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MathGroup Archive 2007

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  • To: mathgroup at smc.vnet.net
  • Subject: [mg74865] question
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Mon, 9 Apr 2007 06:11:15 -0400 (EDT)

BesselI[n, x] can be expressed as (x^n*Hypergeometric0F1Regularized[1
+ n, x^2/4])/2^n

Mathematica knows this and simplifies the hypergeometric function
(applying
FunctionExpand or FullSimplify)

In[53]:=
(x^n*Hypergeometric0F1Regularized[1 + n, x^2/4])/2^n//FunctionExpand

Out[53]=
BesselI[n, x]

The opposite is also possible.

In[54]:=
FullSimplify[BesselI[n, x], ComplexityFunction -> (Count[{#1},
_BesselI, Infinity] & )]

Out[54]=
(x^n*Hypergeometric0F1Regularized[1 + n, x^2/4])/2^n

The same holds true for some special functions I have tried.

Let see a more trivial function: Log[1+x].

Log[1+z] can be expressed as z*Hypergeometric2F1[1, 1, 2, -z]
and Mathematica automatically simplifies the hypergeometric series

In[55]:=
z*Hypergeometric2F1[1, 1, 2, -z]

Out[55]=
Log[1 + z]

Is it possible the opposite through some command(s)?

Something like the following does not look very smart. That's why it
fails!

In[57]:=
FullSimplify[Log[x+1],ComplexityFunction\[Rule]
(Count[{#1},_Log,Infinity]&)]

Out[57]=
Log[1+x]

Thanks
Dimitris



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