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\section{Taylor series}

\begin{Introduction}{TAYLOR}
This short note describes a package of REDUCE procedures that allow
Taylor expansion in one or more variables and efficient manipulation
of the resulting Taylor series. Capabilities include basic operations
(addition, subtraction, multiplication and division) and also
application of certain algebraic and transcendental functions. To a
certain extent, Laurent expansion can be performed as well.
\end{Introduction}

\begin{Operator}{taylor}
  The \name{taylor} operator is used for expanding an expression into a
  Taylor series.

  \begin{Syntax}
    \name{taylor}\(\meta{expression}
                   \name{,}\meta{var}\name{,}
                   \meta{expression}\name{,}\meta{number}\\
                   \{\name{,}\meta{var}\name{,}
                   \meta{expression}\name{,}\meta{number}\}\optional\)
  \end{Syntax}
  \meta{expression} can be any valid REDUCE algebraic expression.
  \meta{var} must be a \nameref{kernel}, and is the expansion
  variable.  The \meta{expression} following it denotes the point
  about which the expansion is to take place. \meta{number} must be a
  non-negative integer and denotes the maximum expansion order.  If
  more than one triple is specified \name{taylor} will expand its
  first argument independently with respect to all the variables.
  Note that once the expansion has been done it is not possible to
  calculate higher orders.

  Instead of a \nameref{kernel}, \meta{var} may also be a list of
  kernels.  In this case expansion will take place in a way so that
  the {\em sum\/} of the degrees of the kernels does not exceed the
  maximum expansion order.  If the expansion point evaluates to the
  special identifier \name{infinity}, \name{taylor} tries to expand in
  a series in 1/\meta{var}.

  The expansion is performed variable per variable, i.e.\ in the
  example above by first expanding 
  \IFTEX{$\exp(x^{2}+y^{2})$}{exp(x^2+y^2)}
  with respect to
  \name{x} and then expanding every coefficient with respect to \name{y}.

  \begin{Examples}
    taylor(e^(x^2+y^2),x,0,2,y,0,2);     &
                1 + Y^{2} + X^{2} + Y^{2}*X^{2} + O(X^{2},Y^{2})   \\
    taylor(e^(x^2+y^2),{x,y},0,2); & 1 + Y^{2} + X^{2} + O(\{X^{2},Y^{2}\})\\
    \explanation{The following example shows the case of a non-analytical function.}\\
    taylor(x*y/(x+y),x,0,2,y,0,2); & ***** Not a unit in argument to QUOTTAYLOR \\
  \end{Examples}

  \begin{Comments}
    Note that it is not generally possible to apply the standard
    reduce operators to a Taylor kernel. For example, \nameref{part},
    \nameref{coeff}, or \nameref{coeffn} cannot be used. Instead, the
    expression at hand has to be converted to standard form first
    using the \nameref{taylortostandard} operator.

    Differentiation of a Taylor expression is possible.  If you
    differentiate with respect to one of the Taylor variables the
    order will decrease by one.

    Substitution is a bit restricted: Taylor variables can only be
    replaced by other kernels.  There is one exception to this rule:
    you can always substitute a Taylor variable by an expression that
    evaluates to a constant.  Note that REDUCE will not always be able
    to determine that an expression is constant: an example is
    sin(acos(4)).

    Only simple taylor kernels can be integrated. More complicated
    expressions that contain Taylor kernels as parts of themselves are
    automatically converted into a standard representation by means of
    the \nameref{taylortostandard} operator. In this case a suitable
    warning is printed.

  \end{Comments}

\end{Operator}


\begin{Switch}{taylorautocombine}
  If you set \name{taylorautocombine} to \name{on}, REDUCE
  automatically combines Taylor expressions during the simplification
  process.  This is equivalent to applying \nameref{taylorcombine} to
  every expression that contains Taylor kernels.  Default is
  \name{on}.
\end{Switch}

\begin{Switch}{taylorautoexpand}
  \name{taylorautoexpand} makes Taylor expressions ``contagious'' in
  the sense that \nameref{taylorcombine} tries to Taylor expand all
  non-Taylor subexpressions and to combine the result with the rest.
  Default is \name{off}.
\end{Switch}

\begin{Operator}{taylorcombine}
  This operator tries to combine all Taylor kernels found in its
  argument into one.  Operations currently possible are:
  \begin{itemize}
    \item Addition, subtraction, multiplication, and division.
    \item Roots, exponentials, and logarithms.
    \item Trigonometric and hyperbolic functions and their inverses.
  \end{itemize}

