This function computes the p-adic expansion of a rational function ex.

The parameter s sets the size of the resulting expression, where "size" means the number of terms of the p-adic expansion which will be printed.  If omitted, it defaults to number 6.

A p-adic expansion is represented in Maple using the unevaluated function call PADIC() whose argument is another unevaluated function of the form p_adic() which has three arguments. The first argument is the polynomial p or 1/x. The second argument is the p-adic order at p. The third argument is the list of coefficients. For example,

represents the p-adic expansion

The print routine print/PADIC is used by the prettyprinter to format the p-adic expansion on screen.

The command with(padic,expansion) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{padic}\right)\:$

$\mathrm{expansion}\left(\frac{x^3+1}{x^2+3x+5}\,x\,x\right)$

${\left({\mathrm{p\_adic}\left(x\,0\,\left[\frac{1}{5}\,-\frac{3}{25}\,\frac{4}{125}\,\frac{128}{625}\,-\frac{404}{3125}\,\frac{572}{15625}\right]\right)}_3\right)}_1+\frac{{\left({\mathrm{p\_adic}\left(x\,0\,\left[\frac{1}{5}\,-\frac{3}{25}\,\frac{4}{125}\,\frac{128}{625}\,-\frac{404}{3125}\,\frac{572}{15625}\right]\right)}_3\right)}_2}{x}+\frac{{\left({\mathrm{p\_adic}\left(x\,0\,\left[\frac{1}{5}\,-\frac{3}{25}\,\frac{4}{125}\,\frac{128}{625}\,-\frac{404}{3125}\,\frac{572}{15625}\right]\right)}_3\right)}_3}{x^2}+\frac{{\left({\mathrm{p\_adic}\left(x\,0\,\left[\frac{1}{5}\,-\frac{3}{25}\,\frac{4}{125}\,\frac{128}{625}\,-\frac{404}{3125}\,\frac{572}{15625}\right]\right)}_3\right)}_4}{x^3}+\frac{{\left({\mathrm{p\_adic}\left(x\,0\,\left[\frac{1}{5}\,-\frac{3}{25}\,\frac{4}{125}\,\frac{128}{625}\,-\frac{404}{3125}\,\frac{572}{15625}\right]\right)}_3\right)}_5}{x^4}+\frac{{\left({\mathrm{p\_adic}\left(x\,0\,\left[\frac{1}{5}\,-\frac{3}{25}\,\frac{4}{125}\,\frac{128}{625}\,-\frac{404}{3125}\,\frac{572}{15625}\right]\right)}_3\right)}_6}{x^5}+\mathrm{O}\left(x^6\right)$

$\mathrm{expansion}\left(\frac{x^3+1}{x^2+3x+5}\,x^2+2\,x\right)$

${\left({\mathrm{p\_adic}\left(x^2+2\,0\,\left[-\frac{x}{3}-\frac{1}{3}\,\frac{4}{9}\,-\frac{4}{81}+\frac{4x}{81}\,\frac{4x}{729}-\frac{16}{729}\,-\frac{8x}{6561}-\frac{4}{6561}\,-\frac{20x}{59049}+\frac{44}{59049}\right]\right)}_3\right)}_1+\frac{{\left({\mathrm{p\_adic}\left(x^2+2\,0\,\left[-\frac{x}{3}-\frac{1}{3}\,\frac{4}{9}\,-\frac{4}{81}+\frac{4x}{81}\,\frac{4x}{729}-\frac{16}{729}\,-\frac{8x}{6561}-\frac{4}{6561}\,-\frac{20x}{59049}+\frac{44}{59049}\right]\right)}_3\right)}_2}{\left(x^2+2\right)}+\frac{{\left({\mathrm{p\_adic}\left(x^2+2\,0\,\left[-\frac{x}{3}-\frac{1}{3}\,\frac{4}{9}\,-\frac{4}{81}+\frac{4x}{81}\,\frac{4x}{729}-\frac{16}{729}\,-\frac{8x}{6561}-\frac{4}{6561}\,-\frac{20x}{59049}+\frac{44}{59049}\right]\right)}_3\right)}_3}{{\left(x^2+2\right)}^2}+\frac{{\left({\mathrm{p\_adic}\left(x^2+2\,0\,\left[-\frac{x}{3}-\frac{1}{3}\,\frac{4}{9}\,-\frac{4}{81}+\frac{4x}{81}\,\frac{4x}{729}-\frac{16}{729}\,-\frac{8x}{6561}-\frac{4}{6561}\,-\frac{20x}{59049}+\frac{44}{59049}\right]\right)}_3\right)}_4}{{\left(x^2+2\right)}^3}+\frac{{\left({\mathrm{p\_adic}\left(x^2+2\,0\,\left[-\frac{x}{3}-\frac{1}{3}\,\frac{4}{9}\,-\frac{4}{81}+\frac{4x}{81}\,\frac{4x}{729}-\frac{16}{729}\,-\frac{8x}{6561}-\frac{4}{6561}\,-\frac{20x}{59049}+\frac{44}{59049}\right]\right)}_3\right)}_5}{{\left(x^2+2\right)}^4}+\frac{{\left({\mathrm{p\_adic}\left(x^2+2\,0\,\left[-\frac{x}{3}-\frac{1}{3}\,\frac{4}{9}\,-\frac{4}{81}+\frac{4x}{81}\,\frac{4x}{729}-\frac{16}{729}\,-\frac{8x}{6561}-\frac{4}{6561}\,-\frac{20x}{59049}+\frac{44}{59049}\right]\right)}_3\right)}_6}{{\left(x^2+2\right)}^5}+\mathrm{O}\left({\left(x^2+2\right)}^6\right)$

$\mathrm{expansion}\left(\frac{x^3+1}{x^2+3x+5}\,\frac{1}{x}\,x\,5\right)$

$\frac{{\left({\mathrm{p\_adic}\left(\frac{1}{x}\,\mathrm{-1}\,\left[1\,\mathrm{-3}\,4\,4\,\mathrm{-32}\right]\right)}_3\right)}_1}{\left(\frac{1}{x}\right)}+\frac{{\left({\mathrm{p\_adic}\left(\frac{1}{x}\,\mathrm{-1}\,\left[1\,\mathrm{-3}\,4\,4\,\mathrm{-32}\right]\right)}_3\right)}_2}{{\left(\frac{1}{x}\right)}^2}+\frac{{\left({\mathrm{p\_adic}\left(\frac{1}{x}\,\mathrm{-1}\,\left[1\,\mathrm{-3}\,4\,4\,\mathrm{-32}\right]\right)}_3\right)}_3}{{\left(\frac{1}{x}\right)}^3}+\frac{{\left({\mathrm{p\_adic}\left(\frac{1}{x}\,\mathrm{-1}\,\left[1\,\mathrm{-3}\,4\,4\,\mathrm{-32}\right]\right)}_3\right)}_4}{{\left(\frac{1}{x}\right)}^4}+\frac{{\left({\mathrm{p\_adic}\left(\frac{1}{x}\,\mathrm{-1}\,\left[1\,\mathrm{-3}\,4\,4\,\mathrm{-32}\right]\right)}_3\right)}_5}{{\left(\frac{1}{x}\right)}^5}+\mathrm{O}\left({\left(\frac{1}{x}\right)}^4\right)$