[Maxima-bugs] [ maxima-Bugs-660948 ] simplification of exp(%i*...)

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Bugs item #660948, was opened at 2003-01-02 05:23
Message generated for change (Settings changed) made by crategus
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Category: Lisp Core - Complex
Group: None
>Status: Closed
>Resolution: Fixed
Priority: 5
Private: No
Submitted By: Stavros Macrakis (macrakis)
Assigned to: Nobody/Anonymous (nobody)
Summary: simplification of exp(%i*...)

Initial Comment:
There are some weirdnesses and inconsistencies in the 
simplification of exp(%i*...).

First, let's look at the case exp(%i*%pi*q).

If q is (m/n) with n in {1,2,3,4,6}, it returns a solution in 
rationals and radicals, e.g. for q=11/3, we get 1/2-SQRT
(3)*%I/2.  For other n's, it reduces q to the range (-1,+1].

If q is a float which is exactly m/n as above, it 
rationalizes it: 

 exp(%i*%pi*.25) => 
   SQRT(2)*%I/2+SQRT(2)/2

I am not comfortable with this, because a floating point 
number is supposed to be an approximation, and we're 
getting an exact result here. It does the same thing for 
x^1.0 and (x^3)^float(1/3), though not for x^0.5 or x^2.0 or 
(x^3)^float(5/3).

However, it does a weird thing if q is very near m/n:

exp(%i*%pi*.166666666) =>
  (%I/2+SQRT(3)/2)*%E^-(6.666666663157628E-10*%I*%
PI)

Of course this is accurate, but it seems pointless and 
bizarre.

It is also inconsistent in its handling of purely floating 
exponents:

  exp(%i*%pi*.333) =>
     %E^(0.333*%I*%PI)        (OK)
but
  exp(%i*%pi*5.333) =>
     %E^-(667*%I*%PI/1000)     (why rationalized here?)

Now let's look at the case exp(%i*q).

If q = 1.0 exactly, it acts as though q=1:

    exp(%i*1.0) => %e^%i.

This follows the (in my opinion anomalous) case x^1.0 
=> x, as above.  I don't know why this should be; after 
all, 2^1.0 => 2.0, not 2.

And for other floats, it does no reduction at all to (-pi,pi]:

  exp(%i*ev(%pi,numer)) =>
    %E^(3.141592653589793*%I)
  exp(%i*ev(%pi*100,numer)) =>
    %E^(314.1592653589793*%I)
  exp(%i*300.0) => exp(%i*300.0)

All this needs to be made consistent and clear.

In my opinion, floats should NEVER be coerced to exact 
numbers, unless the user asks for it (e.g. by converting 
to Rat form).

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>Comment By: Dieter Kaiser (crategus)
Date: 2009-07-26 00:12

Message:
I think the problems of this bug report are no longer present. 
These are the results for the given examples:

No rationalizing  of float numbers:

(%i2) exp(%i*%pi*0.25);
(%o2) %e^(0.25*%i*%pi)

No simplification to an exact result:

(%i3) x^1.0;
(%o3) x^1.0
(%i4) (x^3)^float(1/3);
(%o4) x^1.0

More examples for float numbers:

(%i5) exp(%i*%pi*0.166666666);
(%o5) %e^(0.166666666*%i*%pi)

(%i6) exp(%i*%pi*5.333);
(%o6) %e^(5.333*%i*%pi)

The exp function with a complex float argument evaluate numerically:

(%i7) exp(%i*1.0);
(%o7) .8414709848078965*%i+.5403023058681398

(%i8) exp(%i*ev(%pi,numer));
(%o8) 1.224500708041116E-16*%i-1.0

(%i9) exp(%i*ev(%pi*100,numer));
(%o9) 1.961926030129856E-15*%i+1.0

(%i10) exp(%i*300.0);
(%o10) -.9997558399011495*%i-.02209661927868395

Closing this bug report as fixed.

Dieter Kaiser

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