  \begin{Examples}
    hugo := taylor(exp(x),x,0,2); & HUGO := 1 + X + \rfrac{1}{2}*X^{2} + O(X^{3})\\
    taylorcombine log hugo; & X + O(X^{3})\\
    taylorcombine(hugo + x); & (1 + X + \rfrac{1}{2}*X^{2} + O(X^{3})) + X\\
    on taylorautoexpand; \\
    taylorcombine(hugo + x); & 1 + 2*X + \rfrac{1}{2}*X^{2} + O(X^{3})
  \end{Examples}

  \begin{Comments}
    Application of unary operators like \name{log} and \name{atan}
    will nearly always succeed. For binary operations their arguments
    have to be Taylor kernels with the same template. This means that
    the expansion variable and the expansion point must match.
    Expansion order is not so important, different order usually means
    that one of them is truncated before doing the operation.

    If \nameref{taylorkeeporiginal} is set to \name{on} and if all
    Taylor kernels in its argument have their original expressions
    kept \name{taylorcombine} will also combine these and store the
    result as the original expression of the resulting Taylor kernel.
    There is also the switch \nameref{taylorautoexpand}.

    There are a few restrictions to avoid mathematically undefined
    expressions: it is not possible to take the logarithm of a Taylor
    kernel which has no terms (i.e. is zero), or to divide by such a
    beast.  There are some provisions made to detect singularities
    during expansion: poles that arise because the denominator has
    zeros at the expansion point are detected and properly treated,
    i.e.\ the Taylor kernel will start with a negative power.  (This
    is accomplished by expanding numerator and denominator separately
    and combining the results.)  Essential singularities of the known
    functions (see above) are handled correctly.
  \end{Comments}
\end{Operator}

\begin{Switch}{taylorkeeporiginal}
  \name{taylorkeeporiginal}, if set to \name{on}, forces the
  \nameref{taylor} and all Taylor kernel manipulation operators to
  keep the original expression, i.e.\ the expression that was Taylor
  expanded.  All operations performed on the Taylor kernels are also
  applied to this expression which can be recovered using the operator
  \nameref{taylororiginal}.  Default is \name{off}.
\end{Switch}

\begin{Operator}{taylororiginal}
  Recovers the original expression (the one that was expanded) from
  the Taylor kernel that is given as its argument.

  \begin{Syntax}
    \name{taylororiginal}\(\meta{expression}\) or
    \name{taylororiginal} \meta{simple_expression}
  \end{Syntax}

  \begin{Examples}
    hugo := taylor(exp(x),x,0,2); & HUGO := 1 + X + \rfrac{1}{2}*X^{2} + O(X^{3})\\
    taylororiginal hugo; &
       ***** Taylor kernel doesn't have an original part in TAYLORORIGINAL\\
    on taylorkeeporiginal; \\
    hugo := taylor(exp(x),x,0,2); & HUGO := 1 + X + \rfrac{1}{2}*X^{2} + O(X^{3})\\
    taylororiginal hugo; & E^{X}
  \end{Examples}

  \begin{Comments}
    An error is signalled if the argument is not a Taylor kernel or if
    the original expression was not kept, i.e.\ if
    \nameref{taylorkeeporiginal} was set \name{off} during expansion.
  \end{Comments}

\end{Operator}

\begin{Switch}{taylorprintorder}
  \name{taylorprintorder}, if set to \name{on}, causes the remainder
  to be printed in big-O notation.  Otherwise, three dots are printed.
  Default is \name{on}.
\end{Switch}

\begin{Variable}{taylorprintterms}
  Only a certain number of (non-zero) coefficients are printed. If
  there are more, an expression of the form \name{n terms} is printed
  to indicate how many non-zero terms have been suppressed.  The
  number of terms printed is given by the value of the shared
  algebraic variable \name{taylorprintterms}.  Allowed values are
  integers and the special identifier \name{all}. The latter setting
  specifies that all terms are to be printed. The default setting is
  5.

  \begin{Examples}
    taylor(e^(x^2+y^2),x,0,4,y,0,4); &
           1 + Y^{2} + \rfrac{1}{2}*Y^{4} + X^{2} + Y^{2}*X^{2} +
             (4 terms) + O(X^{5},Y^{5})\\
    taylorprintterms := all; & TAYLORPRINTTERMS := ALL \\
    taylor(e^(x^2+y^2),x,0,4,y,0,4); &
    \begin{multilineoutput}{}
1 + Y^{2} + \rfrac{1}{2}*Y^{4} + X^{2} + Y^{2}*X^{2} +%
 \rfrac{1}{2}*Y^{4}*X^{2} + \rfrac{1}{2}*X^{4} +%
 \rfrac{1}{2}*Y^{2}*X^{4}\\
 + \rfrac{1}{4}*Y^{4}*X^{4} + O(X^{5},Y^{5})
    \end{multilineoutput}
  \end{Examples}

