analyticity conditions for the solution of linear difference equation.

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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
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);

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)

cond := {-2*(eval(diff(f(x), x), {x = 2}))+4*(eval(diff(f(x), x), {x = 1}))-f(2) = 0, -(eval(diff(f(x), x), {x = 2}))+2*(eval(diff(f(x), x), {x = 1}))-f(1) = 0}

(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)

${(1170954982963801582545471524589187104768*f(-7^(1/2)+8)-442579383053441910909142480344512987136*f(-7^(1/2)+8)*7^(1/2)+133891344428376248521019410957347808346112*f(-7^(1/2)+7)-50606171437368160875061851410289106354176*f(-7^(1/2)+7)*7^(1/2))*(-17296647829885592257/156558164414945034240-(45762628672942311391/1095907150904615239680)*7^(1/2)) = 0, (49*f(7^(1/2)+3)-14*7^(1/2)*f(7^(1/2)+3)+479*f(7^(1/2)+2)+159*f(7^(1/2)+2)*7^(1/2))*(1/27-(1/378)*7^(1/2)) = 0}$

(13)

(14)

${((518698944*I)*f(-3-I*2^(1/2))*2^(1/2)-(1868185728*I)*f(-2-I*2^(1/2))*2^(1/2)+631924848*f(-3-I*2^(1/2))-11520304272*f(-2-I*2^(1/2)))*(-27989/1867804898040576-((70405/14942439184324608)*I)*2^(1/2)) = 0, (-631924848*f(-3+I*2^(1/2))-(1868185728*I)*f(-2+I*2^(1/2))*2^(1/2)+(518698944*I)*f(-3+I*2^(1/2))*2^(1/2)+11520304272*f(-2+I*2^(1/2)))*(27989/1867804898040576-((70405/14942439184324608)*I)*2^(1/2)) = 0, f(-2) = 0}$

(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.