Re: Simplifying exponential expressions

Message-ID	<55E8240E.EAF9CB89@freenet.de> (permalink)
Date	2015-09-03 12:42 +0200
From	clicliclic@freenet.de
Newsgroups	sci.math.symbolic
Subject	Re: Simplifying exponential expressions
References	(1 earlier) <mru1an$f15$1@news.albasani.net> <5ff7f1c8-4ddd-44d0-8be4-36467950bf83@googlegroups.com> <d4gefoFgd8vU1@mid.individual.net> <d0ba7f54-c35a-4dbb-bfbb-cc85c7364548@googlegroups.com> <ms05t6$mv9$1@speranza.aioe.org>
Organization	synergetic AG

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Richard Fateman schrieb:
> 
> The original comment from Axel did not have the full context of the
> question, which was posted to a Maple newsgroup.
> 
> For the readers of sci.math.symbolic, here are 3 expressions,
> in Maxima syntax, that are the source of the Maple complaint..
> 
> [F,G,H] :
> 
> [(%e^(-%i*n*t-%i*(1-n)*t)*(%e^%e^(%i*t)+1))/(((%e^%e^(%i*t)-1)/(%e^(2*%e^(%i*t))+1)+1)*cosh(%e^(%i*t))),
> 
> (%e^%e^(%i*t)+%e^(-%e^(%i*t)))/(2*cosh(%e^(%i*t))),
> 
> (%e^(-%i*t)*(-2*cosh(%e^(%i*t))+%e^%e^(%i*t)+%e^(-%e^(%i*t))))/cosh(%e^(%i*t))]
> 
> in Maxima,
> F,exponentialize, ratsimp    returns 2*exp(%i*t)
> G,exponentialize             returns 1
> H,exponentialize             returns 0
> 
> So apparently the solution is similar to the one in Maple,
> to convert to exponentials.
> 

Thanks for the missing context. I am assuming that Maxima returns
2*exp(-%i*t) for the first expression :).

With the factory settings [Trigonometry := Auto, Trigpower := Auto] for
trigonometric simplification, Derive 6.10 transforms [F,G,H] as follows:

[#e^(-#i*n*t-#i*(1-n)*t)*(#e^(#e^(#i*t))+1)/(((#e^(#e^(#i*t))-1)~
/(#e^(2*#e^(#i*t))+1)+1)*COSH(#e^(#i*t))),(#e^(#e^(#i*t))+#e^(-#~
e^(#i*t)))/(2*COSH(#e^(#i*t))),#e^(-#i*t)*(-2*COSH(#e^(#i*t))+#e~
^(#e^(#i*t))+#e^(-#e^(#i*t)))/COSH(#e^(#i*t))]

