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

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.

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

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{Setup}\left(\mathrm{anticommutativepre}=\mathrm{\theta }\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{anticommutativepre}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{anticommutativeprefix}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{anticommutativeprefix}=\left\{\mathrm{\theta }\right\}\right]$

$a\mathrm{\theta }\left[1\right]\mathrm{\theta }\left[2\right]+b$

$a{\mathrm{\theta }}_1{\mathrm{\theta }}_2+b$

$\mathrm{Gtaylor}\left(\,\mathrm{\theta }\left[1\right]\right)$

$a{\mathrm{\theta }}_1{\mathrm{\theta }}_2+b$

$\mathrm{Coefficients}\left(\,\mathrm{\theta }\left[1\right]\right)$

$b,a{\mathrm{\theta }}_2$

$\exp \left(x\mathrm{\theta }\right)$

${\ⅇ}^{x\mathrm{\theta }}$

$\mathrm{Gtaylor}\left(\,\mathrm{\theta }\right)$

$x\mathrm{\theta }+1$

$F\left(x\,y\,\mathrm{\theta }\left[1\right]\,\mathrm{\theta }\left[2\right]\right)$

$F\left(x\,y\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\right)$

$\mathrm{Gtaylor}\left(\,\mathrm{\theta }\left[1\right]\right)$

$F\left(x\,y\,0\,{\mathrm{\theta }}_2\right)+{\mathrm{D}}_3\left(F\right)\left(x\,y\,0\,{\mathrm{\theta }}_2\right){\mathrm{\theta }}_1$

$\mathrm{ToFieldComponents}\left(\right)$

$\mathrm{\_F1}\left(x\,y\right)-{\mathrm{\theta }}_1\mathrm{\_Q1}\left(x\,y\right)-{\mathrm{\theta }}_2\mathrm{\_Q2}\left(x\,y\right)+\mathrm{\_F2}\left(x\,y\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2$

$\mathrm{GrassmannParity}\left(\right)$

$0$

$\mathrm{GrassmannParity}\left(\right)$

$\mathrm{undefined}$

$\mathrm{GrassmannParity}\left(\right)$

$0$

$\mathrm{Gtaylor}\left(\,\mathrm{\theta }\left[1\right]\,0\right)$

$0$

$\mathrm{Gtaylor}\left(\,\mathrm{\theta }\left[1\right]\,0\right)$

$0$

$\mathrm{Gtaylor}\left(\,\mathrm{\theta }\left[1\right]\,1\right)$

$F\left(x\,y\,0\,{\mathrm{\theta }}_2\right)$

$\mathrm{Gtaylor}\left(\,\mathrm{\theta }\left[1\right]\,1\right)$

$\mathrm{\_F1}\left(x\,y\right)-{\mathrm{\theta }}_2\mathrm{\_Q2}\left(x\,y\right)$

$\mathrm{Gtaylor}\left(\,\mathrm{\theta }\left[1\right]\,2\right)$

$F\left(x\,y\,0\,{\mathrm{\theta }}_2\right)+{\mathrm{D}}_3\left(F\right)\left(x\,y\,0\,{\mathrm{\theta }}_2\right){\mathrm{\theta }}_1$

$\mathrm{Gtaylor}\left(\,\mathrm{\theta }\left[1\right]\,2\right)$

$\mathrm{\_F1}\left(x\,y\right)-{\mathrm{\theta }}_1\mathrm{\_Q1}\left(x\,y\right)-{\mathrm{\theta }}_2\mathrm{\_Q2}\left(x\,y\right)+\mathrm{\_F2}\left(x\,y\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2$

$\mathrm{Coefficients}\left(=\,\mathrm{\theta }\left[1\right]\,0\right)$

$F\left(x\,y\,0\,{\mathrm{\theta }}_2\right)=\mathrm{\_F1}\left(x\,y\right)-{\mathrm{\theta }}_2\mathrm{\_Q2}\left(x\,y\right)$

$\mathrm{Coefficients}\left(=\,\mathrm{\theta }\left[1\right]\,1\right)$

${\mathrm{D}}_3\left(F\right)\left(x\,y\,0\,{\mathrm{\theta }}_2\right)=-\mathrm{\_Q1}\left(x\,y\right)+\mathrm{\_F2}\left(x\,y\right){\mathrm{\theta }}_2$

$\mathrm{GrassmannParity}\left(\right)$

$0=1$

$\mathrm{Gtaylor}\left(\,\left[\mathrm{\theta }\left[1\right]\,\mathrm{\theta }\left[2\right]\right]\right)$

$F\left(x\,y\,0\,0\right)+{\mathrm{D}}_3\left(F\right)\left(x\,y\,0\,0\right){\mathrm{\theta }}_1+{\mathrm{D}}_4\left(F\right)\left(x\,y\,0\,0\right){\mathrm{\theta }}_2+{\mathrm{D}}_{3,4}\left(F\right)\left(x\,y\,0\,0\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2$