The poisson function generates a multivariate Taylor series expansion of the input expression f, with respect to the variables v, to order n, using the variable weights w. Trigonometric terms in the coefficients are combined.

The parameters and result are the same type as for the mtaylor function. The only difference is that sines and cosines in the coefficients are combined into the Fourier canonical form.

$f≔\sin \left(3w+x\right)\cos \left(2w-y\right)$

$f≔\sin \left(3w+x\right)\cos \left(2w-y\right)$

$\mathrm{poisson}\left(f\,\left[x\,y\right]\,3\right)$

$\frac{\sin \left(5w\right)}{2}+\frac{\sin \left(w\right)}{2}+\left(-\frac{\sin \left(5w\right)}{4}-\frac{\sin \left(w\right)}{4}\right)y^2+\left(-\frac{\sin \left(5w\right)}{4}-\frac{\sin \left(w\right)}{4}\right)x^2+\left(\frac{\sin \left(5w\right)}{2}-\frac{\sin \left(w\right)}{2}\right)yx+\left(\frac{\cos \left(w\right)}{2}+\frac{\cos \left(5w\right)}{2}\right)x+\left(\frac{\cos \left(w\right)}{2}-\frac{\cos \left(5w\right)}{2}\right)y$

$\mathrm{mtaylor}\left(f\,\left[x\,y\right]\,3\right)$

$\sin \left(3w\right)\cos \left(2w\right)+\sin \left(3w\right)\sin \left(2w\right)y+\cos \left(3w\right)x\cos \left(2w\right)-\frac{\sin \left(3w\right)\cos \left(2w\right)y^2}{2}+\cos \left(3w\right)x\sin \left(2w\right)y-\frac{x^2\sin \left(3w\right)\cos \left(2w\right)}{2}$

$\mathrm{poisson}\left(f\,\left[x\,y\right]\,3\,\left[2\,1\right]\right)$

$\frac{\sin \left(5w\right)}{2}+\frac{\sin \left(w\right)}{2}+\left(-\frac{\sin \left(5w\right)}{4}-\frac{\sin \left(w\right)}{4}\right)y^2+\left(\frac{\cos \left(w\right)}{2}+\frac{\cos \left(5w\right)}{2}\right)x+\left(\frac{\cos \left(w\right)}{2}-\frac{\cos \left(5w\right)}{2}\right)y$