mtaylor - Maple Programming Help

mtaylor

multivariate Taylor series expansion

 Calling Sequence mtaylor(f, v) mtaylor(f, v, n) mtaylor(f, v, n, w)

Parameters

 f - algebraic expression v - list or set of names or equations n - (optional) non-negative integer w - (optional) list of positive integers

Description

 • The mtaylor function computes a truncated multivariate Taylor series expansion of the input expression f, with respect to the variables v, to order n, using the variable weights w.
 • The variables v can be a list or set of names or equations. If ${v}_{i}$ is an equation, then the left-hand side of ${v}_{i}$ is the variable, and the right-hand side is the point of expansion.  If ${v}_{i}$ is a name, then ${v}_{i}=0$ is assumed as the point of expansion.
 • If the third argument n is present then it specifies the truncation order'' of the series. The concept of truncation order'' used is total degree'' in the variables. If n is not present, the truncation order used is the value of the global variable Order, which is 6 by default.
 • If the fourth argument w is present it specifies the variable weights to be used (by default all 1).  A weight of 2 will halve the order in the corresponding variable to which the series is computed.
 • Note:  mtaylor restricts its domain to pure'' Taylor series, those series with non-negative powers in the variables.

Examples

 > $\mathrm{mtaylor}\left({ⅇ}^{{x}^{2}+{y}^{2}},\left[x,y\right],8\right)$
 ${1}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}\frac{{1}}{{2}}{}{{x}}^{{4}}{+}{{y}}^{{2}}{}{{x}}^{{2}}{+}\frac{{1}}{{2}}{}{{y}}^{{4}}{+}\frac{{1}}{{6}}{}{{x}}^{{6}}{+}\frac{{1}}{{2}}{}{{y}}^{{2}}{}{{x}}^{{4}}{+}\frac{{1}}{{2}}{}{{y}}^{{4}}{}{{x}}^{{2}}{+}\frac{{1}}{{6}}{}{{y}}^{{6}}$ (1)
 > $\mathrm{mtaylor}\left(\sqrt{1+{x}^{2}+{y}^{2}},\left[x,y\right],8\right)$
 ${1}{+}\frac{{1}}{{2}}{}{{x}}^{{2}}{+}\frac{{1}}{{2}}{}{{y}}^{{2}}{-}\frac{{1}}{{8}}{}{{x}}^{{4}}{-}\frac{{1}}{{4}}{}{{y}}^{{2}}{}{{x}}^{{2}}{-}\frac{{1}}{{8}}{}{{y}}^{{4}}{+}\frac{{1}}{{16}}{}{{x}}^{{6}}{+}\frac{{3}}{{16}}{}{{y}}^{{2}}{}{{x}}^{{4}}{+}\frac{{3}}{{16}}{}{{y}}^{{4}}{}{{x}}^{{2}}{+}\frac{{1}}{{16}}{}{{y}}^{{6}}$ (2)
 > $\mathrm{mtaylor}\left(\mathrm{sin}\left({x}^{2}+{y}^{2}\right),\left[x,y\right]\right)$
 ${{x}}^{{2}}{+}{{y}}^{{2}}$ (3)
 > $\mathrm{mtaylor}\left(\mathrm{sin}\left({x}^{2}+{y}^{2}\right),\left[x,y\right],8\right)$
 ${{x}}^{{2}}{+}{{y}}^{{2}}{-}\frac{{1}}{{6}}{}{{x}}^{{6}}{-}\frac{{1}}{{2}}{}{{y}}^{{2}}{}{{x}}^{{4}}{-}\frac{{1}}{{2}}{}{{y}}^{{4}}{}{{x}}^{{2}}{-}\frac{{1}}{{6}}{}{{y}}^{{6}}$ (4)
 > $\mathrm{mtaylor}\left(\mathrm{sin}\left({x}^{2}+{y}^{2}\right),\left[x,y\right],8,\left[2,1\right]\right)$
 ${{y}}^{{2}}{+}{{x}}^{{2}}{-}\frac{{1}}{{6}}{}{{y}}^{{6}}$ (5)
 > $\mathrm{mtaylor}\left(\mathrm{sin}\left({x}^{2}+{y}^{2}\right),\left[x=1,y\right],3\right)$
 ${\mathrm{sin}}{}\left({1}\right){+}{2}{}{\mathrm{cos}}{}\left({1}\right){}\left({x}{-}{1}\right){-}\left({2}{}{\mathrm{sin}}{}\left({1}\right){-}{\mathrm{cos}}{}\left({1}\right)\right){}{\left({x}{-}{1}\right)}^{{2}}{+}{\mathrm{cos}}{}\left({1}\right){}{{y}}^{{2}}$ (6)
 > $\mathrm{mtaylor}\left(\mathrm{cos}\left({x}^{2}+{y}^{2}\right),\left[x=1,y=2\right],3\right)$
 ${\mathrm{cos}}{}\left({5}\right){-}{2}{}{\mathrm{sin}}{}\left({5}\right){}\left({x}{-}{1}\right){-}{4}{}{\mathrm{sin}}{}\left({5}\right){}\left({y}{-}{2}\right){-}\left({2}{}{\mathrm{cos}}{}\left({5}\right){+}{\mathrm{sin}}{}\left({5}\right)\right){}{\left({x}{-}{1}\right)}^{{2}}{-}{8}{}{\mathrm{cos}}{}\left({5}\right){}\left({y}{-}{2}\right){}\left({x}{-}{1}\right){-}\left({8}{}{\mathrm{cos}}{}\left({5}\right){+}{\mathrm{sin}}{}\left({5}\right)\right){}{\left({y}{-}{2}\right)}^{{2}}$ (7)
 > $\mathrm{evalf}\left(\right)$
 ${-}{9.305580560}{+}{1.917848549}{}{x}{+}{3.835697099}{}{y}{+}{0.3915999037}{}{\left({x}{-}{1.}\right)}^{{2}}{-}{2.269297484}{}\left({y}{-}{2.}\right){}\left({x}{-}{1.}\right){-}{1.310373209}{}{\left({y}{-}{2.}\right)}^{{2}}$ (8)
 > $\mathrm{mtaylor}\left(f\left(x,y\right),\left[x,y\right],3\right)$
 ${f}{}\left({0}{,}{0}\right){+}{{\mathrm{D}}}_{{1}}{}\left({f}\right){}\left({0}{,}{0}\right){}{x}{+}{{\mathrm{D}}}_{{2}}{}\left({f}\right){}\left({0}{,}{0}\right){}{y}{+}\frac{{1}}{{2}}{}{{\mathrm{D}}}_{{1}{,}{1}}{}\left({f}\right){}\left({0}{,}{0}\right){}{{x}}^{{2}}{+}{{\mathrm{D}}}_{{1}{,}{2}}{}\left({f}\right){}\left({0}{,}{0}\right){}{x}{}{y}{+}\frac{{1}}{{2}}{}{{\mathrm{D}}}_{{2}{,}{2}}{}\left({f}\right){}\left({0}{,}{0}\right){}{{y}}^{{2}}$ (9)