powlog - Maple Help

powseries

 powlog
 logarithm of an expression

 Calling Sequence powlog(p)

Parameters

 p - either formal power series, or polynomial, or function that is acceptable for power series function evalpow

Description

 • The function powlog(p) returns the formal power series that is the natural logarithm of p.
 • The power series p must have a nonzero first term ($p\left(0\right)\ne 0$) for its logarithm to be well-defined.
 • The command with(powseries,powlog) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{powseries}\right):$
 > $t≔\mathrm{powpoly}\left(1+x,x\right):$
 > $s≔\mathrm{powlog}\left(t\right):$
 > $\mathrm{tpsform}\left(s,x,7\right)$
 ${x}{-}\frac{{1}}{{2}}{}{{x}}^{{2}}{+}\frac{{1}}{{3}}{}{{x}}^{{3}}{-}\frac{{1}}{{4}}{}{{x}}^{{4}}{+}\frac{{1}}{{5}}{}{{x}}^{{5}}{-}\frac{{1}}{{6}}{}{{x}}^{{6}}{+}{O}{}\left({{x}}^{{7}}\right)$ (1)
 > $u≔\mathrm{powlog}\left(\mathrm{powdiff}\left(\mathrm{powexp}\left(x\right)\right)\right):$
 > $\mathrm{tpsform}\left(u,x,6\right)$
 ${x}{+}{O}{}\left({{x}}^{{6}}\right)$ (2)
 > $v≔\mathrm{powlog}\left(\mathrm{powadd}\left(\mathrm{powpoly}\left(1+x,x\right),\mathrm{powexp}\left(x\right)\right)\right):$
 > $\mathrm{tpsform}\left(v,x,6\right)$
 ${\mathrm{ln}}{}\left({2}\right){+}{x}{-}\frac{{1}}{{4}}{}{{x}}^{{2}}{+}\frac{{1}}{{6}}{}{{x}}^{{3}}{-}\frac{{3}}{{32}}{}{{x}}^{{4}}{+}\frac{{7}}{{120}}{}{{x}}^{{5}}{+}{O}{}\left({{x}}^{{6}}\right)$ (3)