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ReduceHyperexp

  

a reduction algorithm for hyperexponential functions

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

ReduceHyperexp(H, x, newH)

Parameters

H

-

hyperexponential function of x

H1

-

hyperexponential function of x

H2

-

hyperexponential function of x

x

-

variable

newH

-

(optional) name; assigned a computed equivalence of H

Description

• 

For a specified hyperexponential function H of x, the (H1, H2) := ReduceHyperexp(H, x, newH) calling sequence constructs two hyperexponential functions H1 and H2 such that Hx=ⅆⅆxH1x+H2x and the certificate ⅆⅆxH2xH2x has a differential rational normal form r,s,u,v with v of minimal degree.

• 

The output from ReduceHyperexp is a sequence of two elements H1,H2 each of which is either 0 or written in the form

Hx=Vxⅇ∫Fxⅆx

  

(The form shown above is called a multiplicative decomposition of the hyperexponential function Hx.)

• 

ReduceHyperexp is a generalization of the reduction algorithm for rational functions by Hermite (recall that a rational function is also a hyperexponential function). It also covers the differential Gosper's algorithm.

Examples

withDEtools:

Hⅇ∫2x7x+42ⅆxx6+16x5+103x4+327x3+647x2+737x+194x12x+24x+42

H:=ⅇ∫2x7x+42ⅆxx6+16x5+103x4+327x3+647x2+737x+194x12x+24x+42

(1)

H1,H2ReduceHyperexpH,x,'nH'

H1,H2:=24x3+143x2+292x+216ⅇ∫15x+42ⅆxx1x+23,x3+17x2+88x231ⅇ∫232xx+42ⅆxx1

(2)

nH

x6+16x5+103x4+327x3+647x2+737x+194ⅇ∫15x+42ⅆxx12x+24

(3)

Hⅇ∫2x7x+42ⅆxx2+27x+62x+24x+42

H:=ⅇ∫2x7x+42ⅆxx2+27x+62x+24x+42

(4)

H1,H2ReduceHyperexpH,x

H1,H2:=x2+8x+16ⅇ∫15x+42ⅆxx+23,0

(5)

References

  

Geddes, Keith; Le, Ha; and Li, Ziming. "Differential rational canonical forms and a reduction algorithm for hyperexponential functions." Proceedings of ISSAC 2004. ACM Press. (2004): 183-190.

See Also

DEtools[AreSimilar]

DEtools[Gosper]

DEtools[IsHyperexponential]

DEtools[MultiplicativeDecomposition]

SumTools[Hypergeometric][SumDecomposition]

 


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