Asymptotic Expansions and Summable Series
8.1 Simple example of an asymptotic expansion.
Consider the function \(f(x) = \!\int_x^\infty \! t^{-1}e^{x-t} \, d t\), where \(x\) is real and positive, and the path of integration is the real axis.
By repeated integrations by parts, we obtain \[ f(x)=\frac{1}{x}-\frac{1!}{x^2}+\frac{2!}{x^3}- \cdots +\frac{(-1)^{n-1}(n-1)!}{x^n} + (-1)^n n! \! \int_x^\infty \! \frac{e^{x-t}}{t^{n+1}} dt. \]
In connexion with the function \(f(x)\), we therefore consider the expression \[ u_{n-1} = \frac{(-1)^{n-1}(n-1)!}{x^n}, \] and we shall write \[ \sum_{m=0}^n \frac{1}{x} - \frac{1!}{x^2} + \frac{2!}{x^3}- \cdots + \frac{(-1)^n n!}{x^{n+1}} = S_{n}(x) . \] Then we have \(\left|\, u_m \middle/ u_{m-1} \,\right| = mx^{-1} \rightarrow \infty \) as \(m \rightarrow \infty\). The series \(\sum u_m\) m is therefore divergent for all values of \(x\). In spite of this, however, the series can be used for the calculation of \(f(x)\); this can be seen in the following way.
Take any fixed value for the number \(n\), and calculate the value of \(S_n\). We have \[ f(x) - S_n (x) = (-1)^{n+1} (n + 1) ! \! \int_x^\infty \! \frac{e^{x-t}}{t^{n+2}} dt , \] and therefore, since \(e^{x-t} \leq 1\), \[ \left|\, f(x) - S_n (x) \,\right| = (n + 1) ! \! \int_x^\infty \! \frac{e^{x-t}}{t^{n+2}} dt < (n+1)! \! \int_x^\infty \! \frac{dt}{t^{n+2}} = \frac{n!}{x^{n+1}}. \] For values of \(x\) which are sufficiently large, the right-hand member of this equation is very small. Thus, if we take \(x \geq 2n\), we have \[ \left|\, f(x) - S_n (x) \,\right| < \frac{1}{2^{n+1} n^2} , \] which for large values of \(n\) is very small. It follows therefore that the value of the function \(f(x)\) can be calculated with great accuracy for large values of \(x\), by taking the sum of a suitable number of terms of the series \(\sum u_m\).
Taking even fairly small values of \(x\) and \(n\), \[ S_5 (10) = 0.09152,\,\text{ and } \, 0 < f(10) - S_5 (10) < 0.00012. \]
The series is on this account said to be an asymptotic expansion of the function \(f(x)\). The precise definition of an asymptotic expansion will now be given.
8.2 Definition of an asymptotic expansion.
A divergent series \[ A_0 + \frac{A_1}{x} + \frac{A_2}{x^2} + \cdots + \frac{A_n}{x^n} + \cdots , \]
in which the sum of the first \((n + 1)\) terms is \(S_n (z)\), is said to be an asymptotic expansion of a function \(f(z)\) for a given range of values of \(\arg z\), if the expression \(R_n (z) = z^n \!\left\{f(z) - S_n (z)\right\}\) satisfies the condition \[ \lim_{\left|\;\! z \;\!\right|\:\!\rightarrow\:\! \infty} R_n (z) = 0 \quad\text{(}\:\! n\,\text{ fixed)}, \] even though \[ \lim_{n \:\!\rightarrow\:\! \infty} R_n (z) = 0 \quad\text{(}\:\! z\,\text{ fixed)}. \]
When this is the case, we can make \[ \left|\, z^n \!\left\{f(z) - S_n (z)\right\} \,\right| < \epsilon, \] where \(\epsilon\) can be made arbitrarily small, by taking \(\left|\;\! z \;\!\right|\) sufficiently large.
We denote the fact that the series is the asymptotic expansion of \(f(z)\) by writing \[ f(z) \sim \sum_{n=0}^\infty A_n z^{-n} \]
The definition which has just been given is due to Poincaré.[1] Special asymptotic expansions had, however, been discovered and used in the eighteenth century by Stirling, Maclaurin and Euler. Asymptotic expansions are of great importance in the theory of Linear Differential Equations, and in Dynamical Astronomy; some applications will be given in subsequent chapters of the present work.
