Gtaylor - Maple Help

Physics[Gtaylor] - Taylor series expansion of expressions involving anticommutative variables

 Calling Sequence Gtaylor(f, x, N)

Parameters

 f - algebraic expression, or relation between them, or a set or list of them x - variable, can be a name (expansion point is assumed equal to zero), or an equation  x = a where x is a name or and a the expansion point as an algebraic expression, or a list of them N - optional, default to all, can be an integer, or a range of integers (n..m), or any of the keywords leading, trailing, all; indicates whether to compute all the coefficients or some or only one

Description

 • The Gtaylor command computes a truncated series expansion of f, with respect to the variable x, about the point a, up to order N, where f and x can involve anticommutative variables. Gtaylor is a generalization of taylor, that works with commutative and anticommutative variables (also called Grassmannian variables) in equal footing.
 • The truncated series around $z=a$ is computed according to the standard formula

$\sum _{k=1}^{N-1}\left(\frac{{\left(z-a\right)}^{k}}{k!}\right)\left(\genfrac{}{}{0}{}{\left(\frac{{\partial }^{k}}{\partial {z}^{k}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}f\right)}{\phantom{z=a}}|\genfrac{}{}{0}{}{\phantom{\left(\frac{{\partial }^{k}}{\partial {z}^{k}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}f\right)}}{z=a}\right)$

 • The first argument, f, can also be a relation between algebraic expressions, or a set or list of them, in which case Gtaylor maps itself over the elements of the relation, set or list. For example, if f is an equation, then Gtaylor(p, x) returns the equation obtained by computing Gtaylor(lhs(f), x) = Gtaylor(rhs(f), x), where lhs(f) and rhs(f) respectively represent the left and right hand sides of f.
 • The second argument, x, can be a name, in which case the expansion point is assumed to be equal to zero, or an equation z = a where z is a name and a the expansion point as an algebraic expression. Also,x can be a list of expansion variables or equations, in which case the series is computed recursively: starting with computing the series with respect to the first variable in x, then computing the series of this result with respect to the second variable in x and so on.
 • The third argument, N, is optional, and indicates the "truncation order" of the series expansion, that is, the expansion will be of degree N-1 at most.
 • Unlike series and taylor, Gtaylor return just the polynomial truncated to order N-1, with no O(..) trailing term.
 • Note that the result computed with Gtaylor does not preserve the parity in that, for instance, the expansion of an arbitrary commutative function of anticommutative variables - say $F\left(x,\mathrm{\theta }\right)$ with $\mathrm{\theta }$ anticommutative - is a truncated polynomial that contains both commutative and anticommutative terms. An exact expansion, similar to the one performed by Gtaylor, but that does preserve the parity of the object is obtained using the ToFieldComponents command.
 • Generally speaking, the computation of a series expansion with respect to an anticommutative variable has two terms, the coefficients of orders zero and one. When the expression being expanded is already a polynomial, to compute these coefficients you can also use the Coefficients command.

Examples

 > $\mathrm{with}\left(\mathrm{Physics}\right):$
 > $\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$
 $\left[{\mathrm{mathematicalnotation}}{=}{\mathrm{true}}\right]$ (1)

Set first theta as an identifier to work with type/anticommutative variables (see Setup)

 > $\mathrm{Setup}\left(\mathrm{anticommutativepre}=\mathrm{\theta }\right)$
 $\mathrm{* Partial match of \text{'}}{}\mathrm{anticommutativepre}{}\mathrm{\text{'} against keyword \text{'}}{}\mathrm{anticommutativeprefix}{}\text{'}$
 $\mathrm{_______________________________________________________}$
 $\left[{\mathrm{anticommutativeprefix}}{=}\left\{{\mathrm{\theta }}\right\}\right]$ (2)
 > $a\mathrm{\theta }\left[1\right]\mathrm{\theta }\left[2\right]+b$
 ${a}{}{{\mathrm{\theta }}}_{{1}}{}{{\mathrm{\theta }}}_{{2}}{+}{b}$ (3)

