AnalyticityConditions - Maple Help
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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

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  or 'direction'='right', the conditions on .

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);

(1)
  

and a point A. The solution f(x) is analytic on some open set which contains a set . The procedure returns the set of conditions for the analyticity of f(x) on  or  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 .

Examples

(2)

(3)

(4)

(5)

(6)

(7)

(8)

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

(9)

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

(10)

(11)

(12)

(13)

(14)

(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]

 


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