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Re: Trig Simplifications
*To*: mathgroup at smc.vnet.net
*Subject*: [mg3710] Re: Trig Simplifications
*From*: Harald Berndt <haraldb at nature.berkeley.edu>
*Date*: Wed, 10 Apr 1996 02:11:54 -0400
*Organization*: University of California Forest Prodcts Lab
*Sender*: owner-wri-mathgroup at wolfram.com
Alexander Casti wrote:
>
> Basically my question is how to get mathematica to write
>
> Exp[I x] in place of Cos[x] + I Sin[x]
>
> Suppose I have the expression
>
> In[1]: f[x_]= Cos[x] + I Sin[x] - Cos[2 X] + I Sin[2 X]
>
> I would like mathematica to simplify this into
>
> f[x_]= Exp[I X] - Exp[2 I X]
>
> I tried the substitution command
>
> In[2]: % ./ Cos[X]+I Sin[X]-> Exp[I X] and it returned
> Out[2]: E^{I X} - Cos[2 X] + I Sin[2 X]
>
I believe you don't really want what you've written out, because:
Cos[x] + I Sin[x] - Cos[2 X] + I Sin[2 X] == Exp[I X] - Exp[- 2 I X].
That sign error is one part of the problem: one needs two replacement
rules,
Cos[t_]+I Sin[t_]-> Exp[I t]
and
Cos[t_]-I Sin[t_]-> Exp[-I t].
Using Pattern[t, Blank[]] (the t_) will take care of the different
variable names x and 2 X, but there are still two substitutions
required. One needs to use ReplaceRepeated[] (//.).
Cos[x] + I Sin[x] + Cos[2 X] + I Sin[2 X]//.Cos[t_]+I Sin[t_]-> Exp[
I t]
will work and so will
Cos[x] + I Sin[x] - Cos[2 X] + I Sin[2 X]//.{Cos[t__]+I Sin[t__]->
Exp[I t], -(Cos[t_]-I Sin[t_])-> -Exp[- I t]}
This still seems clumsy and I hope someone will suggest a more
efficient way of writing this.
--
______________________________________________________________________
Harald Berndt, University of California
Research Specialist Forest Products Laboratory
Phone: 510-215-4224 FAX: 510-215-4299
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