Re: laplace transform

• To: mathgroup at smc.vnet.net
• Subject: [mg65130] Re: laplace transform
• From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
• Date: Wed, 15 Mar 2006 06:29:16 -0500 (EST)
• Organization: The Open University, Milton Keynes, UK
• References: <dv68hj$nvj$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Marlies.Goorden at physics.unige.ch wrote:
> Hi,
> I have a problem with the Laplace transform of mathematica. I
> want to know the laplace transform of sin(a*t).
> When I type
> LaplaceTransform[sin(a*t),t,s]
> mathematica gives me
> \sqrt(a^2) sign(a)/(a^2+s^2)
>
> On the other hand my mathematics books gives the answer
> a/(s^2+a^2)
>
> For complex a the answer is not the same. If I choose for
> example
> a=0.3+0.5i and
> s=1
> the two formulas give me
> -0.096+0.65i and 0.51+0.41i respectively.
>
> A numerical integration,
> i.e. NIntegrate[Sin((0.3+0.5i)*t)*Exp[-t],{t,0,Infinity}]
> gives me the same numerical value as the mathematics book
> formula.
> Is the mathematica formula wrong?

No.

Mathematica works over the field of complex numbers by default.
Therefore, it assumes that 'a' is a complex coefficient. On the other
hand, you books assume that 'a' is a real. You cannot use the second
formula/formula from your books with 'a' being a complex (see below).

In[1]:=
laplaceComplex = LaplaceTransform[Sin[a*t], t, s]

Out[1]=
2
Sqrt[a ] Sign[a]
----------------
2    2
a  + s

In[2]:=
laplaceReal = LaplaceTransform[Sin[a*t], t, s,
Assumptions -> a \[Element] Reals]

Out[2]=
a
-------
2    2
a  + s

In[3]:=
laplaceComplex /. a -> I

Out[3]=
1
-(-------)
2
-1 + s

In[4]:=
laplaceReal /. a -> I

Out[4]=
I
-------
2
-1 + s

In[5]:=
laplaceComplex /. a -> 2

Out[5]=
2
------
2
4 + s

In[6]:=
laplaceReal /. a -> 2

Out[6]=
2
------
2
4 + s



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