lcoeffp - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

 orderp
 the order of a p-adic expansion of a rational function
 lcoeffp
 the leading coefficient of a p-adic expansion of a rational function

 Calling Sequence orderp(ex, p, x) lcoeffp(ex, p, x)

Parameters

 ex - rational function p - irreducible (or square-free) polynomial or 1/x (or infinity) x - independent variable

Description

 • The orderp command computes the order at p of the p-adic expansion of a rational function ex in x.
 • The lcoeffp command computes the leading coefficient  at p of the p-adic expansion of a rational function ex in x.

Examples

 > $\mathrm{with}\left(\mathrm{padic}\right):$
 > $\mathrm{expansion}\left(\frac{{x}^{3}+1}{{x}^{2}+3x+5},{x}^{2}+2,x\right)$
 ${\left({{\mathrm{p_adic}}{}\left({{x}}^{{2}}{+}{2}{,}{0}{,}\left[{-}\frac{{x}}{{3}}{-}\frac{{1}}{{3}}{,}\frac{{4}}{{9}}{,}{-}\frac{{4}}{{81}}{+}\frac{{4}{}{x}}{{81}}{,}\frac{{4}{}{x}}{{729}}{-}\frac{{16}}{{729}}{,}{-}\frac{{8}{}{x}}{{6561}}{-}\frac{{4}}{{6561}}{,}{-}\frac{{20}{}{x}}{{59049}}{+}\frac{{44}}{{59049}}\right]\right)}_{{3}}\right)}_{{1}}{+}\frac{{\left({{\mathrm{p_adic}}{}\left({{x}}^{{2}}{+}{2}{,}{0}{,}\left[{-}\frac{{x}}{{3}}{-}\frac{{1}}{{3}}{,}\frac{{4}}{{9}}{,}{-}\frac{{4}}{{81}}{+}\frac{{4}{}{x}}{{81}}{,}\frac{{4}{}{x}}{{729}}{-}\frac{{16}}{{729}}{,}{-}\frac{{8}{}{x}}{{6561}}{-}\frac{{4}}{{6561}}{,}{-}\frac{{20}{}{x}}{{59049}}{+}\frac{{44}}{{59049}}\right]\right)}_{{3}}\right)}_{{2}}}{\left({{x}}^{{2}}{+}{2}\right)}{+}\frac{{\left({{\mathrm{p_adic}}{}\left({{x}}^{{2}}{+}{2}{,}{0}{,}\left[{-}\frac{{x}}{{3}}{-}\frac{{1}}{{3}}{,}\frac{{4}}{{9}}{,}{-}\frac{{4}}{{81}}{+}\frac{{4}{}{x}}{{81}}{,}\frac{{4}{}{x}}{{729}}{-}\frac{{16}}{{729}}{,}{-}\frac{{8}{}{x}}{{6561}}{-}\frac{{4}}{{6561}}{,}{-}\frac{{20}{}{x}}{{59049}}{+}\frac{{44}}{{59049}}\right]\right)}_{{3}}\right)}_{{3}}}{{\left({{x}}^{{2}}{+}{2}\right)}^{{2}}}{+}\frac{{\left({{\mathrm{p_adic}}{}\left({{x}}^{{2}}{+}{2}{,}{0}{,}\left[{-}\frac{{x}}{{3}}{-}\frac{{1}}{{3}}{,}\frac{{4}}{{9}}{,}{-}\frac{{4}}{{81}}{+}\frac{{4}{}{x}}{{81}}{,}\frac{{4}{}{x}}{{729}}{-}\frac{{16}}{{729}}{,}{-}\frac{{8}{}{x}}{{6561}}{-}\frac{{4}}{{6561}}{,}{-}\frac{{20}{}{x}}{{59049}}{+}\frac{{44}}{{59049}}\right]\right)}_{{3}}\right)}_{{4}}}{{\left({{x}}^{{2}}{+}{2}\right)}^{{3}}}{+}\frac{{\left({{\mathrm{p_adic}}{}\left({{x}}^{{2}}{+}{2}{,}{0}{,}\left[{-}\frac{{x}}{{3}}{-}\frac{{1