eulermac - Maple Help

eulermac

Euler-Maclaurin summation

 Calling Sequence eulermac(expr, x)     or eulermac(expr, x = a..b) eulermac(expr, x, n)   or eulermac(expr, x = a..b, n)

Parameters

 expr - expression in x x - independent variable a, b - interval over which the approximation to the sum is computed n - (optional) integer (degree of summation)

Description

 • The forms eulermac(expr, x) and eulermac(expr, x, n) compute asymptotic approximations to sum(expr, x). If F(x) = eulermac(f(x), x), then $F\left(x+1\right)-F\left(x\right)$ is asymptotically equivalent to f(x). The order of the approximation is specified by n, and defaults to Order - 1.
 • The forms eulermac(expr, x=a..b) and eulermac(expr, x=a..b, n) compute nth degree Euler-Maclaurin summation formulas for expr (thus n terms of the expansion are given). If n is not specified, it is assumed to be Order - 1.

Examples

 > $\mathrm{eulermac}\left(\frac{1}{x},x\right)$
 ${\mathrm{ln}}{}\left({x}\right){-}\frac{{1}}{{2}{}{x}}{-}\frac{{1}}{{12}{}{{x}}^{{2}}}{+}\frac{{1}}{{120}{}{{x}}^{{4}}}{-}\frac{{1}}{{252}{}{{x}}^{{6}}}{+}{\mathrm{O}}{}\left(\frac{{1}}{{{x}}^{{8}}}\right)$ (1)
 > $\mathrm{eulermac}\left(\frac{1}{k},k=1..x\right)$
 ${{\int }}_{{1}}^{{x}}\frac{{1}}{{k}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{k}{+}{\mathrm{\gamma }}{+}\frac{{1}}{{2}{}{x}}{-}\frac{{1}}{{12}{}{{x}}^{{2}}}{+}\frac{{1}}{{120}{}{{x}}^{{4}}}{-}\frac{{1}}{{252}{}{{x}}^{{6}}}{+}{\mathrm{O}}{}\left(\frac{{1}}{{{x}}^{{8}}}\right)$ (2)
 > $\mathrm{eulermac}\left(\frac{1}{{x}^{2}},x\right)$
 ${-}\frac{{1}}{{x}}{-}\frac{{1}}{{2}{}{{x}}^{{2}}}{-}\frac{{1}}{{6}{}{{x}}^{{3}}}{+}\frac{{1}}{{30}{}{{x}}^{{5}}}{-}\frac{{1}}{{42}{}{{x}}^{{7}}}{+}{\mathrm{O}}{}\left(\frac{{1}}{{{x}}^{{9}}}\right)$ (3)
 > $\mathrm{eulermac}\left(\frac{1}{{k}^{2}},k=1..x,4\right)$
 ${{\int }}_{{1}}^{{x}}\frac{{1}}{{{k}}^{{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{k}{+}\frac{{{\mathrm{\pi }}}^{{2}}}{{6}}{-}{1}{+}\frac{{1}}{{2}{}{{x}}^{{2}}}{-}\frac{{1}}{{6}{}{{x}}^{{3}}}{+}\frac{{1}}{{30}{}{{x}}^{{5}}}{+}{\mathrm{O}}{}\left(\frac{{1}}{{{x}}^{{7}}}\right)$ (4)