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