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\chapter{ZTRANS: $Z$-transform package}
\label{ZTRANS}
\typeout{{ZTRANS: $Z$-transform package}}

{\footnotesize
\begin{center}
Wolfram Koepf and Lisa Temme \\
Konrad--Zuse--Zentrum f\"ur Informationstechnik Berlin \\
Takustra\"se 7 \\
D--14195 Berlin--Dahlem, Germany \\[0.05in]
e--mail: Koepf@zib.de
\end{center}
}
\ttindex{ZTRANS}

The $Z$-Transform of a sequence $\{f_n\}$ is the discrete analogue
of the Laplace Transform, and
\[{\cal Z}\{f_n\} = F(z) = \sum^\infty_{n=0} f_nz^{-n}\;.\] \\
This series converges in the region outside the circle 
$|z|=|z_0|= \limsup\limits_{n \rightarrow \infty} \sqrt[n]{|f_n|}\;.$
In the same way that a Laplace Transform can be used to
solve differential equations, so $Z$-Transforms can be used
to solve difference equations.

\begin{tabbing}

{\bf SYNTAX:}\ \ {\tt ztrans($f_n$, n,  z)}\ \ \ \ \ \ \ \
  \=where $f_n$ is an expression, and $n$,$z$ \\
  \> are identifiers.\\
\end{tabbing}
\ttindex{ztrans}

\begin{tabbing}
This pack\=age can compute the \= $Z$-Transforms of the \=following 
list of $f_n$, and \\ certain combinations thereof.\\ \\

\>$1$                             
\>$e^{\alpha n}$                  
\>$\frac{1}{(n+k)}$               \\ \\
\>$\frac{1}{n!}$                  
\>$\frac{1}{(2n)!}$               
\>$\frac{1}{(2n+1)!}$             \\ \\
\>$\frac{\sin(\beta n)}{n!}$      
\>$\sin(\alpha n+\phi)$           
\>$e^{\alpha n} \sin(\beta n)$    \\ \\
\>$\frac{\cos(\beta n)}{n!}$      
\>$\cos(\alpha n+\phi)$           
\>$e^{\alpha n} \cos(\beta n)$    \\ \\
\>$\frac{\sin(\beta (n+1))}{n+1}$ 
\>$\sinh(\alpha n+\phi)$          
\>$\frac{\cos(\beta (n+1))}{n+1}$ \\ \\
\>$\cosh(\alpha n+\phi)$          
\>${n+k \choose m}$\\
\end{tabbing}

\begin{tabbing}
\underline {{\bf Other Combinations}}\= \\ \\

\underline {Linearity}
  \>${\cal Z} \{a f_n+b g_n \} = a{\cal Z} \{f_n\}+b{\cal Z}\{g_n\}$
  \\ \\
\underline {Multiplication by $n$}
  \>${\cal Z} \{n^k \cdot f_n\} = -z \frac{d}{dz} \left({\cal Z}\{n^{k-1} \cdot f_n,n,z\} \right)$
  \\ \\
\underline {Multiplication by $\lambda^n$}
  \>${\cal Z} \{\lambda^n \cdot f_n\}=F \left(\frac{z}{\lambda}\right)$
  \\ \\
\underline {Shift Equation}
  \>${\cal Z} \{f_{n+k}\} = 
           z^k \left(F(z) - \sum\limits^{k-1}_{j=0} f_j z^{-j}\right)$
  \\ \\
\underline {Symbolic Sums}

  \> ${\cal Z} \left\{ \sum\limits_{k=0}^{n} f_k \right\} =
                       \frac{z}{z-1} \cdot {\cal Z} \{f_n\}$ \\ \\

  \>${\cal Z} \left\{ \sum\limits_{k=p}^{n+q} f_k \right\}$
  \ \ \ combination of the above \\ \\
  where $k$,$\lambda \in$ {\bf N}$- \{0\}$; and $a$,$b$ are variables
  or fractions; and $p$,$q \in$ {\bf Z} or \\ 
  are functions of $n$; and $\alpha$, $\beta$ and $\phi$ are angles 
  in radians.
\end{tabbing}


The calculation of the Laurent coefficients of a regular function
results in the following inverse formula for the $Z$-Transform:

If $F(z)$ is a regular function in the region $|z|> \rho$ then
$\exists$ a sequence \{$f_n$\} with ${\cal Z} \{f_n\}=F(z)$
given by \[f_n = \frac{1}{2 \pi i}\oint F(z) z^{n-1} dz\]

\begin{tabbing}

{\bf SYNTAX:}\ \ {\tt invztrans($F(z)$, z,  n)}\ \ \ \ \ \ \ \
  \=where $F(z)$ is an expression, \\
  \> and $z$,$n$ are identifiers.
\end{tabbing}
\ttindex{invztrans}

\begin{tabbing}
  This \= package can compute the Inverse \= Z-Transforms of any 
  rational function, \\ whose denominator can be factored over 
  ${\bf Q}$, in addition to the following list \\ of $F(z)$.\\ \\

\> $\sin \left(\frac{\sin (\beta)}{z} \ \right) 
    e^{\left(\frac{\cos (\beta)}{z} \ \right)}$
\> $\cos \left(\frac{\sin (\beta)}{z} \ \right) 
    e^{\left(\frac{\cos (\beta)}{z} \ \right)}$ \\ \\
\> $\sqrt{\frac{z}{A}} \sin \left( \sqrt{\frac{z}{A}} \ \right)$
\> $\cos \left( \sqrt{\frac{z}{A}} \ \right)$ \\ \\
\> $\sqrt{\frac{z}{A}} \sinh \left( \sqrt{\frac{z}{A}} \ \right)$
\> $\cosh \left( \sqrt{\frac{z}{A}} \ \right)$ \\ \\
\> $z \ \log \left(\frac{z}{\sqrt{z^2-A z+B}} \ \right)$
\> $z \ \log \left(\frac{\sqrt{z^2+A z+B}}{z} \ \right)$ \\ \\
\> $\arctan \left(\frac{\sin (\beta)}{z+\cos (\beta)} \ \right)$
\\
\end{tabbing}

here $k$,$\lambda \in$ {\bf N}$ -  \{0\}$ and $A$,$B$ are fractions
or variables ($B>0$) and $\alpha$,$\beta$, \&  $\phi$ are angles 
in radians.

Examples:
\begin{verbatim}
ztrans(sum(1/factorial(k),k,0,n),n,z);

  1/z
 e   *z
--------
 z - 1

invztrans(z/((z-a)*(z-b)),z,n);

  n    n
 a  - b
---------
  a - b
\end{verbatim}