FullSimplify successive transforms

• To: mathgroup at smc.vnet.net
• Subject: [mg117491] FullSimplify successive transforms
• From: Arturas.Acus at tfai.vu.lt
• Date: Sun, 20 Mar 2011 04:55:00 -0500 (EST)

```Dear Group,

How to extract successful transformations of FullSimplify?

In particulary I am interesting what rules were used to transform
BesselI[ ] functions into BesselJ/BesselK in the example below. Of
course, learning the general method to detect sequence of these
transforms would be of great value.

all\[Psi]MathematicaSimplified={(2 k1 Sqrt[2/\[Pi]] \[Kappa]2
BesselJ[n,k1 \[Rho]] BesselK[n,a \[Kappa]2] Sin[n \[Pi]])/(\[Sqrt](2 a
k1^2 \[Pi] \[Kappa]2 BesselI[n,a \[Kappa]2] (n (BesselI[1-n,a
\[Kappa]2]+BesselI[-1+n,a \[Kappa]2]) BesselJ[n,a k1]^2+a \[Kappa]2
BesselI[-n,a \[Kappa]2] BesselJ[-1+n,a k1] BesselJ[1+n,a k1])+\[Pi]
BesselI[n,a \[Kappa]2]^2 (2 a k1 n \[Kappa]2^2 BesselJ[-1+n,a k1]
BesselJ[n,a k1]-4 n^2 (k1^2+\[Kappa]2^2) BesselJ[n,a k1]^2+a^2 k1^2
\[Kappa]2^2 BesselJ[1+n,a k1]^2)+a k1^2 \[Kappa]2 (a \[Pi] \[Kappa]2
((BesselI[1-n,a \[Kappa]2]^2-2 BesselI[1-n,a \[Kappa]2] BesselI[-1+n,a
\[Kappa]2]+BesselI[1+n,a \[Kappa]2]^2) BesselJ[n,a k1]^2-BesselI[-n,a
\[Kappa]2]^2 BesselJ[-1+n,a k1] BesselJ[1+n,a k1])+4 n BesselI[-n,a
\[Kappa]2] BesselJ[n,a k1]^2 BesselK[-1+n,a \[Kappa]2] Sin[n
\[Pi]]))),(2 k1 Sqrt[2/\[Pi]] \[Kappa]2 BesselJ[n,a k1]
BesselK[n,\[Kappa]2 \[Rho]] Sin[n \[Pi]])/(\[Sqrt](2 a k1^2 \[Pi]
\[Kappa]2 BesselI[n,a \[Kappa]2] (n (BesselI[1-n,a
\[Kappa]2]+BesselI[-1+n,a \[Kappa]2]) BesselJ[n,a k1]^2+a \[Kappa]2
BesselI[-n,a \[Kappa]2] BesselJ[-1+n,a k1] BesselJ[1+n,a k1])+\[Pi]
BesselI[n,a \[Kappa]2]^2 (2 a k1 n \[Kappa]2^2 BesselJ[-1+n,a k1]
BesselJ[n,a k1]-4 n^2 (k1^2+\[Kappa]2^2) BesselJ[n,a k1]^2+a^2 k1^2
\[Kappa]2^2 BesselJ[1+n,a k1]^2)+a k1^2 \[Kappa]2 (a \[Pi] \[Kappa]2
((BesselI[1-n,a \[Kappa]2]^2-2 BesselI[1-n,a \[Kappa]2] BesselI[-1+n,a
\[Kappa]2]+BesselI[1+n,a \[Kappa]2]^2) BesselJ[n,a k1]^2-BesselI[-n,a
\[Kappa]2]^2 BesselJ[-1+n,a k1] BesselJ[1+n,a k1])+4 n BesselI[-n,a
\[Kappa]2] BesselJ[n,a k1]^2 BesselK[-1+n,a \[Kappa]2] Sin[n \[Pi]])))}

all\[Psi]MathematicaSimplifiedNoHyper=Assuming[a>0&&\[Kappa]2>0&&k1>0&&n>0,FullSimplify[all\[Psi]MathematicaSimplified,ComplexityFunction->(10000*Count[#,_BesselI|_Hypergeometric0F1Regularized|_Hypergeometric0F1,{0,Infinity}]+LeafCount[#]&)]]

Sincerely, Arturas Acus

```

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