}}{{3}}{,}\frac{{4}}{{9}}{,}{-}\frac{{4}}{{81}}{+}\frac{{4}{}{x}}{{81}}{,}\frac{{4}{}{x}}{{729}}{-}\frac{{16}}{{729}}{,}{-}\frac{{8}{}{x}}{{6561}}{-}\frac{{4}}{{6561}}{,}{-}\frac{{20}{}{x}}{{59049}}{+}\frac{{44}}{{59049}}\right]\right)}_{{3}}\right)}_{{5}}}{{\left({{x}}^{{2}}{+}{2}\right)}^{{4}}}{+}\frac{{\left({{\mathrm{p_adic}}{}\left({{x}}^{{2}}{+}{2}{,}{0}{,}\left[{-}\frac{{x}}{{3}}{-}\frac{{1}}{{3}}{,}\frac{{4}}{{9}}{,}{-}\frac{{4}}{{81}}{+}\frac{{4}{}{x}}{{81}}{,}\frac{{4}{}{x}}{{729}}{-}\frac{{16}}{{729}}{,}{-}\frac{{8}{}{x}}{{6561}}{-}\frac{{4}}{{6561}}{,}{-}\frac{{20}{}{x}}{{59049}}{+}\frac{{44}}{{59049}}\right]\right)}_{{3}}\right)}_{{6}}}{{\left({{x}}^{{2}}{+}{2}\right)}^{{5}}}{+}{\mathrm{O}}{}\left({\left({{x}}^{{2}}{+}{2}\right)}^{{6}}\right)$ (1)
 > $\mathrm{orderp}\left(\frac{{x}^{3}+1}{{x}^{2}+3x+5},{x}^{2}+2,x\right)$
 ${0}$ (2)
 > $\mathrm{lcoeffp}\left(\frac{{x}^{3}+1}{{x}^{2}+3x+5},{x}^{2}+2,x\right)$
 ${-}\frac{{x}}{{3}}{-}\frac{{1}}{{3}}$ (3)
 > $\mathrm{expansion}\left(\frac{{x}^{3}+1}{{x}^{2}+3x+5},\frac{1}{x},x\right)$
 $\frac{{\left({{\mathrm{p_adic}}{}\left(\frac{{1}}{{x}}{,}{-1}{,}\left[{1}{,}{-3}{,}{4}{,}{4}{,}{-32}{,}{76}\right]\right)}_{{3}}\right)}_{{1}}}{\left(\frac{{1}}{{x}}\right)}{+}\frac{{\left({{\mathrm{p_adic}}{}\left(\frac{{1}}{{x}}{,}{-1}{,}\left[{1}{,}{-3}{,}{4}{,}{4}{,}{-32}{,}{76}\right]\right)}_{{3}}\right)}_{{2}}}{{\left(\frac{{1}}{{x}}\right)}^{{2}}}{+}\frac{{\left({{\mathrm{p_adic}}{}\left(\frac{{1}}{{x}}{,}{-1}{,}\left[{1}{,}{-3}{,}{4}{,}{4}{,}{-32}{,}{76}\right]\right)}_{{3}}\right)}_{{3}}}{{\left(\frac{{1}}{{x}}\right)}^{{3}}}{+}\frac{{\left({{\mathrm{p_adic}}{}\left(\frac{{1}}{{x}}{,}{-1}{,}\left[{1}{,}{-3}{,}{4}{,}{4}{,}{-32}{,}{76}\right]\right)}_{{3}}\right)}_{{4}}}{{\left(\frac{{1}}{{x}}\right)}^{{4}}}{+}\frac{{\left({{\mathrm{p_adic}}{}\left(\frac{{1}}{{x}}{,}{-1}{,}\left[{1}{,}{-3}{,}{4}{,}{4}{,}{-32}{,}{76}\right]\right)}_{{3}}\right)}_{{5}}}{{\left(\frac{{1}}{{x}}\right)}^{{5}}}{+}\frac{{\left({{\mathrm{p_adic}}{}\left(\frac{{1}}{{x}}{,}{-1}{,}\left[{1}{,}{-3}{,}{4}{,}{4}{,}{-32}{,}{76}\right]\right)}_{{3}}\right)}_{{6}}}{{\left(\frac{{1}}{{x}}\right)}^{{6}}}{+}{\mathrm{O}}{}\left({\left(\frac{{1}}{{x}}\right)}^{{5}}\right)$ (4)
 > $\mathrm{orderp}\left(\frac{{x}^{3}+1}{{x}^{2}+3x+5},\frac{1}{x},x\right)$
 ${-1}$ (5)
 > $\mathrm{lcoeffp}\left(\frac{{x}^{3}+1}{{x}^{2}+3x+5},\frac{1}{x},x\right)$
 ${1}$ (6)