1\chapter{ZTRANS: $Z$-transform package}
2\label{ZTRANS}
3\typeout{{ZTRANS: $Z$-transform package}}
4
5{\footnotesize
6\begin{center}
7Wolfram Koepf and Lisa Temme \\
8Konrad--Zuse--Zentrum f\"ur Informationstechnik Berlin \\
9Takustrass 7 \\
10D--14195 Berlin--Dahlem, Germany \\[0.05in]
11e--mail: Koepf@zib.de
12\end{center}
13}
14\ttindex{ZTRANS}
15
16The $Z$-Transform of a sequence $\{f_n\}$ is the discrete analogue
17of the Laplace Transform, and
18\[{\cal Z}\{f_n\} = F(z) = \sum^\infty_{n=0} f_nz^{-n}\;.\] \\
19This series converges in the region outside the circle
20$|z|=|z_0|= \limsup\limits_{n \rightarrow \infty} \sqrt[n]{|f_n|}\;.$
21In the same way that a Laplace Transform can be used to
22solve differential equations, so $Z$-Transforms can be used
23to solve difference equations.
24
25\begin{tabbing}
26
27{\bf SYNTAX:}\ \ {\tt ztrans($f_n$, n,  z)}\ \ \ \ \ \ \ \
28  \=where $f_n$ is an expression, and $n$,$z$ \\
29  \> are identifiers.\\
30\end{tabbing}
31\ttindex{ztrans}
32
33\begin{tabbing}
34This pack\=age can compute the \= $Z$-Transforms of the \=following
35list of $f_n$, and \\ certain combinations thereof.\\ \\
36
37\>$1$
38\>$e^{\alpha n}$
39\>$\frac{1}{(n+k)}$               \\ \\
40\>$\frac{1}{n!}$
41\>$\frac{1}{(2n)!}$
42\>$\frac{1}{(2n+1)!}$             \\ \\
43\>$\frac{\sin(\beta n)}{n!}$
44\>$\sin(\alpha n+\phi)$
45\>$e^{\alpha n} \sin(\beta n)$    \\ \\
46\>$\frac{\cos(\beta n)}{n!}$
47\>$\cos(\alpha n+\phi)$
48\>$e^{\alpha n} \cos(\beta n)$    \\ \\
49\>$\frac{\sin(\beta (n+1))}{n+1}$
50\>$\sinh(\alpha n+\phi)$
51\>$\frac{\cos(\beta (n+1))}{n+1}$ \\ \\
52\>$\cosh(\alpha n+\phi)$
53\>${n+k \choose m}$\\
54\end{tabbing}
55
56\begin{tabbing}
57\underline {{\bf Other Combinations}}\= \\ \\
58
59\underline {Linearity}
60  \>${\cal Z} \{a f_n+b g_n \} = a{\cal Z} \{f_n\}+b{\cal Z}\{g_n\}$
61  \\ \\
62\underline {Multiplication by $n$}
63  \>${\cal Z} \{n^k \cdot f_n\} = -z \frac{d}{dz} \left({\cal Z}\{n^{k-1} \cdot f_n,n,z\} \right)$
64  \\ \\
65\underline {Multiplication by $\lambda^n$}
66  \>${\cal Z} \{\lambda^n \cdot f_n\}=F \left(\frac{z}{\lambda}\right)$
67  \\ \\
68\underline {Shift Equation}
69  \>${\cal Z} \{f_{n+k}\} =
70           z^k \left(F(z) - \sum\limits^{k-1}_{j=0} f_j z^{-j}\right)$
71  \\ \\
72\underline {Symbolic Sums}
73
74  \> ${\cal Z} \left\{ \sum\limits_{k=0}^{n} f_k \right\} =
75                       \frac{z}{z-1} \cdot {\cal Z} \{f_n\}$ \\ \\
76
77  \>${\cal Z} \left\{ \sum\limits_{k=p}^{n+q} f_k \right\}$
78  \ \ \ combination of the above \\ \\
79  where $k$,$\lambda \in$ {\bf N}$- \{0\}$; and $a$,$b$ are variables
80  or fractions; and $p$,$q \in$ {\bf Z} or \\
81  are functions of $n$; and $\alpha$, $\beta$ and $\phi$ are angles
82  in radians.
83\end{tabbing}
84
85
86The calculation of the Laurent coefficients of a regular function
87results in the following inverse formula for the $Z$-Transform:
88
89If $F(z)$ is a regular function in the region $|z|> \rho$ then
90$\exists$ a sequence \{$f_n$\} with ${\cal Z} \{f_n\}=F(z)$
91given by \[f_n = \frac{1}{2 \pi i}\oint F(z) z^{n-1} dz\]
92
93\begin{tabbing}
94
95{\bf SYNTAX:}\ \ {\tt invztrans($F(z)$, z,  n)}\ \ \ \ \ \ \ \
96  \=where $F(z)$ is an expression, \\
97  \> and $z$,$n$ are identifiers.
98\end{tabbing}
99\ttindex{invztrans}
100
101\begin{tabbing}
102  This \= package can compute the Inverse \= Z-Transforms of any
103  rational function, \\ whose denominator can be factored over
104  ${\bf Q}$, in addition to the following list \\ of $F(z)$.\\ \\
105
106\> $\sin \left(\frac{\sin (\beta)}{z} \ \right)
107    e^{\left(\frac{\cos (\beta)}{z} \ \right)}$
108\> $\cos \left(\frac{\sin (\beta)}{z} \ \right)
109    e^{\left(\frac{\cos (\beta)}{z} \ \right)}$ \\ \\
110\> $\sqrt{\frac{z}{A}} \sin \left( \sqrt{\frac{z}{A}} \ \right)$
111\> $\cos \left( \sqrt{\frac{z}{A}} \ \right)$ \\ \\
112\> $\sqrt{\frac{z}{A}} \sinh \left( \sqrt{\frac{z}{A}} \ \right)$
113\> $\cosh \left( \sqrt{\frac{z}{A}} \ \right)$ \\ \\
114\> $z \ \log \left(\frac{z}{\sqrt{z^2-A z+B}} \ \right)$
115\> $z \ \log \left(\frac{\sqrt{z^2+A z+B}}{z} \ \right)$ \\ \\
116\> $\arctan \left(\frac{\sin (\beta)}{z+\cos (\beta)} \ \right)$
117\\
118\end{tabbing}
119
120here $k$,$\lambda \in$ {\bf N}$ -  \{0\}$ and $A$,$B$ are fractions
121or variables ($B>0$) and $\alpha$,$\beta$, \&  $\phi$ are angles
122in radians.
123
124Examples:
125\begin{verbatim}
126ztrans(sum(1/factorial(k),k,0,n),n,z);
127
128  1/z
129 e   *z
130--------
131 z - 1
132
133invztrans(z/((z-a)*(z-b)),z,n);
134
135  n    n
136 a  - b
137---------
138  a - b
139\end{verbatim}
140
141