not know.

But when doing the Laplace transform, Mathematica must have first tried table lookup for each term. It must saw the 1/t term there. Using the lookup table, the Laplace transform for $t^n$ for negative $n$ is Gamma[n+1]/(s^(n+1)) reference and for $n=-1$ this gives Gamma[0] which is not defined. Hence LaplaceTransform gave up.

Basically what seems to have happened, is that LaplaceTransform does lookup first (for speed). It does not call Integrate to do the integration right away unless needed (else why have LaplaceTransform function in first place, no need to be calling a function which will just call integrate right away).

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1  
We cannot be sure what happens, but this seems one of the most plausible explanations. +1 from me. –  halirutan Nov 20 '13 at 23:35

If you don't want to resort to tricks, you can differentiate the transform over s first, which would bring -t downstairs and cancel 1/t. You can then take the transform for:

LaplaceTransform[(1 - Cos[t]), t, s]

(* 1/s - s/(1 + s^2) *)

and then integrate this over s (with a negative sign, since differentiation produced an extra -1):

-Integrate[1/s - s/(1 + s^2), s]

(* -Log[s] + 1/2 Log[1 + s^2] *)

In principle, this was done up to an additive constant, but you can see that this constant is zero, by inspecting the behavior of the original transform, and also the result, at $s\to\infty$ - both tend to zero in this limit.

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Another quick way is to use the following trig identity

In[13]:= TrigFactor[(1-Cos[2t])]
Out[13]= 2 Sin[t]^2

If t==2*p, then we have:

LaplaceTransform[2 Sin[p]^2/p, p, s]

(* 1/2 (-2 Log[s]+Log[-2 I+s]+Log[2 I+s]) *)
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