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MathGroup Archive 2006

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Re: Pattern Matching for Exponentials

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67508] Re: [mg67502] Pattern Matching for Exponentials
  • From: "Carl K. Woll" <carlw at wolfram.com>
  • Date: Thu, 29 Jun 2006 00:09:11 -0400 (EDT)
  • References: <200606280752.DAA03521@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Rick Eller wrote:
> I am looking for a pattern-matching replacement rule which will 
> transform exponential functions of the form exp(n i t) into 
> (1/(n^2-1))exp(n i t) where i = sqrt[-1]. For example, exp(2 i t) 
> should convert to (1/3)exp(2 i t). I've tried the following code without 
> success :
> 
> In:     Exp[2 i t]   /.  Exp[n_ i t] -> Exp[n i t]/(n^2-1)
> 
> Out:  e^2 i t
> 
> I would appreciate any suggestions as to how this code should be 
> modified.
> 
> Thanks,
> 
> Rick Eller

In cases where pattern matching does not work as expected, you should 
check the FullForm of the expression:

In[5]:= FullForm[Exp[2 I t]]

Out[5]//FullForm= Power[E, Times[Complex[0, 2], t]]

Your factor of 2 is absorbed into Complex. Integer, rational and inexact 
numbers get absorbed into Complex, and in these cases the following will 
work:

In[6]:= Exp[2 I t] /. Exp[Complex[0, n_] t] :> 1/(n^2 - 1) Exp[n I t]

Out[6]= (1/3)*E^(2*I*t)

If n is an exact number other than an integer or rational, such as 
Sqrt[3], then you will need to work a bit harder:

In[8]:= Exp[Sqrt[3] I t] /.
  Exp[f_. Complex[0, n_] t] :> 1/((f n)^2 - 1) Exp[f n I t]

Out[8]= (1/2)*E^(I*Sqrt[3]*t)

This latter rule should work in all cases.

Carl Woll
Wolfram Research


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