Rational Functions

The product of two formal power series

and

is the formal power series given by

where the coefficients satisfy

for

If the leading coefficient of equal 1, then the above formula gives

Thus if the formal power series

satisfies

then its coefficients can be computed with formula (), with the understanding for . For instance, if , , then





















We shall show that if a formal power series ``converges'' on an interval, then the resulting function has derivatives of all orders and its coefficients are given by, exactly like the polynomials,

More is true: if takes the form then for . This fact suggests a computational procedure to find for a rational function first, express in the form with and . Notice that cannot be zero, or else would not be defined. Once is factored out from , takes the form

Now the above method of expansion is applicable. Once the expansion

is found is calculated from the formula

Example: Compute for in the expansion

Exercise:

(a) Find the power series expansion of up to

Let . Find for

N-th Order Derivatives of Particular Rational Functions

The expansion

holds (the geometric series). Therefore, if then for all

Replacing by , we have the expansion

Therefore, if then for all . To find the expansion for functions of the form we write it in the form to obtain the expansion

Therefore, if then . By differentiating the geometric series on both sides, we obtain

Therefore, if then . Again, differentiating this series we obtain

Therefore, if then

Exercise:

Find out if . What if What if What if

If the rational function has the partial fractions decomposition in the form

the above method may be applied to find