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 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 $L\left(f\left(x\right)\right)=0$ 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 $\Re \left(x\right) or 'direction'='right', the conditions on $A\le \Re \left(x\right)$.

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);
 ${L}{:=}{\sum }_{{i}{=}{1}}^{{d}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{{a}}_{{i}}{}\left({x}\right){}{{E}}^{{i}}$ (1)
 and a point A. The solution f(x) is analytic on some open set which contains a set $A<=\mathrm{Re}\left(x\right). The procedure returns the set of conditions for the analyticity of f(x) on $\Re \left(x\right) or $A\le \Re \left(x\right)$ 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<=\mathrm{Re}\left(x\right).

Examples

 > $\mathrm{with}\left(\mathrm{LREtools}\right):$
 > $\mathrm{L1}≔\left(x-3\right){E}^{2}+{x}^{3}E+\left(x+2\right)\left(x+\frac{53}{18}\right){\left(x-\frac{7}{2}\right)}^{2}$
 ${\mathrm{L1}}{:=}\left({x}{-}{3}\right){}{{E}}^{{2}}{+}{{x}}^{{3}}{}{E}{+}\left({x}{+}{2}\right){}\left({x}{+}\frac{{53}}{{18}}\right){}{\left({x}{-}\frac{{7}}{{2}}\right)}^{{2}}$ (2)
 > $\mathrm{AnalyticityConditions}\left(\mathrm{L1},E,f\left(x\right),'\mathrm{HalfInterval}'=-1\right)$
 $\left\{{f}{}\left({-}{1}\right){=}{0}{,}{f}{}\left({0}\right){=}{0}{,}{f}{}\left(\frac{{1}}{{18}}\right){=}{-}\frac{{6716052847}}{{4293017172}}{}{f}{}\left({-}\frac{{17}}{{18}}\right)\right\}$ (3)
 > $\mathrm{AnalyticityConditions}\left(\mathrm{L1},E,f\left(x\right)\right)$
 $\left\{{f}{}\left({0}\right){=}{0}{,}{f}{}\left({1}\right){=}{0}{,}{f}{}\left(\frac{{19}}{{18}}\right){=}{-}\frac{{1077057743867711}}{{154496079692388}}{}{f}{}\left(\frac{{1}}{{18}}\right)\right\}$ (4)
 > $\mathrm{AnalyticityConditions}\left(\mathrm{L1},E,f\left(x\right),'\mathrm{HalfInterval}'=-1,'\mathrm{direction}'='\mathrm{left}'\right)$
 $\left\{{f}{}\left({0}\right){=}{-}\frac{{8}}{{5}}{}{f}{}\left({-}{1}\right){,}{f}{}\left(\frac{{1}}{{18}}\right){=}{-}\frac{{6716052847}}{{4293017172}}{}{f}{}\left({-}\frac{{17}}{{18}}\right)\right\}$ (5)
 > $\mathrm{AnalyticityConditions}\left(\mathrm{L1},E,f\left(x\right),'\mathrm{HalfInterval}'=-1,'\mathrm{direction}'='\mathrm{right}'\right)$
 $\left\{{f}{}\left({0}\right){=}{-}\frac{{80951794875}}{{29374512824}}{}{f}{}\left({-}{1}\right)\right\}$ (6)
 > $\mathrm{L2}≔\left(-25{x}^{2}-4-15{x}^{3}-16x-3{x}^{4}\right){E}^{2}+\left(38{x}^{2}+8+6{x}^{4}+28x+24{x}^{3}\right)E-3{x}^{4}-7{x}^{2}-9{x}^{3}$
 ${\mathrm{L2}}{:=}\left({-}{3}{}{{x}}^{{4}}{-}{15}{}{{x}}^{{3}}{-}{25}{}{{x}}^{{2}}{-}{16}{}{x}{-}{4}\right){}{{E}}^{{2}}{+}\left({6}{}{{x}}^{{4}}{+}{24}{}{{x}}^{{3}}{+}{38}{}{{x}}^{{2}}{+}{28}{}{x}{+}{8}\right){}{E}{-}{3}{}{{x}}^{{4}}{-}{7}{}{{x}}^{{2}}{-}{9}{}{{x}}^{{3}}$ (7)
 > $\mathrm{cond}≔\mathrm{AnalyticityConditions}\left(\mathrm{L2},E,f\left(x\right),'\mathrm{HalfInterval}'=1\right)$
 ${\mathrm{cond}}{:=}\left\{{2}{}\left(\genfrac{}{}{0}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}{1}}\right){-}\left(\genfrac{}{}{0}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}{2}}\right){-}{f}{}\left({1}\right){=}{0}{,}{4}{}\left(\genfrac{}{}{0}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}{1}}\right){-}{2}{}\left(\genfrac{}{}{0}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}{2}}\right){-}{f}{}\left({2}\right){=}{0}\right\}$ (8)

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

 > $f≔x→x:$
 > ${\mathrm{map}\left(\mathrm{evalb},\mathrm{cond}\right)}_{[]}$
 ${\mathrm{true}}$ (9)