[16*(#e^(10*COS(t))*COS(t)+#e^(9*COS(t))*(COS(2*SIN(t))*(COS(t)*~
COS(SIN(t))-SIN(t)*SIN(SIN(t)))+SIN(2*SIN(t))*(SIN(t)*COS(SIN(t)~
)+COS(t)*SIN(SIN(t)))+COS(t)*COS(SIN(t))-SIN(t)*SIN(SIN(t)))+#e^~
(8*COS(t))*(2*COS(2*SIN(t))*(COS(t)*COS(SIN(t))^2-SIN(t)*SIN(SIN~
(t))*COS(SIN(t))+COS(t))+2*SIN(2*SIN(t))*SIN(SIN(t))*(COS(t)*COS~
(SIN(t))-SIN(t)*SIN(SIN(t)))+2*COS(t)*COS(SIN(t))^2+2*SIN(t)*SIN~
(SIN(t))*COS(SIN(t))-COS(t))+#e^(7*COS(t))*(COS(4*SIN(t))*(COS(t~
)*COS(SIN(t))-SIN(t)*SIN(SIN(t)))+SIN(4*SIN(t))*(SIN(t)*COS(SIN(~
t))+COS(t)*SIN(SIN(t)))+6*COS(t)*COS(2*SIN(t))*COS(SIN(t))+(SIN(~
t)*COS(SIN(t))-COS(t)*SIN(SIN(t)))*(COS(SIN(t))*(SIN(2*SIN(t))*C~
OS(SIN(t))-3*SIN(SIN(t)))+SIN(SIN(t))*(SIN(2*SIN(t))*SIN(SIN(t))~
-COS(SIN(t))))-SIN(2*SIN(t))*(SIN(t)*COS(SIN(t))+COS(t)*SIN(SIN(~
t)))+COS(t)*COS(SIN(t))+SIN(t)*SIN(SIN(t)))+#e^(6*COS(t))*(2*COS~
(4*SIN(t))*COS(SIN(t))*(COS(t)*COS(SIN(t))-SIN(t)*SIN(SIN(t)))+2~
*SIN(4*SIN(t))*SIN(SIN(t))*(COS(t)*COS(SIN(t))-SIN(t)*SIN(SIN(t)~
))+6*COS(2*SIN(t))*COS(SIN(t))*(COS(t)*COS(SIN(t))+SIN(t)*SIN(SI~
N(t)))+SIN(t)*(COS(SIN(t))*(SIN(2*SIN(t))*COS(SIN(t))-3*SIN(SIN(~
t)))+SIN(SIN(t))*(SIN(2*SIN(t))*SIN(SIN(t))-COS(SIN(t))))-SIN(2*~
SIN(t))*(2*SIN(t)*COS(SIN(t))^2-2*COS(t)*SIN(SIN(t))*COS(SIN(t))~
-SIN(t))+2*COS(t))+#e^(5*COS(t))*(COS(4*SIN(t))*(3*COS(t)*COS(SI~
N(t))+SIN(t)*SIN(SIN(t)))-SIN(4*SIN(t))*(SIN(t)*COS(SIN(t))+COS(~
t)*SIN(SIN(t)))+2*COS(2*SIN(t))*((SIN(t)*COS(SIN(t))-COS(t)*SIN(~
SIN(t)))*(COS(SIN(t))*(SIN(2*SIN(t))*COS(SIN(t))-2*SIN(SIN(t)))+~
SIN(SIN(t))*(SIN(2*SIN(t))*SIN(SIN(t))-COS(SIN(t))))+COS(t)*COS(~
SIN(t))^3+SIN(t)*SIN(SIN(t))*COS(SIN(t))^2+COS(t)*COS(SIN(t))+SI~
N(t)*SIN(SIN(t)))+2*COS(t)*SIN(2*SIN(t))*SIN(SIN(t))+5*COS(t)*CO~
S(SIN(t))+SIN(t)*SIN(SIN(t)))+#e^(4*COS(t))*(COS(4*SIN(t))*(2*CO~
S(t)*COS(SIN(t))^2+2*SIN(t)*SIN(SIN(t))*COS(SIN(t))+COS(t))-SIN(~
4*SIN(t))*(2*SIN(t)*COS(SIN(t))^2-2*COS(t)*SIN(SIN(t))*COS(SIN(t~
))-SIN(t))+2*COS(2*SIN(t))*(SIN(t)*(COS(SIN(t))*(SIN(2*SIN(t))*C~
OS(SIN(t))-2*SIN(SIN(t)))+SIN(SIN(t))*(SIN(2*SIN(t))*SIN(SIN(t))~
-COS(SIN(t))))+COS(t)*COS(SIN(t))^2)+2*SIN(2*SIN(t))*SIN(SIN(t))~
*(COS(t)*COS(SIN(t))-SIN(t)*SIN(SIN(t)))+4*COS(t)*COS(SIN(t))^2+~
4*SIN(t)*SIN(SIN(t))*COS(SIN(t))+COS(t))+#e^(3*COS(t))*(COS(4*SI~
