An exact simplification challenge - 27 (Ei, hypergeom)... help Mr. Cisson!
Vladimir Bondarenko
Over the floor, crumpled pieces of paper are scattered.
Streaming with perspiration, a gentleman keeps jotting
down new and new formulas...
Today, Pythagoras Secret Society Chairman Mr. Cisson
discovered right at the very center of his circular
table a strip of paper bearing a long cryptic message.
Mr. Cisson has an absolutely precise feeling that is
is important for the PSS but is unfortunately unable
to decipher it...
Is there a cryptanalyst to help suffering Mr. Cisson
via inventing a way to decode this message
------------------------------------------------------
*** TOP SECRET ***
-26460*I*Ei(1,1+I)*exp(1)-26460*Ei(1,1-I)*exp(1)+26460
*I*Ei(1,1-I)*exp(1)-26460*Ei(1,1+I)*exp(1)-52920*gamma
*exp(1)+13230*Pi*exp(1)+8820*exp(2)*cos(1)+196*hyperge
om([3/2],[7/4, 9/4, 5/2, 5/2],-1/64)*exp(1)-32*hyperge
om([7/4],[9/4, 5/2, 11/4, 11/4],-1/64)*exp(1)+2205*hyp
ergeom([1,1],[5/4, 3/2, 7/4, 2, 2],-1/64)*exp(1)-26460
*ln(2)*exp(1)+4410*exp(2)*sin(1)
------------------------------------------------------
using a sequence of CAS commands ?
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
dimitris
Could you post also the relevat Mathematica code?
Thank you very much!
Greetings from the sunny Athens!
Dimitris
/ Vladimir Bondarenko :
Vladimir Bondarenko
DA> Could you post also the relevat Mathematica code?
Hi syntrofo Dimitris!
Here you are,
-26460 I ExpIntegralE[1,1+I] Exp[1]-26460 ExpIntegralE[1,1-I]
Exp[1]+26460 I ExpIntegralE[1,1-I] Exp[1] - 26460 ExpIntegralE
[1,1+I] Exp[1] - 52920 EulerGamma Exp[1]+13230 Pi Exp[1]+8820*
Exp[2]*Cos[1]+196*HypergeometricPFQ[{3/2},{7/4,9/4,5/2,5/2},-1
/64]*Exp[1]-32*HypergeometricPFQ[{7/4},{9/4,5/2,11/4,11/4},-1/
64]*Exp[1]+2205*HypergeometricPFQ[{1,1},{5/4,3/2,7/4,2,2},-1/
64]*Exp[1]-26460*Log[2]*Exp[1]+4410*Exp[2]*Sin[1]
= 23319.901552771110279
Please help the wretched Mr. Cisson to recover his poise ;)
Greetings from the sunny Simferopol!
Vladimir
dimitris
I must admit that I didn't try many things.
Mma 5.2 fails to simplify this expression more than
In[16]:=
o = FullSimplify[-26460*I*ExpIntegralE[1, 1 + I]*Exp[1] -
26460*ExpIntegralE[1, 1 - I]*Exp[1] +
26460*I*ExpIntegralE[1, 1 - I]*Exp[1] - 26460*ExpIntegralE[1, 1 +
I]*Exp[1] - 52920*EulerGamma*Exp[1] + 13230*Pi*Exp[1] +
8820*Exp[2]*Cos[1] + 196*HypergeometricPFQ[{3/2}, {7/4, 9/4, 5/2,
5/2}, -64^(-1)]*Exp[1] -
32*HypergeometricPFQ[{7/4}, {9/4, 5/2, 11/4, 11/4},
-64^(-1)]*Exp[1] +
2205*HypergeometricPFQ[{1, 1}, {5/4, 3/2, 7/4, 2, 2},
-64^(-1)]*Exp[1] - 26460*Log[2]*Exp[1] + 4410*Exp[2]*Sin[1]]
Out[16]=
E*(-52920*EulerGamma + 13230*Pi - (26460 - 26460*I)*ExpIntegralE[1, 1
- I] - (26460 + 26460*I)*ExpIntegralE[1, 1 + I] +
196*HypergeometricPFQ[{3/2}, {7/4, 9/4, 5/2, 5/2}, -(1/64)] -
32*HypergeometricPFQ[{7/4}, {9/4, 5/2, 11/4, 11/4}, -(1/64)] +
2205*(HypergeometricPFQ[{1, 1}, {5/4, 3/2, 7/4, 2, 2}, -(1/64)] -
12*Log[2]) + 4410*E*(2*Cos[1] + Sin[1]))
I will give an idea on how someone could work
but this approach does not mean that we will get
surely desirable (i.e. more simplification) results.
