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Re: laplace transform


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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