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Science Forum Index » Math - Symbolic Forum » An exact simplification challenge - 12
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Message |
| Vladimir Bondarenko |
 Posted: Sun Apr 29, 2007 5:31 am |
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Guest
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Hello all computer algebra fans,
Neither Maple nor Mathematica nor MuPAD can
simplify these expressions directly...
exp(cosh(z))^sinh(z)-exp(sinh(z))^cosh(z)
exp(sinh(z+I*(2*z-1)))^cosh(z+I*(2*z-1))
-exp(cosh(z+I*(2*z-1)))^sinh(z+I*(2*z-1))
Is there a simplification soul who is able
to show us how via using a set of a CAS
commands to get to the exact answers?
Best wishes,
Vladimir Bondarenko
VM and GEMM architect
Co-founder, CEO, Mathematical Director
http://www.cybertester.com/ Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/ CAS Testing |
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| G. A. Edgar |
| Posted: Sun Apr 29, 2007 5:53 am |
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Guest
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In article <1177842708.507175.299620@c35g2000hsg.googlegroups.com>,
Vladimir Bondarenko <vb@cybertester.com> wrote:
Quote: Hello all computer algebra fans,
Neither Maple nor Mathematica nor MuPAD can
simplify these expressions directly...
exp(cosh(z))^sinh(z)-exp(sinh(z))^cosh(z)
Maple...
T := exp(cosh(z))^sinh(z)-exp(sinh(z))^cosh(z);
sinh(z) cosh(z)
(exp(cosh(z))) - (exp(sinh(z)))
simplify(T,symbolic);
0
You have to say "symbolic", since in actuality that simplification
is incorrect in some cases...
subs(z=1.86+I*1.57,T);
sinh(1.86 + 1.57 I)
(exp(cosh(1.86 + 1.57 I)))
cosh(1.86 + 1.57 I)
- (exp(sinh(1.86 + 1.57 I)))
evalf(%);
-11868.83163 + 0.2411699102 I
or:
simplify(T) assuming z::real;
0
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/ |
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| Vladimir Bondarenko |
| Posted: Sun Apr 29, 2007 6:15 am |
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On Apr 29, 3:53 am, "G. A. Edgar" <e...@math.ohio-state.edu.invalid>
writes:
GAE> You have to say "symbolic", since in actuality that
simplification
GAE> is incorrect in some cases...
Having seen your amazing speed, it is not once already that I thought
that
you have already a multi-tons stock of solutions at your fingertips :)
GAE> You have to say "symbolic", since in actuality that
simplification
GAE> is incorrect in some cases...
Is it possible to build an exact formula for a set of intervals for
which this
expression is non-zero? (at first sight it does not look like this,
but...)
How you guessed that there are intervals where the original expression
is non-zero?
Quote: In article <1177842708.507175.299...@c35g2000hsg.googlegroups.com>,
Vladimir Bondarenko <v...@cybertester.com> wrote:
Hello all computer algebra fans,
Neither Maple nor Mathematica nor MuPAD can
simplify these expressions directly...
exp(cosh(z))^sinh(z)-exp(sinh(z))^cosh(z)
Maple...
T := exp(cosh(z))^sinh(z)-exp(sinh(z))^cosh(z);
sinh(z) cosh(z)
(exp(cosh(z))) - (exp(sinh(z)))
simplify(T,symbolic);
0
You have to say "symbolic", since in actuality that simplification
is incorrect in some cases...
subs(z=1.86+I*1.57,T);
sinh(1.86 + 1.57 I)
(exp(cosh(1.86 + 1.57 I)))
cosh(1.86 + 1.57 I)
- (exp(sinh(1.86 + 1.57 I)))
evalf(%);
-11868.83163 + 0.2411699102 I
or:
simplify(T) assuming z::real;
0
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/ |
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| Denis Feldmann |
| Posted: Sun Apr 29, 2007 6:37 am |
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G. A. Edgar a écrit :
Quote: In article <1177842708.507175.299620@c35g2000hsg.googlegroups.com>,
Vladimir Bondarenko <vb@cybertester.com> wrote:
Hello all computer algebra fans,
Neither Maple nor Mathematica nor MuPAD can
simplify these expressions directly...
exp(cosh(z))^sinh(z)-exp(sinh(z))^cosh(z)
Maple...
