1.1 Introduction and Notational Conventions

 

representing a Suppose that we are given a power series

function f(z), so that This expansion is the fundamental starting point of any analysis using Pade for the given set of coefficients, and /(2) is the associated function. A Padé approximant is a rational fraction  which has a Maclaurin expansion which agrees with (1.1) as far as possible. We give a more complete and precise definition of Pade approximants in Section 1.4. Notice that in (1.1) there are L+ 1 numerator coefficients and M+ 1 denominator coefficients. There is a more or less irrelevant common factor between them, and for definiteness we take b0 = 1. This choice turns out to be an essential part of the precise definition, and (1.2) is our conventional notation with this choice for b0. So there are L+ 1 independent numerator coefficients and M independent denominator coefficients, mak­ing L + A/+1 unknown coefficients in all. This number suggests that normally the [L/M] ought to fit the power series (1.1) through the orders approximants. Throughout this work we reserve the notation

In the notation of formal power series,

 

(1.3)

 Example.

 Returning to (1.3) and cross-multiplying, we find that

 (1.4)

 

Equating the coefficients of

we find

 (1.5)

 

If j<0, we define cy =0 for consistency. Since bQ = \, Equations (1.5) become a set of M linear equations for the M unknown denominator coefficients:

 (1.6)

 from which the bt may be found. The numerator coefficients,

follow immediately from (1.4) by equating the coefficients of 1,

(1.7)

Thus (1.6) and (1.7) normally determine the Pade numerator and denomina­tor and are called the Pade eauations: we have constructed an \L/M1 Pade through order

Because the starting point of these manipulations is the given power series, we do not

ever need to know about the existence of any function /(z) with

approximant which agrees with

 its Maclaurin series, as in (1.1). Of course, we expect that a well-chosen sequence of Pade approximants will normally approximate a function /(z) with the Maclaurin expansion but it is important to distinguish  between problems of convergence of Pade approximants and problems of construction of Pade approximants. Given the power series, (1.6) and (1.7) show how the Pade approximants are constructed.

 

the power series series converges, and if | it does not. If

Every power series has a circle of convergence  represents an analytic function (functions analytic everywhere we often call entire) and the series may be summed directly for any value of z to yield the

 

function /(z). If

 

the power series is undoubtedly formal. It contains

information about /(z), but just how this information is to be used is not immediately clear. However, if a sequence of Pade approximants of the

for

 

then we may

formal power series converges to a function

 

 

then a

 

with

 

where 6D is a

 

domain larger than

. We will then have extended our domain of

reasonably conclude that g(z) is a function with the given power series. In certain circumstances (see Chapter 5) we make such statements precise and prove them. Nevertheless, in this book we will not be hampered by a lack of rigorous justification of any technique, and empirical convergence is re­garded as entirely satisfactory within its limitations. If the given power

series converges to the same function for sequence of Pade approximants may converge for

convergence. This is frequently a practical approach to what amounts to analytic continuation. The method of expansion and reexpansion due to Weierstrass is more suited to principle than practice. As an example of how well Pade approximants may work in their natural context, we consider an example.

shows remarkable accuracy for a function with a radius of convergence of 5, using just three terms of the series.

 

 

In particular,

 

and in Figure 1 we compare this with

and

giving 8% accuracy at infinity. This example

There is one feature of the calculation of Pade approximations to be emphasized at the start—these calculations require more numerical accu­racy than one might at first expect. The Pade approximant exploits the differences of the coefficients to do its long-range extrapolation, and so the differences must all be accurate. We consider the problem of deciding how much numerical accuracy is needed to calculate an [L/M] Pade approxi­mant in Section 2.4.

 

 

(1.8)

 

We take (1.8) to define

and use this convention throughout.

Thus far, we have assumed that Pade approximants are calculated directly from (1.6) and (1.7) without implying any particular method. If Cramer's

 

from (1.6) and hence the

 

rule is used, we may calculate

denominator of (1.2). Aside from a common factor, the result is

 

Again, recall that c = 0 ify<0. Now consider

times the second

 

 

 

By subtracting

times the first row from the last,

 

row from the last, etc., up to

times the penultimate row from the last,

 

 we reduce the series in the last row. They become lacunary series, with a gap

of M terms missing. Using the initial terms of these series, we define

 

(1.9)

 

 

 

 

and that the

Again, (1.9) is our notational convention. We now prove our first theorem. Theorem 1.1.1. With the definitions (1.8) and (1.9),

(1.10)

 

Proof. We note that remainder is this end, consider

This is called a Hankel determinant, because of the systematic way in which its rows are formed from the given coefficients c(. Notice that if (9' //W'(0) t^O, then the linear equations (1.6) are nonsingular and the solution given by (1.8) is unambiguous. Furthermore, we may divide (1.10) yielding

 

 

This result has proved our second theorem:

Theorem 1.1.2 [Jacobi, 1846]. With the definitions (1.8) and (1.9), the

 

[L/M] Pade approximant oj

is given by

 

(1.12)

 

provided

 

 

when

We extend the notation [L/M] of (1.12) as [L/M] f to

 

to emphasize the

z-dependence. We will thus have the various forms

 

 

 

The only difficulties, which we defer to Section 1.4, are those occurring

emphasize approximation of /(z), and as

available for convenience. It is common practice to display the approxi- mants in a table, called the Pade table, shown as Table 1.

Tahlp 1. The PaHe tahleTable 2. Part of the Pade table of exp( z) [Pade, 1892],

 

Among other things, we prove in Section 1.2 that part of the Pade table of exp(z) is given by the entries in Table 2.