The taylor series with respect to ${\mathrm{\theta }}_{1}$ of a polynomial in ${\mathrm{\theta }}_{1}$ of degree 1 is the polynomial itself;

 > $\mathrm{Gtaylor}\left(,\mathrm{\theta }\left[1\right]\right)$
 ${a}{}{{\mathrm{\theta }}}_{{1}}{}{{\mathrm{\theta }}}_{{2}}{+}{b}$ (4)

The coefficients of ${\mathrm{\theta }}_{1}$ entering this series can be computed with Coefficients

 > $\mathrm{Coefficients}\left(,\mathrm{\theta }\left[1\right]\right)$
 ${b}{,}{a}{}{{\mathrm{\theta }}}_{{2}}$ (5)

The series expansion of a mathematical function

 > $\mathrm{exp}\left(x\mathrm{\theta }\right)$
 ${{ⅇ}}^{{x}{}{\mathrm{\theta }}}$ (6)
 > $\mathrm{Gtaylor}\left(,\mathrm{\theta }\right)$
 ${x}{}{\mathrm{\theta }}{+}{1}$ (7)

The expansion of an arbitrary function and a comparison with the exact expansion performed with ToFieldComponents

 > $F\left(x,y,\mathrm{\theta }\left[1\right],\mathrm{\theta }\left[2\right]\right)$
 ${F}{}\left({x}{,}{y}{,}{{\mathrm{\theta }}}_{{1}}{,}{{\mathrm{\theta }}}_{{2}}\right)$ (8)
 > $\mathrm{Gtaylor}\left(,\mathrm{\theta }\left[1\right]\right)$
 ${F}{}\left({x}{,}{y}{,}{0}{,}{{\mathrm{\theta }}}_{{2}}\right){+}{{\mathrm{D}}}_{{3}}{}\left({F}\right){}\left({x}{,}{y}{,}{0}{,}{{\mathrm{\theta }}}_{{2}}\right){}{{\mathrm{\theta }}}_{{1}}$ (9)
 > $\mathrm{ToFieldComponents}\left(\right)$
 ${\mathrm{_F1}}{}\left({x}{,}{y}\right){-}{{\mathrm{\theta }}}_{{1}}{}{\mathrm{_Q1}}{}\left({x}{,}{y}\right){-}{{\mathrm{\theta }}}_{{2}}{}{\mathrm{_Q2}}{}\left({x}{,}{y}\right){+}{\mathrm{_F2}}{}\left({x}{,}{y}\right){}{{\mathrm{\theta }}}_{{1}}{}{{\mathrm{\theta }}}_{{2}}$ (10)

Note that the expansion performed with Gtaylor does not preserve the parity while the one performed with ToFieldComponents does:

 > $\mathrm{GrassmannParity}\left(\right)$
 ${0}$ (11)
 > $\mathrm{GrassmannParity}\left(\right)$
 ${\mathrm{undefined}}$ (12)
 > $\mathrm{GrassmannParity}\left(\right)$
 ${0}$ (13)

Taking (10) as an exact expansion of (9), compare now the Gtaylor expansion for different orders of these two equivalent expressions

