The function ztrans finds the Z transformation of f(n) with respect to z. Formally,

ztrans recognizes and specially handles a large class of expressions, and only resorts to using the definition to calculate the transformation if the given expression has an unknown form. If the Z transform of the given expression cannot be found in a closed form, then the left-hand side of the formal definition is returned, rather than the right-hand side.

The functions referred to in the literature as Delta and Step may be simulated in this function as charfcn[0](...) and Heaviside(...), respectively.

$\mathrm{ztrans}\left(f\left(n+1\right)\,n\,z\right)$

$z\mathrm{ztrans}\left(f\left(n\right)\,n\,z\right)-f\left(0\right)z$

$\mathrm{ztrans}\left(\sin \left(\frac{\mathrm{\pi }}{2}t\right)\,t\,z\right)$

$\frac{z}{z^2+1}$

$\mathrm{ztrans}\left(\frac{3^n}{n!}\,n\,w\right)$

${\ⅇ}^{\frac{3}{w}}$

$\mathrm{ztrans}\left(\mathrm{\beta }{5}^n\left(n^2+3n+1\right)\,n\,z\right)$

$\mathrm{\beta }\left(\frac{z\left(\frac{z}{5}+1\right)}{5{\left(\frac{z}{5}-1\right)}^3}+\frac{3z}{5{\left(\frac{z}{5}-1\right)}^2}+\frac{z}{5\left(\frac{z}{5}-1\right)}\right)$

$\mathrm{ztrans}\left(\mathrm{charfcn}\left[5\right]\left(t\right)\mathrm{\Psi }\left(t\right)\,t\,w\right)$

$\frac{\frac{25}{12}-\mathrm{\gamma }}{w^5}$

$\mathrm{ztrans}\left(n\mathrm{Heaviside}\left(n-3\right)\,n\,z\right)$

$\frac{3z-2}{z^2{\left(z-1\right)}^2}$

$\mathrm{ztrans}\left(\mathrm{invztrans}\left(f\left(z\right)\,z\,n\right)\,n\,z\right)$

$f\left(z\right)$