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

 > $f≔x→\frac{1}{{x}^{2}}:$
 > ${\mathrm{map}\left(\mathrm{evalb},\mathrm{cond}\right)}_{[]}$
 ${\mathrm{false}}$ (10)
 > $\mathrm{unassign}\left(f\right)$
 > $\mathrm{L3}≔{x}^{2}{E}^{2}-\left(3x-3\right)E+{\left(x+3\right)}^{5}:$
 > $\mathrm{AnalyticityConditions}\left(\mathrm{L3},E,f\left(x\right),'\mathrm{HalfInterval}'=-2\right)$
 $\left\{{-}\left(\genfrac{}{}{0}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{-}{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}{-}{2}}\right){=}{0}{,}{-}\left(\genfrac{}{}{0}{}{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{-}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)}}{{x}{=}{-}{1}}\right){=}{0}{,}{-}\frac{{3}}{{4}}{}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{-}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}}{{x}{=}{-}{1}}{-}\left(\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{-}{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}}{{x}{=}{-}{2}}\right){=}{0}{,}\frac{{5}}{{4}}{}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{-}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}}{{x}{=}{-}{1}}{-}\frac{{4}}{{3}}{}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{-}{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{f}{}\left({x}\right)}}{{x}{=}{-}{2}}{-}\left(\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{-}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{f}{}\left({x}\right)}}{{x}{=}{-}{1}}\right){=}{0}{,}{2}{}\left(\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{-}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)}}{{x}{=}{-}{1}}\right){-}\frac{{20}}{{9}}{}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{-}{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}{}{f}{}\left({x}\right)}}{{x}{=}{-}{2}}{-}\frac{{4}}{{3}}{}\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{-}{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}{}{f}{}\left({x}\right)}}{{x}{=}{-}{2}}{-}\left(\genfrac{}{}{0}{}{\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}{}{f}{}\left({x}\right)}{\phantom{{x}{=}{-}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{|}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}{}{f}{}\left({x}\right)}}{{x}{=}{-}{1}}\right){=}{0}{,}{f}{}\left({-}{2}\right){=}{0}{,}{f}{}\left({-}{1}\right){=}{0}\right\}$ (11)
 > $\mathrm{L4}≔\left(x-3\right){E}^{2}+{x}^{3}E+{x}^{2}-7$
 ${\mathrm{L4}}{:=}\left({x}{-}{3}\right){}{{E}}^{{2}}{+}{{x}}^{{3}}{}{E}{+}{{x}}^{{2}}{-}{7}$ (12)
 > $\mathrm{AnalyticityConditions}\left(\mathrm{L4},E,f\left(x\right),'\mathrm{HalfInterval}'=4\right)$
 $\left\{\frac{{1}}{{403217015040}}{}\left({18619523901029928552906399674728}{}\sqrt{{7}}{}{f}{}\left({-}\sqrt{{7}}{+}{7}\right){+}{162838190813653966125567194983}{}{f}{}\left({-}\sqrt{{7}}{+}{8}\right){}\sqrt{{7}}{-}{49262629772548413762714911006611}{}{f}{}\left({-}\sqrt{{7}}{+}{7}\right){-}{430829356836610952477937688279}{}{f}{}\left({-}\sqrt{{7}}{+}{8}\right)\right){}\left({121076534809199145799}{+}{45762628672942311391}{}\sqrt{{7}}\right){=}{0}{,}{-}\frac{{1}}{{378}}{}\left({159}{}\sqrt{{7}}{}{f}{}\left(\sqrt{{7}}{+}{2}\right){-}{14}{}\sqrt{{7}}{}{f}{}\left(\sqrt{{7}}{+}{3}\right){+}{479}{}{f}{}\left(\sqrt{{7}}{+}{2}\right){+}{49}{}{f}{}\left(\sqrt{{7}}{+}{3}\right)\right){}\left({-}{14}{+}\sqrt{{7}}\right){=}{0}\right\}$ (13)
 > $\mathrm{L5}≔2\left({x}^{2}+2\right)\left(x-3\right){E}^{2}-\left(3x+7\right)\left(x-3\right)E+\left(x+3\right)\left(x+1\right)$
 ${\mathrm{L5}}{:=}{2}{}\left({{x}}^{{2}}{+}{2}\right){}\left({x}{-}{3}\right){}{{E}}^{{2}}{-}\left({3}{}{x}{+}{7}\right){}\left({x}{-}{3}\right){}{E}{+}\left({x}{+}{3}\right){}\left({x}{+}{1}\right)$ (14)
 > $\mathrm{AnalyticityConditions}\left(\mathrm{L5},E,f\left(x\right),'\mathrm{HalfInterval}'=-3\right)$
 $\left\{{-}\frac{{1}}{{34588979593344}}{}\left({1200692}{}{I}{}\sqrt{{2}}{}{f}{}\left({-}{3}{-}{I}{}\sqrt{{2}}\right){-}{4324504}{}{I}{}\sqrt{{2}}{}{f}{}\left({-}{2}{-}{I}{}\sqrt{{2}}\right){+}{1462789}{}{f}{}\left({-}{3}{-}{I}{}\sqrt{{2}}\right){-}{26667371}{}{f}{}\left({-}{2}{-}{I}{}\sqrt{{2}}\right)\right){}\left({70405}{}{I}{}\sqrt{{2}}{+}{223912}\right){=}{0}{,}\frac{{1}}{{34588979593344}}{}\left({4324504}{}{I}{}\sqrt{{2}}{}{f}{}\left({-}{2}{+}{I}{}\sqrt{{2}}\right){-}{1200692}{}{I}{}\sqrt{{2}}{}{f}{}\left({-}{3}{+}{I}{}\sqrt{{2}}\right){-}{26667371}{}{f}{}\left({-}{2}{+}{I}{}\sqrt{{2}}\right){+}{1462789}{}{f}{}\left({-}{3}{+}{I}{}\sqrt{{2}}\right)\right){}\left({70405}{}{I}{}\sqrt{{2}}{-}{223912}\right){=}{0}{,}{f}{}\left({-}{2}\right){=}{0}\right\}$ (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.