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Using "Limit" when the limit is a delta function
*To*: mathgroup at smc.vnet.net
*Subject*: [mg85199] Using "Limit" when the limit is a delta function
*From*: Jim Rockford <jim.rockford1 at gmail.com>
*Date*: Fri, 1 Feb 2008 02:20:34 -0500 (EST)
I'm just wondering whether I can get Mathematica to understand the
following limit (I'm using version 6.01).
The standard solution for u(x,t) of the diffusion equation u_t = D
u_xx over an infinite domain with initial condition u(x,0) =
delta(x) (Dirac delta) is
u[x_,t_] = (1/Sqrt[4 Pi D t]) Exp[-x^2/(4 D t)]
This solution, in fact, defines a family of functions whose limit as
t-->0 yields the Dirac delta, as the initial condition dictates. In
Mathematica I tried
Limit[u[x, t], t -> 0]
and got nothing from it. Can I get Mathematica to handle this sort of
limit? I realize that it involves an essential singularity at t=0,
but I would've guessed that Mathematica is equipped to deal with it.
An extra option needs to be specified or something?
Thanks,
Jim
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