
<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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 <P> <H3> 
syntax: </H3>
<em>taylor</em>(&lt;expression&gt; 
 <em>,</em>&lt;var&gt;<em>,</em> 
 &lt;expression&gt;<em>,</em>&lt;number&gt; 
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{<em>,</em>&lt;var&gt;<em>,</em> 
 &lt;expression&gt;<em>,</em>&lt;number&gt;}*) 
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&lt;expression&gt; can be any valid REDUCE algebraic expression. 
 &lt;var&gt; must be a 
<A HREF=r37_0002.html>kernel</A>, and is the expansion 
 variable. The &lt;expression&gt; following it denotes the point 
 about which the expansion is to take place. &lt;number&gt; 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>, &lt;var&gt; 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/&lt;var&gt;. 
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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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 <P> <H3> 
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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