Clearly, since we are representing function and by power series, the function must necessarily be holomorphic; that is, it is differentiable as a complex-valued function on an open subset of the complex plane. (It follows that must also be holomorphic.) Holomorphic functions include the elementary functions studied in calculus such as , , and .
For the method to work smoothly, it is best to assume that we want to invert at around the origin, and its value there is zero. There is no loss of generality, since if , we can apply series inversion to the function defined by . Then ; we obtain the power series for centred at .
Also, it must be true that , otherwise will not even be invertible around the origin.
We explain the method by an example, for . In the following, we will consistently use the notation to denote a holomorphic function whose power series expansion begins with . And similarly when is replaced with the variable .
First, we start with the well-known power series expansion for :
The number of explicit terms in the power series expansion determines the number of terms that we will be able to obtain in the power series expansion of . So in this case, we are going to seek an expansion of up to (but excluding) the term.
A simple rearrangement of (1) gives
Now we substitute equation (2) into itself. Of course, usually when we substitute an equation into itself we do not get anything, but here it works because we can perform simplications using the notation. So for instance, in the following, in second term on the right of equation (2), we put in equation (2) simplified to . Why we should choose this simplication will be clear in a moment:
In equation (5) we used the fact that the expansion for must begin with a term, i.e. . Also note that we are guaranteed that the and terms have non-zero coefficients, because . (Otherwise we would not be able to isolate in equation (2).)
Now equation (5) simplifies to
which is already an achievement, because we have identified exactly what the term must be.
To summarize the procedure in general, we start with the expansion
and rearrange it to,
So we know that , and we can substitute this into the term of equation (11). At the end we will get an equation of the form , and we can substitute this into the terms and of (11). And what ever results we will substitute back into the terms , , of equation (11). We can repeat this process until we have all the terms of that we need.
We probably should normalize the functions so that to make the computations easier.
- 1 Lars V. Ahlfors. Complex Analysis. McGraw-Hill, 1979.
|Date of creation||2013-03-22 15:39:09|
|Last modified on||2013-03-22 15:39:09|
|Last modified by||stevecheng (10074)|