Suppose that
In[23]:=
hp =HypergeometricPFQ[{1, s, k + t}, {k + s + 1, t + 1}, 1];
Then
In[37]:=
FullSimplify[hp]
FunctionExpand[hp]
Out[37]=
HypergeometricPFQ[{1, s, k + t}, {1 + k + s, 1 + t}, 1]
Out[38]=
t/(-s + t) + (s*t)/(k*(-s + t)) + (Gamma[1 + k + s]*Gamma[1 + t])/
(k*(s - t)*Gamma[s]*Gamma[k + t])
Another hint that for situations like this FunctionExpand is much
preferable
than FullSimplify.
Let's go now to
http://functions.wolfram.com/07.27.03.0020.01
we see that
HypergeometricPFQ[{1, a, b}, {d, 2 + a + b - d}, 1] == ((1 + a + b -
d)/((1 + a - d) (1 + b - d))) (1 - d + (Gamma[d] Gamma[1 + a + b - d])/
(Gamma[a] Gamma[b]))
Using this as rule we have
In[40]:=
Expand[hp /. HypergeometricPFQ[{1, a_, b_}, {d_, e_}, 1] :>
((a + b - d + 1)*(-d + (Gamma[d]*Gamma[a + b - d + 1])/
(Gamma[a]*Gamma[b]) + 1))/((a - d + 1)*(b - d + 1)) /;
e == a + b - d + 2]
Out[40]=
t/(-s + t) + (s*t)/(k*(-s + t)) - (t*Gamma[1 + k + s]*Gamma[t])/(k*(-s
+ t)*Gamma[s]*Gamma[k + t])
which is exactly the result obtaing by FunctionExpand.
Of course, in this case we didn't do anything important
because FunctionExpand itself knows this rule.
But finding some similar identities from above source or
other sources one may get a result with much less ComplexityFunction.
Speaking personally I must admit this is not exactly my cup of tea!
Dimitris
Vladimir Bondarenko :
dimitris
Suppose that
Then
http://functions.wolfram.com/07.27.03.0020.01
we see that
Dimitris
Vladimir Bondarenko :
dimitris
when someone execute commands like
FunctionExpand[HypergeometricPFQ[{1, s, k + t}, {k + s + 1, t + 1},
1]]
To this end,
In[2]:=
Begin["System`HypergeometricDump`"] ;
In[3]:=
Trace[FunctionExpand[ HypergeometricPFQ[{1,
s, k + t}, {k + s + 1, t + 1}, 1]],TraceInternal\[Rule]True]
In the length output someone can see
a number of rules being tried such as F32.
In[10]:=
?F32
(*ommiting output*)
For example
In[11]:=
?F32Formula50
Here we can see which rule specifically was applied
In[13]:=
Trace[FunctionExpand[HypergeometricPFQ[{1, s, t}, {s + 1, t + 1}, 1]],
F32]
Out[13]=
{ {{{{{{HoldForm[F32[{1, s, t}, {1 + s, 1 + t}, 1]], HoldForm[(((1 +
s) - 1)*((1 + t) - 1)*(PolyGamma[0, (1 + s) - 1] -
PolyGamma[0, (1 + t) - 1]))/((1 + s) - (1 + t))]}}}}}}}
Dimitris
dimitris :
Vladimir Bondarenko
Vladimir Bondarenko
wonder slept immediately... and he saw a Dream He
Desired! The long-for encoding!
So he thrown his red blanket to the right, and his
crumpled pillow to the left, and jumped out his
bed right into his rickety chair near the table!
Bang!...
Too strong was this jump, alas... so upon sitting at
his table he was able to remember a bit of his Dream
only....
8820*
Sad he is sitting near the table and keep trying to
recollect the rest of his Dream...
Could you please help to the poor thing?
Vladimir Bondarenko
... this was the dream the poor Mr. Cisson saw
today... and he is in total despair because he
forgot what were those abcdefghijk...
SOS! Heeelp the poor fellow!