N(t))*(COS(t)*COS(SIN(t))+SIN(t)*SIN(SIN(t)))+SIN(4*SIN(t))*(COS~
(t)*SIN(SIN(t))-SIN(t)*COS(SIN(t)))+COS(2*SIN(t))*(5*COS(t)*COS(~
SIN(t))+3*SIN(t)*SIN(SIN(t)))+(SIN(t)*COS(SIN(t))-COS(t)*SIN(SIN~
(t)))*(COS(SIN(t))*(SIN(2*SIN(t))*COS(SIN(t))-SIN(SIN(t)))+SIN(S~
IN(t))*(SIN(2*SIN(t))*SIN(SIN(t))-COS(SIN(t))))-SIN(2*SIN(t))*(S~
IN(t)*COS(SIN(t))+COS(t)*SIN(SIN(t)))+2*COS(t)*COS(SIN(t))+2*SIN~
(t)*SIN(SIN(t)))+#e^(2*COS(t))*(COS(2*SIN(t))*(2*COS(t)*COS(SIN(~
t))^2+2*SIN(t)*SIN(SIN(t))*COS(SIN(t))+3*COS(t))+SIN(t)*(COS(SIN~
(t))*(SIN(2*SIN(t))*COS(SIN(t))-SIN(SIN(t)))+SIN(SIN(t))*(SIN(2*~
SIN(t))*SIN(SIN(t))-COS(SIN(t))))-SIN(2*SIN(t))*(2*SIN(t)*COS(SI~
N(t))^2-2*COS(t)*SIN(SIN(t))*COS(SIN(t))-SIN(t)))+#e^COS(t)*(COS~
(2*SIN(t))*(COS(t)*COS(SIN(t))+SIN(t)*SIN(SIN(t)))+SIN(2*SIN(t))~
*(COS(t)*SIN(SIN(t))-SIN(t)*COS(SIN(t)))+COS(t)*COS(SIN(t))+SIN(~
t)*SIN(SIN(t)))+COS(t))/(8*#e^(10*COS(t))+16*#e^(9*COS(t))*(COS(~
2*SIN(t))*COS(SIN(t))+SIN(2*SIN(t))*SIN(SIN(t)))+8*#e^(8*COS(t))~
*(4*COS(2*SIN(t))+1)+16*#e^(7*COS(t))*(COS(4*SIN(t))*COS(SIN(t))~
+SIN(4*SIN(t))*SIN(SIN(t))+COS(2*SIN(t))*COS(SIN(t))*(4*COS(SIN(~
t))^2-1)+SIN(SIN(t))*(COS(SIN(t))*(SIN(2*SIN(t))*COS(SIN(t))+SIN~
(SIN(t)))-SIN(SIN(t))*(SIN(2*SIN(t))*SIN(SIN(t))-COS(SIN(t)))+SI~
N(2*SIN(t))*(2*COS(SIN(t))^2-1)))+16*#e^(6*COS(t))*(COS(4*SIN(t)~
)*(2*COS(SIN(t))^2-1)+2*SIN(4*SIN(t))*SIN(SIN(t))*COS(SIN(t))+CO~
S(2*SIN(t))*(1-2*(SIN(SIN(t))*(COS(SIN(t))*(COS(SIN(t))*(SIN(2*S~
IN(t))*COS(SIN(t))-SIN(SIN(t)))+SIN(SIN(t))*(SIN(2*SIN(t))*SIN(S~
IN(t))-COS(SIN(t))))+SIN(SIN(t)))-COS(SIN(t))^2))+1)+4*#e^(5*COS~
(t))*(2*COS(4*SIN(t))*COS(SIN(t))*(8*COS(SIN(t))^2-3)+8*SIN(4*SI~
N(t))*SIN(SIN(t))*(2*COS(SIN(t))^2-1)+4*COS(2*SIN(t))*(2*COS(SIN~
(t))-SIN(SIN(t))*(COS(SIN(t))*(SIN(2*SIN(t))*COS(SIN(t))-SIN(SIN~
(t)))+SIN(SIN(t))*(SIN(2*SIN(t))*SIN(SIN(t))-COS(SIN(t)))))+SIN(~
2*SIN(t))*SIN(SIN(t))*(8*COS(SIN(t))^2+1)+2*COS(SIN(t))*(COS(SIN~
(t))^2+2))+#e^(4*COS(t))*(12*COS(4*SIN(t))+3*COS(2*SIN(t))*(6*CO~
S(SIN(t))^2+7)-4*SIN(SIN(t))^4+29)+16*#e^(3*COS(t))*(COS(4*SIN(t~
))*COS(SIN(t))+SIN(4*SIN(t))*SIN(SIN(t))+COS(2*SIN(t))*COS(SIN(t~
))-SIN(2*SIN(t))*SIN(SIN(t))+2*COS(SIN(t)))+8*#e^(2*COS(t))*(4*C~
OS(2*SIN(t))+1)+16*#e^COS(t)*(COS(2*SIN(t))*COS(SIN(t))+SIN(2*SI~
N(t))*SIN(SIN(t)))+8)-8*#i*(2*#e^(10*COS(t))*SIN(t)+2*#e^(9*COS(~
