TAYLORCOMBINE _ _ _ _ _ _ _ _ _ _ _ _ operator

This operator tries to combine all Taylor kernels found in its argument into one. Operations currently possible are:

_ _ _ Addition, subtraction, multiplication, and division.

_ _ _ Roots, exponentials, and logarithms.

_ _ _ Trigonometric and hyperbolic functions and their inverses.

examples:

    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

Application of unary operators like log and 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 taylorkeeporiginal is set to on and if all Taylor kernels in its argument have their original expressions kept taylorcombine will also combine these and store the result as the original expression of the resulting Taylor kernel. There is also the switch 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.