The example discussed in §8.l clearly satisfies the definition just given: for, when \(x\) is positive, \[ \left|\, x^n \!\left\{f(x) - S_n (x)\right\} \,\right| < n! \:\! x^{-1} \rightarrow 0 \, \text{ as }\, x \rightarrow \infty . \] For the sake of simplicity, in this chapter we shall for the most part consider asymptotic expansions only in connexion with real positive values of the argument. The theory for complex values of the argument may be discussed by an extension of the analysis.
8.21 Another example of an asymptotic expansion.
As a second example, consider the function \(f(x)\), represented by the series \[ f(x)= \sum_{k=1}^\infty \frac{c^k}{x+k} , \] where \(x > 0\) and \(0 < c < 1\).
The ratio of the \(k\)-th term of this series to the \((k - 1)\)th is less than \(c\), and consequently the series converges for all positive values of \(x\). We shall confine our attention to positive values of \(x\). We have, when \(x > k\), \[ \frac{1}{x+k}=\frac{1}{x} - \frac{k}{x^2} + \frac{k^2}{x^3} - \frac{k^3}{x^4} + \frac{k^4}{x^5} -\cdots . \] If, therefore, it were allowable to expand each fraction \(\dfrac{1}{x+k}\), in this way,[2] and to rearrange the series for \(f(x)\) in descending powers of \(x\), we should obtain the formal series \[ \frac{A_1}{x} + \frac{A_2}{x^2} + \cdots + \frac{A_n}{x^n} + \cdots , \] where \[ A_n = (-1)^{n-1} \sum_{k=1}^\infty k^{n-1} c^k . \] But this procedure is not legitimate, and in fact \(\sum_{n=1}^\infty A_n x^{-n}\) diverges.[3] We can, however, shew that it is an asymptotic expansion of \(f(x)\).
For let \[ S_n (x) = \frac{A_1}{x} + \frac{A_2}{x^2} + \cdots + \frac{A_{n+1}}{x^{n+1}} \] Then \[ \begin{align*} S_n (x) &= \sum_{k=1}^\infty \left( \frac{c^k}{x} - \frac{kc^k}{x^2} + \frac{k^2 c^k}{x^3} + \cdots + \frac{(-1)^n k^n c^k}{x^{n+1}} \right) \\ &= \sum_{k=1}^\infty \left\{ 1- \left( -\frac{k}{x} \right)^{n+1} \!\right\} \frac{c^k}{x+k}; \end{align*} \] so that \[ \left|\, f(x)-S_n (x) \,\right| = \left|\, \sum_{k=1}^\infty \left( -\frac{k}{x}\right)^{n+1}\! \frac{c^k}{x+k} \,\right| < x^{-n-2} \sum_{k=1}^\infty k^{n+1} c^k . \]
Now \(\sum\limits_{k=1}^\infty k^{n+1} c^k\) converges for any given value of \(n\) and is equal to \(C_n\), say; and hence \(\left|\, f(x)-S_n (x) \,\right| < C_n x^{-n-2} \).
Consequently \[ f(x) \sim \sum_{n=1}^\infty A_n x^{-n} . \]
Example. If \(f(x) = \displaystyle \!\int_x^\infty \! e^{x^2-t^2} dt\), where \(x\) is positive and the path of integration is the real axis, prove that \[ f(x) \sim \frac{1}{2x} - \frac{1}{2^2 x^3} + \frac{1\cdot 3}{2^3 x^5} - \frac{1\cdot 3 \cdot 5}{2^4 x^7} + \cdots . \] [In fact, it was shewn by Stokes in 1857 that \[ \int_0^\infty \! e^{x^2-t^2} dt \,\sim\, \pm \frac{1}{2}e^{x^2}\! \sqrt{\pi} - \left( \frac{1}{2x} - \frac{1}{2^2 x^3} + \frac{1\cdot 3}{2^3 x^5} - \frac{1\cdot 3 \cdot 5}{2^4 x^7} + \cdots \right); \] the upper or lower sign is to be taken according as \(-\frac{1}{2}\pi < \arg x < \frac{1}{2}\pi\) or \(\frac{1}{2}\pi < \arg x < \frac{3}{2}\pi\).]
8.3 Multiplication of asymptotic expansions.
We shall now shew that two asymptotic expansions, valid for a common range of values of \(\arg z\), can be multiplied together in the same way as ordinary series, the result being a new asymptotic expansion.