t))*(COS(2*SIN(t))*(SIN(t)*COS(SIN(t))+COS(t)*SIN(SIN(t)))+SIN(2~
*SIN(t))*(SIN(t)*SIN(SIN(t))-COS(t)*COS(SIN(t)))+SIN(t)*COS(SIN(~
t))+COS(t)*SIN(SIN(t)))+#e^(8*COS(t))*(4*COS(2*SIN(t))*(SIN(t)*C~
OS(SIN(t))^2+COS(t)*SIN(SIN(t))*COS(SIN(t))+SIN(t))+SIN(2*SIN(t)~
)*(4*SIN(t)*SIN(SIN(t))*COS(SIN(t))+COS(t)*(4*SIN(SIN(t))^2-1))+~
4*SIN(t)*COS(SIN(t))^2-2*COS(t)*SIN(SIN(t))*COS(SIN(t))-2*SIN(t)~
)+2*#e^(7*COS(t))*(COS(4*SIN(t))*(SIN(t)*COS(SIN(t))+COS(t)*SIN(~
SIN(t)))+SIN(4*SIN(t))*(SIN(t)*SIN(SIN(t))-COS(t)*COS(SIN(t)))+6~
*SIN(t)*COS(2*SIN(t))*COS(SIN(t))-(COS(t)*COS(SIN(t))+SIN(t)*SIN~
(SIN(t)))*(COS(SIN(t))*(SIN(2*SIN(t))*COS(SIN(t))-3*SIN(SIN(t)))~
+SIN(SIN(t))*(SIN(2*SIN(t))*SIN(SIN(t))-COS(SIN(t))))+SIN(2*SIN(~
t))*(COS(t)*COS(SIN(t))-SIN(t)*SIN(SIN(t)))+SIN(t)*COS(SIN(t))-C~
OS(t)*SIN(SIN(t)))+2*#e^(6*COS(t))*(2*COS(4*SIN(t))*COS(SIN(t))*~
(SIN(t)*COS(SIN(t))+COS(t)*SIN(SIN(t)))+2*SIN(4*SIN(t))*SIN(SIN(~
t))*(SIN(t)*COS(SIN(t))+COS(t)*SIN(SIN(t)))+6*COS(2*SIN(t))*COS(~
SIN(t))*(SIN(t)*COS(SIN(t))-COS(t)*SIN(SIN(t)))-COS(t)*(COS(SIN(~
t))*(SIN(2*SIN(t))*COS(SIN(t))-3*SIN(SIN(t)))+SIN(SIN(t))*(SIN(2~
*SIN(t))*SIN(SIN(t))-COS(SIN(t))))+SIN(2*SIN(t))*(2*COS(t)*COS(S~
IN(t))^2+2*SIN(t)*SIN(SIN(t))*COS(SIN(t))-COS(t))+2*SIN(t))+2*#e~
^(5*COS(t))*(COS(4*SIN(t))*(3*SIN(t)*COS(SIN(t))-COS(t)*SIN(SIN(~
t)))+SIN(4*SIN(t))*(COS(t)*COS(SIN(t))-SIN(t)*SIN(SIN(t)))-2*COS~
(2*SIN(t))*((COS(t)*COS(SIN(t))+SIN(t)*SIN(SIN(t)))*(COS(SIN(t))~
*(SIN(2*SIN(t))*COS(SIN(t))-2*SIN(SIN(t)))+SIN(SIN(t))*(SIN(2*SI~
N(t))*SIN(SIN(t))-COS(SIN(t))))-SIN(t)*COS(SIN(t))^3+COS(t)*SIN(~
SIN(t))*COS(SIN(t))^2-SIN(t)*COS(SIN(t))+COS(t)*SIN(SIN(t)))+2*S~
IN(t)*SIN(2*SIN(t))*SIN(SIN(t))+5*SIN(t)*COS(SIN(t))-COS(t)*SIN(~
SIN(t)))+2*#e^(4*COS(t))*(COS(4*SIN(t))*(2*SIN(t)*COS(SIN(t))^2-~
2*COS(t)*SIN(SIN(t))*COS(SIN(t))+SIN(t))+SIN(4*SIN(t))*(2*COS(t)~
*COS(SIN(t))^2+2*SIN(t)*SIN(SIN(t))*COS(SIN(t))-COS(t))+COS(2*SI~
N(t))*(SIN(t)-2*(COS(t)*(COS(SIN(t))*(SIN(2*SIN(t))*COS(SIN(t))-~
2*SIN(SIN(t)))+SIN(SIN(t))*(SIN(2*SIN(t))*SIN(SIN(t))-COS(SIN(t)~
)))-SIN(t)*COS(SIN(t))^2))+SIN(2*SIN(t))*(2*SIN(t)*SIN(SIN(t))*C~
OS(SIN(t))+COS(t)*(2*SIN(SIN(t))^2-1))+2*(SIN(t)*COS(SIN(t))^2-C~
OS(t)*SIN(SIN(t))*COS(SIN(t))+SIN(t)))+2*#e^(3*COS(t))*(COS(4*SI~
