LREtools
AnalyticityConditions
analyticity conditions for the solution of linear difference equation.
Calling Sequence
Parameters
Description
Examples
References
AnalyticityConditions(L, E, fun, HalfInt_opt, Direction_opt)
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 ℜx<A+d or 'direction'='right', the conditions on A≤ℜx.
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=1daixEi
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 ℜx<A+d or A≤ℜx 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.
withLREtools:
L1≔x−3E2+x3E+x+2x+5318x−722
AnalyticityConditionsL1,E,fx,HalfInterval=−1
f−1=0,f0=0,f118=−6716052847f−17184293017172
AnalyticityConditionsL1,E,fx
f0=0,f1=0,f1918=−1077057743867711f118154496079692388
AnalyticityConditionsL1,E,fx,HalfInterval=−1,direction=left
f0=−8f−15,f118=−6716052847f−17184293017172
AnalyticityConditionsL1,E,fx,HalfInterval=−1,direction=right
f0=−80951794875f−129374512824
L2≔−25x2−4−15x3−16x−3x4E2+38x2+8+6x4+28x+24x3E−3x4−7x2−9x3
L2≔−3x4−15x3−25x2−16x−4E2+6x4+24x3+38x2+28x+8E−3x4−7x2−9x3
cond≔AnalyticityConditionsL2,E,fx,HalfInterval=1
cond≔2ⅆⅆxfxx=1|ⅆⅆxfxx=1−ⅆⅆxfxx=2|ⅆⅆxfxx=2−f1=0,4ⅆⅆxfxx=1|ⅆⅆxfxx=1−2ⅆⅆxfxx=2|ⅆⅆxfxx=2−f2=0
solution f(x) = x is analytic everywhere on C:
f≔x↦x:
mapevalb,cond
true
solution f(x) = x->1/x^2 is not analytic anywhere on C:
f≔x↦1x2:
false
unassignf
L3≔x2E2−3x−3E+x+35:
AnalyticityConditionsL3,E,fx,HalfInterval=−2
−ⅆⅆxfxx=−2|ⅆⅆxfxx=−2=0,−ⅆⅆxfxx=−1|ⅆⅆxfxx=−1=0,−3ⅆ2ⅆx2fxx=−1|ⅆ2ⅆx2fxx=−14−ⅆ2ⅆx2fxx=−2|ⅆ2ⅆx2fxx=−2=0,5ⅆ2ⅆx2fxx=−1|ⅆ2ⅆx2fxx=−14−4ⅆ3ⅆx3fxx=−2|ⅆ3ⅆx3fxx=−23−ⅆ3ⅆx3fxx=−1|ⅆ3ⅆx3fxx=−1=0,2ⅆ2ⅆx2fxx=−1|ⅆ2ⅆx2fxx=−1−20ⅆ3ⅆx3fxx=−2|ⅆ3ⅆx3fxx=−29−4ⅆ4ⅆx4fxx=−2|ⅆ4ⅆx4fxx=−23−ⅆ4ⅆx4fxx=−1|ⅆ4ⅆx4fxx=−1=0,f−2=0,f−1=0
L4≔x−3E2+x3E+x2−7
AnalyticityConditionsL4,E,fx,HalfInterval=4
−2847570073663+10766829668417101688272435223861f−7+87+915038971234759964687f−7+7+8271571450251894539f−7+85976888153870054527741134080=0,5593+1747752474f7+2+39053f7+3−14497f7+3719835172=0
L5≔2x2+2x−3E2−3x+7x−3E+x+3x+1
AnalyticityConditionsL5,E,fx,HalfInterval=−3
−−300568+159517I227363716If−3+I22+797212393f−2+I2−52604455f−3+I21545648142946688=0,300568+159517I2−27363716If−3−I22+797212393f−2−I2−52604455f−3−I21545648142946688=0,f−2=0
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[IsDesingularizable]
LREtools[ValuesAtPoint]
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