<A NAME=taylor>
<TITLE>taylor</TITLE></A>
<b><a href=r37_idx.html>INDEX</a></b><p><p>
<B>TAYLOR</B> _ _ _ _ _ _ _ _ _ _ _ _ <B>operator</B><P>
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The <em>taylor</em> operator is used for expanding an expression into a
Taylor series.
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syntax: </H3>
<em>taylor</em>(<expression>
<em>,</em><var><em>,</em>
<expression><em>,</em><number>
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{<em>,</em><var><em>,</em>
<expression><em>,</em><number>}*)
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<expression> can be any valid REDUCE algebraic expression.
<var> must be a
<A HREF=r37_0002.html>kernel</A>, and is the expansion
variable. The <expression> following it denotes the point
about which the expansion is to take place. <number> must be a
non-negative integer and denotes the maximum expansion order. If
more than one triple is specified <em>taylor</em> 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.
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Instead of a
<A HREF=r37_0002.html>kernel</A>, <var> may also be a list of
kernels. In this case expansion will take place in a way so that
the sum/ of the degrees of the kernels does not exceed the
maximum expansion order. If the expansion point evaluates to the
special identifier <em>infinity</em>, <em>taylor</em> tries to expand in
a series in 1/<var>.
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The expansion is performed variable per variable, i.e. in the
example above by first expanding
exp(x^2+y^2)
with respect to
<em>x</em> and then expanding every coefficient with respect to <em>y</em>.
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examples: </H3>
<P><PRE><TT>
taylor(e^(x^2+y^2),x,0,2,y,0,2);
2 2 2 2 2 2
1 + Y + X + Y *X + O(X ,Y )
taylor(e^(x^2+y^2),{x,y},0,2);
2 2 2 2
1 + Y + X + O({X ,Y })
</TT></PRE><P>The following example shows the case of a non-analytical function.
<P><PRE><TT>
taylor(x*y/(x+y),x,0,2,y,0,2);
***** Not a unit in argument to QUOTTAYLOR
</TT></PRE><P>
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Note that it is not generally possible to apply the standard
reduce operators to a Taylor kernel. For example,
<A HREF=r37_0169.html>part</A>,
<A HREF=r37_0141.html>coeff</A>, or
<A HREF=r37_0142.html>coeffn</A> cannot be used. Instead, the
expression at hand has to be converted to standard form first
using the
<A HREF=r37_0546.html>taylortostandard</A> operator.
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Differentiation of a Taylor expression is possible. If you
differentiate with respect to one of the Taylor variables the
order will decrease by one.
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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)).
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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
<A HREF=r37_0546.html>taylortostandard</A> operator. In this case a suitable
warning is printed.
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