N(t))*(SIN(t)*COS(SIN(t))-COS(t)*SIN(SIN(t)))+SIN(4*SIN(t))*(COS~
(t)*COS(SIN(t))+SIN(t)*SIN(SIN(t)))+COS(2*SIN(t))*(5*SIN(t)*COS(~
SIN(t))-3*COS(t)*SIN(SIN(t)))-(COS(t)*COS(SIN(t))+SIN(t)*SIN(SIN~
(t)))*(COS(SIN(t))*(SIN(2*SIN(t))*COS(SIN(t))-SIN(SIN(t)))+SIN(S~
IN(t))*(SIN(2*SIN(t))*SIN(SIN(t))-COS(SIN(t))))+SIN(2*SIN(t))*(C~
OS(t)*COS(SIN(t))-SIN(t)*SIN(SIN(t)))+2*SIN(t)*COS(SIN(t))-2*COS~
(t)*SIN(SIN(t)))+2*#e^(2*COS(t))*(COS(2*SIN(t))*(2*SIN(t)*COS(SI~
N(t))^2-2*COS(t)*SIN(SIN(t))*COS(SIN(t))+3*SIN(t))-COS(t)*(COS(S~
IN(t))*(SIN(2*SIN(t))*COS(SIN(t))-SIN(SIN(t)))+SIN(SIN(t))*(SIN(~
2*SIN(t))*SIN(SIN(t))-COS(SIN(t))))+SIN(2*SIN(t))*(2*COS(t)*COS(~
SIN(t))^2+2*SIN(t)*SIN(SIN(t))*COS(SIN(t))-COS(t)))+2*#e^COS(t)*~
(COS(2*SIN(t))*(SIN(t)*COS(SIN(t))-COS(t)*SIN(SIN(t)))+SIN(2*SIN~
(t))*(COS(t)*COS(SIN(t))+SIN(t)*SIN(SIN(t)))+SIN(t)*COS(SIN(t))-~
COS(t)*SIN(SIN(t)))+2*SIN(t))/(8*#e^(10*COS(t))+16*#e^(9*COS(t))~
*(COS(2*SIN(t))*COS(SIN(t))+SIN(2*SIN(t))*SIN(SIN(t)))+8*#e^(8*C~
OS(t))*(4*COS(2*SIN(t))+1)+16*#e^(7*COS(t))*(COS(4*SIN(t))*COS(S~
IN(t))+SIN(4*SIN(t))*SIN(SIN(t))+COS(2*SIN(t))*COS(SIN(t))*(4*CO~
S(SIN(t))^2-1)+SIN(SIN(t))*(COS(SIN(t))*(SIN(2*SIN(t))*COS(SIN(t~
))+SIN(SIN(t)))-SIN(SIN(t))*(SIN(2*SIN(t))*SIN(SIN(t))-COS(SIN(t~
)))+SIN(2*SIN(t))*(2*COS(SIN(t))^2-1)))+16*#e^(6*COS(t))*(COS(4*~
SIN(t))*(2*COS(SIN(t))^2-1)+2*SIN(4*SIN(t))*SIN(SIN(t))*COS(SIN(~
t))+COS(2*SIN(t))*(1-2*(SIN(SIN(t))*(COS(SIN(t))*(COS(SIN(t))*(S~
IN(2*SIN(t))*COS(SIN(t))-SIN(SIN(t)))+SIN(SIN(t))*(SIN(2*SIN(t))~
*SIN(SIN(t))-COS(SIN(t))))+SIN(SIN(t)))-COS(SIN(t))^2))+1)+4*#e^~
(5*COS(t))*(2*COS(4*SIN(t))*COS(SIN(t))*(8*COS(SIN(t))^2-3)+8*SI~
N(4*SIN(t))*SIN(SIN(t))*(2*COS(SIN(t))^2-1)+4*COS(2*SIN(t))*(2*C~
OS(SIN(t))-SIN(SIN(t))*(COS(SIN(t))*(SIN(2*SIN(t))*COS(SIN(t))-S~
IN(SIN(t)))+SIN(SIN(t))*(SIN(2*SIN(t))*SIN(SIN(t))-COS(SIN(t))))~
)+SIN(2*SIN(t))*SIN(SIN(t))*(8*COS(SIN(t))^2+1)+2*COS(SIN(t))*(C~
OS(SIN(t))^2+2))+#e^(4*COS(t))*(12*COS(4*SIN(t))+3*COS(2*SIN(t))~
*(6*COS(SIN(t))^2+7)-4*SIN(SIN(t))^4+29)+16*#e^(3*COS(t))*(COS(4~
*SIN(t))*COS(SIN(t))+SIN(4*SIN(t))*SIN(SIN(t))+COS(2*SIN(t))*COS~
(SIN(t))-SIN(2*SIN(t))*SIN(SIN(t))+2*COS(SIN(t)))+8*#e^(2*COS(t)~
)*(4*COS(2*SIN(t))+1)+16*#e^COS(t)*(COS(2*SIN(t))*COS(SIN(t))+SI~
N(2*SIN(t))*SIN(SIN(t)))+8),1,0]

