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DEtools

  

Desingularize

  

desingularize a linear differential operator

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

Desingularize(L, Dx, x, func)

Parameters

L

-

polynomial in Dx with coefficients that are polynomials in x

Dx

-

variable, denoting the differential operator w.r.t. x

x

-

variable

func

-

(optional) procedure

Description

• 

Let L be a linear differential operator, given as a polynomial in Dx with univariate polynomial coefficients in x over a field k of characteristic zero. The command Desingularize(L,Dx,x) constructs a linear differential operator R such that any solution of Ly=0 is also a solution of Ry=0 and R has no apparent singularities.  The operator R is said to maximally desingularize L, and will be right divisible by L over the field kx.

• 

An apparent singularity is a point p where the leading coefficient of L vanishes, yet p is not a pole of any holomorphic solution of Ly=0. In this case there will exist d linearly independent solutions at p where d is the order of L.

• 

A function may be specified using the optional argument func. It is applied to the coefficients of the collected result. Often simplify or factor will be used.

Examples

withDEtools:

For the given differential operator L

L24x318x4+x8+6x5x6Dx7+6x5+72x330x48x772x2Dx6+144x2+36x6+72x32x7+144x18x4Dx5+24x3+36x6+144x14472x2120x58x7x10+x8Dx4+24x5x106x7+x8+18x6Dx3+36x56x672x4+2x9Dx2+36x4+12x510x8+2x9Dx+64x712x432x8+8x9+x12x10

Lx8x6+6x518x4+24x3Dx7+8x7+6x530x4+72x372x2Dx6+2x7+36x618x4+72x3144x2+144xDx5+x10+x88x7+36x6120x5+24x372x2+144x144Dx4+x10+x86x7+18x624x5Dx3+2x96x6+36x572x4Dx2+2x910x8+12x536x4Dx+64x712x432x8+8x9+x12x10

(1)

compute a desingularizing operator for L:

MDesingularizeL,Dx,x,factor

M1728252Dx8+54154x7+161694x6+263753x5+452649x4324882x3+1728252Dx7+433232x61293552x52218332x42969808x3+974646x2+1728252Dx6+108308x6+1626156x5+5185170x4+9891042x3+9684162x21949292x+1728252Dx5+54154x9161694x8263753x7560957x6+976266x52924106x410379136x327461604x245113328x+3677544Dx4x54154x8+161694x7+263753x6+452649x5324882x4+1728252x+13826016Dx3+108308x8+323388x7+635814x6+253914x5+974646x41728252x210369512x20739024Dx2+108308x8218152x7981126x62600232x5520824x41728252x26913008x10369512Dx+54154x11+161694x10+263753x9+560957x8759650x7+538258x6+2595900x5+10387932x4+31684620x31728252x23456504x3456504

(2)

Q,RopDEtools'rightdivision'M,L,Dx,x:

Hence, R=Q·L+R where

Q

1728252Dxx3x5x3+6x218x+24+54154x7+161694x6+263753x5+452649x4324882x3+1728252x3x5x3+6x218x+24

(3)

R

0

(4)

References

  

Tsai, H. "Weyl closure of a linear differential operator." Journal of Symbolic Computation Vol. 29 No. 4-5 (2000): 747-775.

  

Chyzak, F.; Dumas, P.; Le, H.Q.; Martins, J.; Mishna, M.; Salvy, B. "Taming apparent singularities via Ore closure." In preparation.

Compatibility

• 

The DEtools[Desingularize] command was introduced in Maple 15.

• 

For more information on Maple 15 changes, see Updates in Maple 15.

See Also

DEtools/Closure

Groebner

Ore_algebra