 > $\mathrm{Gtaylor}\left(,\mathrm{\theta }\left[1\right],0\right)$
 ${0}$ (14)
 > $\mathrm{Gtaylor}\left(,\mathrm{\theta }\left[1\right],0\right)$
 ${0}$ (15)
 > $\mathrm{Gtaylor}\left(,\mathrm{\theta }\left[1\right],1\right)$
 ${F}{}\left({x}{,}{y}{,}{0}{,}{{\mathrm{\theta }}}_{{2}}\right)$ (16)
 > $\mathrm{Gtaylor}\left(,\mathrm{\theta }\left[1\right],1\right)$
 ${\mathrm{_F1}}{}\left({x}{,}{y}\right){-}{{\mathrm{\theta }}}_{{2}}{}{\mathrm{_Q2}}{}\left({x}{,}{y}\right)$ (17)
 > $\mathrm{Gtaylor}\left(,\mathrm{\theta }\left[1\right],2\right)$
 ${F}{}\left({x}{,}{y}{,}{0}{,}{{\mathrm{\theta }}}_{{2}}\right){+}{{\mathrm{D}}}_{{3}}{}\left({F}\right){}\left({x}{,}{y}{,}{0}{,}{{\mathrm{\theta }}}_{{2}}\right){}{{\mathrm{\theta }}}_{{1}}$ (18)
 > $\mathrm{Gtaylor}\left(,\mathrm{\theta }\left[1\right],2\right)$
 ${\mathrm{_F1}}{}\left({x}{,}{y}\right){-}{{\mathrm{\theta }}}_{{1}}{}{\mathrm{_Q1}}{}\left({x}{,}{y}\right){-}{{\mathrm{\theta }}}_{{2}}{}{\mathrm{_Q2}}{}\left({x}{,}{y}\right){+}{\mathrm{_F2}}{}\left({x}{,}{y}\right){}{{\mathrm{\theta }}}_{{1}}{}{{\mathrm{\theta }}}_{{2}}$ (19)

Computing the Coefficients helps comparing more precisely

 > $\mathrm{Coefficients}\left(=,\mathrm{\theta }\left[1\right],0\right)$
 ${F}{}\left({x}{,}{y}{,}{0}{,}{{\mathrm{\theta }}}_{{2}}\right){=}{\mathrm{_F1}}{}\left({x}{,}{y}\right){-}{{\mathrm{\theta }}}_{{2}}{}{\mathrm{_Q2}}{}\left({x}{,}{y}\right)$ (20)
 > $\mathrm{Coefficients}\left(=,\mathrm{\theta }\left[1\right],1\right)$
 ${{\mathrm{D}}}_{{3}}{}\left({F}\right){}\left({x}{,}{y}{,}{0}{,}{{\mathrm{\theta }}}_{{2}}\right){=}{-}{\mathrm{_Q1}}{}\left({x}{,}{y}\right){+}{\mathrm{_F2}}{}\left({x}{,}{y}\right){}{{\mathrm{\theta }}}_{{2}}$ (21)

Recall however that in (21), the left-hand-side has parity zero (commutative algebraic expression) while the right-hand-side has parity 1 (anticommutative algebraic expression):

 > $\mathrm{GrassmannParity}\left(\right)$
 ${0}{=}{1}$ (22)

To obtain a full multivariable expansion you can pass the (multi) variables in a list. This gives the multivariable expansion of (8) with respect to ${\mathrm{\theta }}_{1},{\mathrm{\theta }}_{2}$:

 > $\mathrm{Gtaylor}\left(,\left[\mathrm{\theta }\left[1\right],\mathrm{\theta }\left[2\right]\right]\right)$
 ${F}{}\left({x}{,}{y}{,}{0}{,}{0}\right){+}{{\mathrm{D}}}_{{3}}{}\left({F}\right){}\left({x}{,}{y}{,}{0}{,}{0}\right){}{{\mathrm{\theta }}}_{{1}}{+}{{\mathrm{D}}}_{{4}}{}\left({F}\right){}\left({x}{,}{y}{,}{0}{,}{0}\right){}{{\mathrm{\theta }}}_{{2}}{+}{{\mathrm{D}}}_{{3}{,}{4}}{}\left({F}\right){}\left({x}{,}{y}{,}{0}{,}{0}\right){}{{\mathrm{\theta }}}_{{1}}{}{{\mathrm{\theta }}}_{{2}}$ (23)
 > 

Compatibility

 • The Physics[Gtaylor] command was introduced in Maple 16.