analyticity conditions for the solution of linear difference equation. - Maple Help

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LREtools[AnalyticityConditions] - analyticity conditions for the solution of linear difference equation.

Calling Sequence

AnalyticityConditions(L, E, fun, HalfInt_opt, Direction_opt)

Parameters

L

-

linear difference operator in E with coefficients which are polynomials in x

E

-

name of the shift operator acting on x

fun

-

function f(x) that is a solution of Lfx=0

HalfInt_opt

-

(optional) 'HalfInterval'= A, A is a rational number, 0 by default

Direction_opt

-

(optional) 'direction'='left' -- the procedure returns the conditions for analyticity of f(x) on x<A+d or 'direction'='right', the conditions on Ax.

Description

• 

The AnalyticityConditions command returns the set of conditions for the analyticity of f(x).

• 

The input includes a difference operator

L := sum(a[i](x)* E^i,i=1..d);

L:=i&equals;1daixEi

(1)
  

and a point A. The solution f(x) is analytic on some open set which contains a set A<=Rex<A+d. The procedure returns the set of conditions for the analyticity of f(x) on x<A+d or Ax if the option Direction_Opt is given or on the whole C otherwise. The conditions are linear relations of f(x) and, perhaps, several derivatives of f(x) at the points into A<=Rex<A+d.

Examples

withLREtools&colon;

L1:=x3E2&plus;x3E&plus;x&plus;2x&plus;5318x722

L1:=x3E2&plus;x3E&plus;x&plus;2x&plus;5318x722

(2)

AnalyticityConditionsL1&comma;E&comma;fx&comma;&apos;HalfInterval&apos;&equals;1

f1&equals;0&comma;f0&equals;0&comma;f118&equals;67160528474293017172f1718

(3)

AnalyticityConditionsL1&comma;E&comma;fx

f0&equals;0&comma;f1&equals;0&comma;f1918&equals;1077057743867711154496079692388f118

(4)

AnalyticityConditionsL1&comma;E&comma;fx&comma;&apos;HalfInterval&apos;&equals;1&comma;&apos;direction&apos;&equals;&apos;left&apos;

f0&equals;85f1&comma;f118&equals;67160528474293017172f1718

(5)

AnalyticityConditionsL1&comma;E&comma;fx&comma;&apos;HalfInterval&apos;&equals;1&comma;&apos;direction&apos;&equals;&apos;right&apos;

f0&equals;8095179487529374512824f1

(6)

L2:=25x2415x316x3x4E2&plus;38x2&plus;8&plus;6x4&plus;28x&plus;24x3E3x47x29x3

L2:=3x415x325x216x4E2&plus;6x4&plus;24x3&plus;38x2&plus;28x&plus;8E3x47x29x3

(7)

cond:=AnalyticityConditionsL2&comma;E&comma;fx&comma;&apos;HalfInterval&apos;&equals;1

cond:=2&DifferentialD;&DifferentialD;xfxx&equals;1|&DifferentialD;&DifferentialD;xfxx&equals;1&DifferentialD;&DifferentialD;xfxx&equals;2|&DifferentialD;&DifferentialD;xfxx&equals;2f1&equals;0&comma;4&DifferentialD;&DifferentialD;xfxx&equals;1|&DifferentialD;&DifferentialD;xfxx&equals;12&DifferentialD;&DifferentialD;xfxx&equals;2|&DifferentialD;&DifferentialD;xfxx&equals;2f2&equals;0

(8)

solution f(x) = x is analytic everywhere on C:

f:=x&rarr;x&colon;

mapevalb&comma;cond&lsqb;&rsqb;

true

(9)

solution f(x) = x->1/x^2 is not analytic anywhere on C:

f:=x&rarr;1x2&colon;

mapevalb&comma;cond&lsqb;&rsqb;

false

(10)

unassignf

L3:=x2E23x3E&plus;x&plus;35&colon;

AnalyticityConditionsL3&comma;E&comma;fx&comma;&apos;HalfInterval&apos;&equals;2

&DifferentialD;&DifferentialD;xfxx&equals;2|&DifferentialD;&DifferentialD;xfxx&equals;2&equals;0&comma;&DifferentialD;&DifferentialD;xfxx&equals;1|&DifferentialD;&DifferentialD;xfxx&equals;1&equals;0&comma;34&DifferentialD;2&DifferentialD;x2fxx&equals;1|&DifferentialD;2&DifferentialD;x2fxx&equals;1&DifferentialD;2&DifferentialD;x2fxx&equals;2|&DifferentialD;2&DifferentialD;x2fxx&equals;2&equals;0&comma;54&DifferentialD;2&DifferentialD;x2fxx&equals;1|&DifferentialD;2&DifferentialD;x2fxx&equals;143&DifferentialD;3&DifferentialD;x3fxx&equals;2|&DifferentialD;3&DifferentialD;x3fxx&equals;2&DifferentialD;3&DifferentialD;x3fxx&equals;1|&DifferentialD;3&DifferentialD;x3fxx&equals;1&equals;0&comma;2&DifferentialD;2&DifferentialD;x2fxx&equals;1|&DifferentialD;2&DifferentialD;x2fxx&equals;1209&DifferentialD;3&DifferentialD;x3fxx&equals;2|&DifferentialD;3&DifferentialD;x3fxx&equals;243&DifferentialD;4&DifferentialD;x4fxx&equals;2|&DifferentialD;4&DifferentialD;x4fxx&equals;2&DifferentialD;4&DifferentialD;x4fxx&equals;1|&DifferentialD;4&DifferentialD;x4fxx&equals;1&equals;0&comma;f2&equals;0&comma;f1&equals;0

(11)

L4:=x3E2&plus;x3E&plus;x27

L4:=x3E2&plus;x3E&plus;x27

(12)

AnalyticityConditionsL4&comma;E&comma;fx&comma;&apos;HalfInterval&apos;&equals;4

13781597f7&plus;2147f7&plus;3&plus;479f7&plus;2&plus;49f7&plus;314&plus;7&equals;0&comma;140321701504018619523901029928552906399674728f7&plus;77&plus;162838190813653966125567194983f7&plus;8749262629772548413762714911006611f7&plus;7430829356836610952477937688279f7&plus;8121076534809199145799&plus;457626286729423113917&equals;0

(13)

L5:=2x2&plus;2x3E23x&plus;7x3E&plus;x&plus;3x&plus;1

L5:=2x2&plus;2x3E23x&plus;7x3E&plus;x&plus;3x&plus;1

(14)

AnalyticityConditionsL5&comma;E&comma;fx&comma;&apos;HalfInterval&apos;&equals;3

1345889795933441200692I2f3&plus;I24324504I2f2&plus;I21462789f3&plus;I2&plus;26667371f2&plus;I270405I2223912&equals;0&comma;1345889795933441200692If3I224324504If2I22&plus;1462789f3I226667371f2I270405I2&plus;223912&equals;0&comma;f2&equals;0

(15)

See Also

LREtools, LREtools[IsDesingularizable], LREtools[ValuesAtPoint]

References

  

Abramov, S.A., and van Hoeij, M. "Set of Poles of Solutions of Linear Difference Equations with Polynomial Coefficients." Computation Mathematics and Mathematical Physics. Vol. 43 No. 1. (2003): 57-62.


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