T := exp(cosh(z))^sinh(z)-exp(sinh(z))^cosh(z);
sinh(z) cosh(z)
(exp(cosh(z))) - (exp(sinh(z)))
simplify(T,symbolic);
0
You have to say "symbolic", since in actuality that simplification
is incorrect in some cases...
subs(z=1.86+I*1.57,T);
sinh(1.86 + 1.57 I)
(exp(cosh(1.86 + 1.57 I)))
cosh(1.86 + 1.57 I)
- (exp(sinh(1.86 + 1.57 I)))
evalf(%);
-11868.83163 + 0.2411699102 I
or:
simplify(T) assuming z::real;
0
Bet you will not get any answer, as usual. So why bother? |
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| Denis Feldmann |
| Posted: Sun Apr 29, 2007 6:39 am |
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Denis Feldmann a écrit :
Quote: G. A. Edgar a écrit :
Bet you will not get any answer, as usual. So why bother?
Oups, sorry. OTOH, Vladimir's answer doesn't strike me as even slightly
repentant (as in "Maple... is unable to simplify this...") |
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| G. A. Edgar |
| Posted: Sun Apr 29, 2007 8:27 am |
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Guest
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In article <1177845331.425291.109590@n76g2000hsh.googlegroups.com>,
Vladimir Bondarenko <vb@cybertester.com> wrote:
Quote: How you guessed that there are intervals where the original expression
is non-zero?
Because the incorrect simplification (a^b)^c -> a^(b*c) is a
common error for complex numbers.
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/ |
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| Robert Israel |
| Posted: Sun Apr 29, 2007 1:15 pm |
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Vladimir Bondarenko <vb@cybertester.com> writes:
Quote: On Apr 29, 3:53 am, "G. A. Edgar" <e...@math.ohio-state.edu.invalid
writes:
GAE> You have to say "symbolic", since in actuality that
simplification
GAE> is incorrect in some cases...
Having seen your amazing speed, it is not once already that I thought
that
you have already a multi-tons stock of solutions at your fingertips :)
GAE> You have to say "symbolic", since in actuality that
simplification
GAE> is incorrect in some cases...
Is it possible to build an exact formula for a set of intervals for
which this
expression is non-zero? (at first sight it does not look like this,
but...)
.... the expression being
Quote: T := exp(cosh(z))^sinh(z)-exp(sinh(z))^cosh(z);
By definition, a^b = exp(b ln(a)), so with s = cosh(z) and t = sinh(z),
exp(s)^t = exp(t ln(exp(s))) = exp(t s + 2 pi i t n(s)) where
n(s) = (ln(exp(s))-s)/(2 pi) is an integer. Similarly,
exp(t)^s = exp(t s + 2 pi i s n(t)). These are equal iff
exp(2 pi i (t n(s) - s n(t))) = 1, i.e. iff t n(s) - s n(t) is an
integer. Now if f(z) = sinh(z) n(cosh(z)) - cosh(z) n(sinh(z)),
since sinh(z), cosh(z) and 1 are linearly independent as analytic
functions, the only way for f(z) to be 0 on a nonempty open subset
of the complex plane is n(cosh(z)) = n(sinh(z)) = 0.
If, as Maple does, you use the principal branch of the logarithm,
n(s) = 0 iff -pi < Im(s) <= pi. Now Im(cosh(z)) = sinh(Re(z)) sin(Im(z))
and Im(sinh(z)) = cosh(Re(z)) sin(Im(z)). In particular,
|Im(cosh(z))| < |Im(sinh(z))| = cosh(Re(z)) |sin(Im(z))|. Thus the
interior of {z: T = 0} can be written as
{ x + I y: cosh(x) |sin(y)| < pi}.
--
Robert Israel israel@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada |
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| Dana |
| Posted: Sun May 06, 2007 12:15 pm |
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Guest
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Hi. Mathematica 6 does not reduce the equation to zero since it "assumes" z
could be Complex:
equ = Exp[Cosh[z]]^Sinh[z] - Exp[Sinh[z]]^Cosh[z];
Both:
Simplify[equ]
and
Assuming[Element[z, Complexes], Simplify[equ]]
<< Returns unevaluated.>>
However:
Assuming[Element[z, Reals], Simplify[equ]]
0
Love following your posts. Keep up the good work! :>)
--
Dana
"Vladimir Bondarenko" <vb@cybertester.com> wrote in message
news:1177842708.507175.299620@c35g2000hsg.googlegroups.com...
Quote: Hello all computer algebra fans,
Neither Maple nor Mathematica nor MuPAD can
simplify these expressions directly...
exp(cosh(z))^sinh(z)-exp(sinh(z))^cosh(z)
exp(sinh(z+I*(2*z-1)))^cosh(z+I*(2*z-1))
-exp(cosh(z+I*(2*z-1)))^sinh(z+I*(2*z-1))
Is there a simplification soul who is able
to show us how via using a set of a CAS
commands to get to the exact answers?
Best wishes,
Vladimir Bondarenko
VM and GEMM architect
Co-founder, CEO, Mathematical Director
http://www.cybertester.com/ Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/ CAS Testing
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