For let \[ f(z) \sim \sum_{m=0}^\infty A_m z^{-m}, \quad \phi(z) \sim \sum_{m=0}^\infty B_m z^{-m}, \] and let \(S_n (z)\) and \(T_n (z)\) be the sums of their first \((n + 1)\) terms; so that, \(n\) being fixed, \[ f(z) - S_n(z) = o(z^{-n}), \quad \phi(z) - T_n(z) = o(z^{-n}). \] Then, if \(C_m = A_0 B_m+A_1 B_{m-1}+\cdots +A_m B_0\), it is obvious that[4] \[ S_n (z)T_n (z) = \sum_{m=0}^n C_m z^{-m} + o(z^{-n}). \] But \[\begin{align*} f(z) \phi (z) &= \left\{ S_n (z) + o(z^{-n}) \right\} \! \left\{ T_n(z)+ o(z^{-n}) \right\} \\ &= S_n(z)T_n(z) + o(z^{-n}) \vphantom{\sum^n} \\ &= \sum_{m=0}^n C_m z^{-m} + o(z^{-n}). \end{align*}\]
This result being true for any fixed value of \(n\), we see that \[ f(z)\phi(z) \sim \sum_{m=0}^\infty C_m z^{-m}. \]
8.31 Integration of asymptotic expansions.
We shall now shew that it is permissible to integrate an asymptotic expansion term by term, the resulting series being the asymptotic expansion of the integral of the function represented by the original series.[5]
For let \(f(x) \sim \sum\limits_{m=2}^\infty A_m x^{-n}\), and let \(S_n (x) = \sum\limits_{m=2}^n A_m x^{-n}\). Then, given any positive number \(\epsilon\), we can find \(x_0\) such that \[ \left|\, f(x) - S_n (x) \,\right| < \epsilon \:\! \left|\, x \,\right|^{-n} \,\text{ when }\, x > x_0 , \] and therefore \[\begin{align*} \left| \int_x^\infty \! f(x) \, d x - \! \int_x^\infty \! S_n (x) \, d x \, \right| &\leq \! \int_x^\infty \! \left| \, f(x) - S_n (x) \,\right| \, d x\\ &< \frac{\epsilon}{(n-1)\:\! x^{n-1}} . \end{align*}\] But \[ \int_x^\infty \! S_n (x) \, d x = \frac{A_2}{x} + \frac{A_3}{2x^2} + \cdots + \frac{A_n}{(n-1)\:\! x^{n-1}}, \] and therefore \[ \int_x^\infty \! f(x) \, d x \sim \sum_{m=2}^\infty \frac{A_m}{(m-1)\:\! x^{m-1}}. \]
On the other hand, it is not in general permissible to differentiate an asymptotic expansion;[6] this may be seen by considering \(e^{-x} \sin (e^x)\).
8.32 Uniqueness of an asymptotic expansion.
A question naturally suggests itself, as to whether a given series can be the asymptotic expansion of several distinct functions. The answer to this is in the affirmative. To shew this, we first observe that there are functions \(L \:\!(x)\) which are represented asymptotically by a series all of whose terms are zero, i.e. functions such that \(\lim\limits_{x \rightarrow \infty} x^n L \:\! (x) = 0\) for every fixed value of \(n\). The function \(e^{-x}\) is such a function when \(x\) is positive. The asymptotic expansion of a function \(J (x)\)[7] is therefore also the asymptotic expansion of \[ J(x) + L \:\!(x). \]
On the other hand, a function cannot be represented by more than one distinct asymptotic expansion over the whole of a given range of values of \(z\); for, if \[ f(z) \sim \sum_{m=0}^\infty A_m z^{-m}, \quad f(z) \sim \sum_{m=0}^\infty B_m z^{-m}, \] then \[ \lim_{z \rightarrow \infty} z^n \left( A_0 + \frac{a_1}{z} + \cdots + \frac{A_n}{z^n}-B_0 - \frac{B_1}{z} - \cdots - \frac{B_n}{z^n} \right) = 0 , \] which can only be if \(A_0 = B_0;\; A_1 = B_1, \cdots \).
Important examples of asymptotic expansions will be discussed later, in connexion with the Gamma-function (Chapter xii) and Bessel functions (Chapter xvii).