\end{Variable}


\begin{Operator}{taylorrevert}
  \name{taylorrevert} allows reversion of a Taylor series of a
  function f, i.e., to compute the first terms of the expansion of the
  inverse of $f$ from the expansion of $f$.

  \begin{Syntax}
    \name{taylorrevert}\(\meta{expression}\name{,}
                         \meta{var}\name{,}\meta{var}\)
  \end{Syntax}

  The first argument must evaluate to a Taylor kernel with the second
  argument being one of its expansion variables. 

  \begin{Examples}
    taylor(u - u**2,u,0,5); &    U - U^{2} + O(U^{6}) \\
    taylorrevert (ws,u,x);  &    X + X^{2} + 2*X^{3} + 5*X^{4} + 14*X^{5} + O(X^{6})
  \end{Examples}

\end{Operator}

\begin{Operator}{taylorseriesp}
  This operator may be used to determine if its argument is a Taylor
  kernel.

  \begin{Syntax}
    \name{taylorseriesp}\(\meta{expression}\) or \name{taylorseriesp}
    \meta{simple_expression}
  \end{Syntax}

  \begin{Examples}
    hugo := taylor(exp(x),x,0,2); & HUGO := 1 + X + \rfrac{1}{2}*X^{2} + O(X^{3})\\
    if taylorseriesp hugo then OK;& OK \\
    if taylorseriesp(hugo + y) then OK else NO; & NO
  \end{Examples}

  \begin{Comments}
    Note that this operator is subject to the same restrictions as,
    e.g., \name{ordp} or \name{numberp}, i.e.\ it may only be used in
    boolean expressions in \name{if} or \name{let} statements.
  \end{Comments}

\end{Operator}


\begin{Operator}{taylortemplate}
  The template of a Taylor kernel, i.e.\ the list of all variables
  with respect to which expansion took place together with expansion
  point and order can be extracted using

  \begin{Syntax}
    \name{taylortemplate}\(\meta{expression}\) or
    \name{taylortemplate} \meta{simple_expression}
  \end{Syntax}

  This returns a list of lists with the three elements
  (VAR,VAR0,ORDER).  An error is signalled if the argument is not a
  Taylor kernel.

  \begin{Examples}
    hugo := taylor(exp(x),x,0,2); & HUGO := 1 + X + \rfrac{1}{2}*X^{2} + O(X^{3})\\
    taylortemplate hugo; & \{\{X,0,2\}\}
  \end{Examples}

\end{Operator}

\begin{Operator}{taylortostandard}
  This operator converts all Taylor kernels in its argument into
  standard form and resimplifies the result.

\begin{Syntax}
  \name{taylortostandard}\(\meta{expression}\) or
  \name{taylortostandard} \meta{simple_expression}
\end{Syntax}

  \begin{Examples}
    hugo := taylor(exp(x),x,0,2); & HUGO := 1 + X + \rfrac{1}{2}*X^{2} + O(X^{3})\\
    taylortostandard hugo; & \rfrac{X^{2} + 2*X + 2}{2}
  \end{Examples}

\end{Operator}

\endinput

\section{Warnings and error messages}
\index{errors ! TAYLOR package}
\begin{itemize}

\item \name{Branch point detected in ...}\\
    This occurs if you take a rational power of a Taylor kernel
    and raising the lowest order term of the kernel to this
    power yields a non analytical term (i.e.\ a fractional power).

\item \name{Cannot expand further... truncation done}\\
    You will get this warning if you try to expand a Taylor kernel to
    a higher order.

\item \name{Converting Taylor kernels to standard representation}\\
    This warning appears if you try to integrate an expression that
    contains Taylor kernels.

\item \name{Error during expansion (possible singularity)}\\
    The expression you are trying to expand caused an error.
    As far as I know this can only happen if it contains a function
    with a pole or an essential singularity at the expansion point.
    (But one can never be sure.)

\item \name{Essential singularity in ...}\\
    An essential singularity was detected while applying a
    special function to a Taylor kernel.
    This error occurs, for example, if you try to take
    the logarithm of a Taylor kernel that starts with a negative
    power in one of its variables, i.e.\ that has a pole
    at the expansion point.