Thus F is transformed into a trigonometric mess of separated real and
imaginary parts, while G and H reduce to 1 and 0. But when Trigonometry
is set to Expand, the desired simplification of [F,G,H] happens right
away

[#e^(-#i*n*t-#i*(1-n)*t)*(#e^(#e^(#i*t))+1)/(((#e^(#e^(#i*t))-1)~
/(#e^(2*#e^(#i*t))+1)+1)*COSH(#e^(#i*t))),(#e^(#e^(#i*t))+#e^(-#~
e^(#i*t)))/(2*COSH(#e^(#i*t))),#e^(-#i*t)*(-2*COSH(#e^(#i*t))+#e~
^(#e^(#i*t))+#e^(-#e^(#i*t)))/COSH(#e^(#i*t))]

Trigonometry:=Expand

[2*#e^(-#i*t),1,0]

and reveals the trigonometric mess to equal 2*COS(t) - 2*#i*SIN(t).

Martin.

Back to sci.math.symbolic | Previous | Next — Previous in thread | Find similar

Thread

Re: Simplifying exponential expressions Axel Vogt <&noreply@axelvogt.de> - 2015-08-30 15:24 +0200
  Re: Simplifying exponential expressions Dr Huang <drhuang57@gmail.com> - 2015-08-30 15:46 -0700
    Re: Simplifying exponential expressions Richard Fateman <fateman@cs.berkeley.edu> - 2015-08-30 17:08 -0700
      Re: Simplifying exponential expressions clicliclic@freenet.de - 2015-09-03 12:42 +0200

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