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AnalyticityConditions

  

analyticity conditions for the solution of linear difference equation.

 

Calling Sequence

Parameters

Description

Examples

References

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);

LaixE+Ed+11+E

(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;

L1x3E2+x3E+x+2x+5318x722

L1x3E2+x3E+x+2x+5318x722

(2)

AnalyticityConditionsL1&comma;E&comma;fx&comma;HalfInterval=1

f−1=0&comma;f0=0&comma;f118=6716052847f17184293017172

(3)

AnalyticityConditionsL1&comma;E&comma;fx

f0=0&comma;f1=0&comma;f1918=1077057743867711f118154496079692388

(4)

AnalyticityConditionsL1&comma;E&comma;fx&comma;HalfInterval=1&comma;direction=left

f0=8f−15&comma;f118=6716052847f17184293017172

(5)

AnalyticityConditionsL1&comma;E&comma;fx&comma;HalfInterval=1&comma;direction=right

f0=80951794875f−129374512824

(6)

L225x2415x316x3x4E2+38x2+8+6x4+28x+24x3E3x47x29x3

L23x415x325x216x4E2+6x4+24x3+38x2+28x+8E3x47x29x3

(7)

condAnalyticityConditionsL2&comma;E&comma;fx&comma;HalfInterval=1

cond2&DifferentialD;&DifferentialD;xfxx=1|&DifferentialD;&DifferentialD;xfxx=1&DifferentialD;&DifferentialD;xfxx=2|&DifferentialD;&DifferentialD;xfxx=2f1=0&comma;4&DifferentialD;&DifferentialD;xfxx=1|&DifferentialD;&DifferentialD;xfxx=12&DifferentialD;&DifferentialD;xfxx=2|&DifferentialD;&DifferentialD;xfxx=2f2=0

(8)

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

fxx&colon;

mapevalb&comma;cond

true

(9)

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

fx1x2&colon;

mapevalb&comma;cond

false

(10)

unassignf

L3x2E23x3E+x+35&colon;

AnalyticityConditionsL3&comma;E&comma;fx&comma;HalfInterval=2

&DifferentialD;&DifferentialD;xfxx=−2|&DifferentialD;&DifferentialD;xfxx=−2=0&comma;&DifferentialD;&DifferentialD;xfxx=−1|&DifferentialD;&DifferentialD;xfxx=−1=0&comma;3&DifferentialD;2&DifferentialD;x2fxx=−1|&DifferentialD;2&DifferentialD;x2fxx=−14&DifferentialD;2&DifferentialD;x2fxx=−2|&DifferentialD;2&DifferentialD;x2fxx=−2=0&comma;5&DifferentialD;2&DifferentialD;x2fxx=−1|&DifferentialD;2&DifferentialD;x2fxx=−144&DifferentialD;3&DifferentialD;x3fxx=−2|&DifferentialD;3&DifferentialD;x3fxx=−23&DifferentialD;3&DifferentialD;x3fxx=−1|&DifferentialD;3&DifferentialD;x3fxx=−1=0&comma;2&DifferentialD;2&DifferentialD;x2fxx=−1|&DifferentialD;2&DifferentialD;x2fxx=−120&DifferentialD;3&DifferentialD;x3fxx=−2|&DifferentialD;3&DifferentialD;x3fxx=−294&DifferentialD;4&DifferentialD;x4fxx=−2|&DifferentialD;4&DifferentialD;x4fxx=−23&DifferentialD;4&DifferentialD;x4fxx=−1|&DifferentialD;4&DifferentialD;x4fxx=−1=0&comma;f−2=0&comma;f−1=0

(11)

L4x3E2+x3E+x27

L4x3E2+x3E+x27

(12)

AnalyticityConditionsL4&comma;E&comma;fx&comma;HalfInterval=4

2847570073663+10766829668417101688272435223861f7+87+915038971234759964687f7+7+8271571450251894539f7+85976888153870054527741134080=0&comma;5593+1747752474f7+2+39053f7+314497f7+3719835172=0

(13)

L52x2+2x3E23x+7x3E+x+3x+1

L52x2+2x3E23x+7x3E+x+3x+1

(14)

AnalyticityConditionsL5&comma;E&comma;fx&comma;HalfInterval=3

18752+44001I227363716If2+I22+5349609f3+I252604455f2+I221896249215104=0&comma;300568+159517I227363716If3I22+797212393f2I252604455f3I21545648142946688=0&comma;f−2=0

(15)

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.

See Also

LREtools

LREtools[IsDesingularizable]

LREtools[ValuesAtPoint]