Nested Derivatives: A simple method for computing series expansions of inverse functions

We give an algorithm to compute the series expansion for the inverse of a given function. The algorithm is extremely easy to implement and gives the first $N$ terms of the series. We show several examples of its application in calculating the inverses of some special functions.


Introduction
"One must always invert." Carl G. J. Jacobi The existence of series expansions for inverses of analytic functions is a well-known result of complex analysis [17]. The standard inverse function theorem, a proof of which can be found, for example, in [12], states that Theorem 1.1 Let h(x) be analytic for |x − x 0 | < R where h ′ (x 0 ) = 0. Then z = h(x) has an analytic inverse x = H(z) in some ε-neighborhood of z 0 = h(x 0 ).
where |·| ≡ det (·) . In Example 10, we show how to get the b n in term of the a n using our method.
A computer system like Maple can reverse the power series of h(x), provided h(x) is not too complicated, by using the command where N is the number of terms wanted. Fast algorithms of order (n log n) 3/2 for reversion of series have been analyzed by Brent and Kung [6], [5]. The multivariate case has been studied by several authors [4], [8], [14], [21] and Wright [35] has studied the connection between reversion of power series and "rooted trees".
The second and more direct method is Lagrange's inversion formula [1], (1.1) Unfortunately, more direct doesn't necessarily mean easier and, except for some simple cases, Lagrange's formula (1.1) is extremely complicated for practical applications. The q-analog (a mathematical expression parametrized by q which generalizes an expression and reduces to it in the limit q → 1 + ) of (1.1) has been studied by various authors [2], [18], [19], [20] and a unified approach to both the regular and q-analog formulas have been obtained by Krattenthaler [23]. There has also been a great deal of attention to the asymptotic expansion of inverses [27], [28], [31], [32].
In this note, we present a simple, easy to implement method for computing the series expansion for the inverse of any function satisfying the conditions of Theorem 1.1, although the method is especially powerful when h(x) has the form and g(x) is some function simpler than h(x). Since this is the case for many special functions, we will present several such examples. This note is organized as follows: In section 2 we define a sequence of functions D n [f ] (x), obtained from a given one f (x), that we call "nested derivatives", for reasons which will be clear from the definition. We give a computer code for generating the nested derivatives and examples of how D n [f ] (x) look for some elementary functions. Section 3 shows how to compute the nested derivatives by using generating functions. We present some examples and compare the results with those obtained in Section 1.
Section 4 contains our main result of the use of nested derivatives to compute power series of inverses. We test our result with some known results and we apply the method for obtaining expansions for the inverse of the error function, the incomplete Gamma function, the sine integral, and other special functions.

Definitions
Definition 2.1 We define D n [f ] (x), the n th nested derivative of the function f (x), by the following recursion: The nested derivative D n [f ] (x) satisfies the following basic properties.
(3) For n ≥ 1, D n [f ] (x) has the following integral representation: where C k is a small loop around x in the complex plane.
Proof. Properties (1) and (2) follow immediately from the definition of D n [f ] (x).
To prove (3) we use induction on n. For n = 1 the result follows from Cauchy's formula Assuming that the result is true for n and using (2.1), it follows that

Algorithm 2.3
The D algorithm. The following Maple procedure implements the recurrence The power function f (x) = x r , r = 1.

Generating functions
Generating functions provide a valuable method for computing sequences of functions defined by an iterative process; we will use them to calculate D n [f ] (x). In the sequel, we shall implicitly assume that the generating function series converges in some small disc around z = 0.
Hence, the generating function G(x, z) satisfies the PDE where g(z) is an arbitrary analytic function. Invoking the boundary condition G( and therefore and the theorem follows. Here h(x) = 1 x dx = ln(x), H(x) = e x , and from (3.1) it follows that We could obtain the same result from Example 2.4 by summing the series

Example 3.3
The power function f (x) = x r , r = 1.
Expanding in series around z = 0, we recover the result from Example 2.5.
Given the particular form of the function h(x) in Theorem 2, we can get alternative expressions for (3.1) which sometimes are easier to employ.
and therefore (ii) Using the chain rule and the conclusion follows from part (i).

Applications
We now state our main result.
Proof. Let's first observe that since h(a) = 0, it follows that H(0) = a and from (3.3) Hence, and (4.1) implies that Therefore, where B k are the Bernoulli numbers [1]. Comparing (4.2) and (4.3) we can see the highly nonlinear behavior of the nested derivatives, since even the addition of 1 to f (x) creates a completely different sequence of values, far more complex than the original.
We now start testing our result on some classical functions.
Example 4.9 The Error Function, erf(x). We now have and (2.2) gives
We will now extend (4.1) to a more general result.
Proof. We consider the function which satisfies u(b) = 0, and its inverse U(x) = u −1 (x). Since f (b) = 0, ±∞, we can apply (4.1) to u(x) and conclude that All that is left is to see the relation between U(z) and H(z). Suppose that u(x) = y. Then and therefore U(y) = H(y + z 0 ) or and (4.5) follows.
We will now use our results to get some power series expansions that have not been studied before.

Conclusion 4.15
We have presented a simple method for computing the series expansion for the inverses of functions and given a Maple procedure to generate the coefficients in these expansions. We showed several examples of the method applied to elementary and special functions, and stated the first few terms of the series in each case.