\item \name{Expansion point lies on branch cut in ...}\\
    The only functions with branch cuts this package knows of are
    (natural) logarithm, inverse circular and hyperbolic tangent and
    cotangent.  The branch cut of the logarithm is assumed to lie on
    the negative real axis.  Those of the arc tangent and arc
    cotangent functions are chosen to be compatible with this: both
    have essential singularities at the points $\pm i$.  The branch
    cut of arc tangent is the straight line along the imaginary axis
    connecting $+1$ to $-1$ going through $\infty$ whereas that of arc
    cotangent goes through the origin.  Consequently, the branch cut
    of the inverse hyperbolic tangent resp.\ cotangent lies on the
    real axis and goes from $-1$ to $+1$, that of the latter across
    $0$, the other across $\infty$.
    
    The error message can currently only appear when you try to
    calculate the inverse tangent or cotangent of a Taylor
    kernel that starts with a negative degree.
    The case of a logarithm of a Taylor kernel whose constant term
    is a negative real number is not caught since it is
    difficult to detect this in general.

\item \name{Not a unity in ...}\\
    This will happen if you try to divide by or take the logarithm of 
    a Taylor series whose constant term vanishes.

\item \name{Not implemented yet (...)}\\
    Sorry, but I haven't had the time to implement this feature.
    Tell me if you really need it, maybe I have already an improved
    version of the package.

\item \name{Reversion of Taylor series not possible: ...}\\
\ttindex{TAYLORREVERT}
    You tried to call the \name{TAYLORREVERT} operator with
    inappropriate arguments. The second half of this error message
    tells you why this operation is not possible.

\item \name{Substitution of dependent variables ...}\\
    You tried to substitute a variable that is already present in the
    Taylor kernel or on which one of the Taylor variables depend.

\item \name{Taylor kernel doesn't have an original part}\\
\ttindex{TAYLORORIGINAL} \ttindex{TAYLORKEEPORIGINAL}
    The Taylor kernel upon which you try to use \name{TAYLORORIGINAL}
    was created with the switch \name{TAYLORKEEPORIGINAL}
    set to \name{OFF}
    and does therefore not keep the original expression.

\item \name{Wrong number of arguments to TAYLOR}\\
    You try to use the operator \name{TAYLOR} with a wrong number of
    arguments.

\item \name{Zero divisor in TAYLOREXPAND}\\
    A zero divisor was found while an expression was being expanded.
    This should not normally occur.

\item \name{Zero divisor in Taylor substitution}\\
    That's exactly what the message says.  As an example consider the
    case of a Taylor kernel containing the term \name{1/x} and you try
    to substitute \name{x| by \verb|0}.

\item \name{... invalid as kernel}\\
    You tried to expand with respect to an expression that is not a
    kernel.

\item \name{... invalid as order of Taylor expansion}\\
    The order parameter you gave to \name{TAYLOR} is not an integer.

\item \name{... invalid as Taylor kernel}\\
\ttindex{TAYLORORIGINAL} \ttindex{TAYLORTEMPLATE}
    You tried to apply \name{TAYLORORIGINAL| or \verb|TAYLORTEMPLATE}
    to an expression that is not a Taylor kernel.

\item \name{... invalid as Taylor variable}\\
    You tried to substitute a Taylor variable by an expression that is
    not a kernel.

\item \name{... invalid as value of TaylorPrintTerms}\\
\ttindex{TAYLORPRINTTERMS} 
    You have assigned an invalid value to \name{TAYLORPRINTTERMS}.
    Allowed values are: an integer or the special identifier
    \name{ALL}.

\item \name{TAYLOR PACKAGE (...): this can't happen ...}\\
    This message shows that an internal inconsistency was detected.
    This is not your fault, at least as long as you did not try to
    work with the internal data structures of \REDUCE. Send input
    and output to me, together with the version information that is
    printed out.

\end{itemize}

\section{Comparison to other packages}

At the moment there is only one \REDUCE{} package that I know of:
the truncated power series package by Alan Barnes and Julian Padget.
In my opinion there are two major differences:
\begin{itemize}
  \item The interface. They use the domain mechanism for their power
        series, I decided to invent a special kind of kernel. Both
        approaches have advantages and disadvantages: with domain
        modes, it is easier
        to do certain things automatically, e.g., conversions.
  \item The concept of a truncated series. Their idea is to remember
        the original expression and to compute more coefficients when
        more of them are needed. My approach is to truncate at a
        certain order and forget how the unexpanded expression
        looked like.  I think that their method is more widely
        usable, whereas mine is more efficient when you know in
        advance exactly how many terms you need.
\end{itemize}

\end{document}
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