<A NAME=taylorcombine>
<TITLE>taylorcombine</TITLE></A>
<b><a href=r37_idx.html>INDEX</a></b><p><p>
<B>TAYLORCOMBINE</B> _ _ _ _ _ _ _ _ _ _ _ _ <B>operator</B><P>
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This operator tries to combine all Taylor kernels found in its
argument into one. Operations currently possible are:
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_ _ _ Addition, subtraction, multiplication, and division.
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_ _ _ Roots, exponentials, and logarithms.
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_ _ _ Trigonometric and hyperbolic functions and their inverses.
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examples: </H3>
<P><PRE><TT>
hugo := taylor(exp(x),x,0,2);
1 2 3
HUGO := 1 + X + -*X + O(X )
2
taylorcombine log hugo;
3
X + O(X )
taylorcombine(hugo + x);
1 2 3
(1 + X + -*X + O(X )) + X
2
on taylorautoexpand;
taylorcombine(hugo + x);
1 2 3
1 + 2*X + -*X + O(X )
2
</TT></PRE><P>Application of unary operators like <em>log</em> and <em>atan</em>
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.
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If
<A HREF=r37_0539.html>taylorkeeporiginal</A> is set to <em>on</em> and if all
Taylor kernels in its argument have their original expressions
kept <em>taylorcombine</em> will also combine these and store the
result as the original expression of the resulting Taylor kernel.
There is also the switch
<A HREF=r37_0537.html>taylorautoexpand</A>.
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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.
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