Asymptotic Series

(1148) |

where is a function whose behavior for large values of is known. If is singular as then the previous series clearly diverges. Nevertheless, under certain circumstances, the series may still be useful. In fact, this is the case if the difference between and the first terms is of order , so that for sufficiently large this difference becomes vanishingly small. More precisely, the series is said to represent asymptotically, that is

(1149) |

provided that

(1150) |

In other words, for a given value of , the sum of the first terms of the series may be made as close as desired to the ratio by making sufficiently large. For each value of and there is an error in the series representation of which is of order . Because the series actually diverges, there is an optimum number of terms in the series used to represent for a given value of . Associated with this is an unavoidable error. As increases, the optimal number of terms increases, and the error decreases.

Consider a simple example. The exponential integral is defined

(1151) |

The asymptotic series for this function can be generated via a series of partial integrations. For example,

(1152) |

A continuation of this procedure yields

(1153) |

The infinite series obtained by taking the limit diverges, because the Cauchy convergence test yields

(1154) |

Note that two successive terms in the series become equal in magnitude for , indicating that the optimum number of terms for a given is roughly the nearest integer to . To prove that the series is asymptotic, we need to show that

(1155) |

This immediately follows, because

(1156) |

Thus, the error involved in using the first terms in the series is less than , which is the magnitude of the next term in the series. We can see that, as increases, this estimate of the error first decreases, and then increases without limit. In order to visualize this phenomenon more exactly, let , and let

(1157) |

be the asymptotic series representation of this function that contains terms. Figure 21 shows the relative error in the asymptotic series plotted as a function of the approximate number of terms in the series, , for . It can be seen that as increases the error initially falls, reaches a minimum value at about , and then increases rapidly. Clearly, the optimum number of terms in the asymptotic series used to represent is about 10.

Asymptotic series are fundamentally different to conventional power law expansions, such as

(1158) |

This series representation of converges absolutely for all finite values of . Thus, at any , the error associated with the series can be made as small as is desired by including a sufficiently large number of terms. In other words, unlike an asymptotic series, there is no intrinsic, or unavoidable, error associated with a convergent series. It follows that a convergent power law series representation of a function is unique within the domain of convergence of the series. On the other hand, an asymptotic series representation of a function is not unique. It is perfectly possible to have two different asymptotic series representations of the same function, as long as the difference between the two series is less than the intrinsic error associated with each series. Furthermore, it is often the case that different asymptotic series are used to represent the same single-valued analytic function in different regions of the complex plane.

For example, consider the asymptotic expansion of the confluent hypergeometric function . This function is the solution of the differential equation

(1159) |

which is analytic at [in fact, ]. Here, denotes . The asymptotic expansion of takes the form:

for , and

for , and

for , et cetera. Here,

(1163) |

is a so-called Gamma function. This function has the property that , where is a non-negative integer. It can be seen that the expansion consists of a linear combination of two asymptotic series (only the first term in each series is shown). For , the first series is exponentially larger than the second whenever . The first series is said to be

Consider a region in which
one or other of the series is dominant. Strictly speaking, it is
not mathematically consistent to include the subdominant series in
the asymptotic expansion, because its contribution is actually
less than the intrinsic error associated with the dominant series
[this error is
times the dominant series, because we are only
including the first term in this series]. Thus, at a general point
in the complex plane, the asymptotic expansion simply consists
of the dominant series. However, this is not the case
in the immediate vicinity of the lines
, which are
called *anti-Stokes lines*. When an anti-Stokes line is
crossed, a dominant series becomes subdominant, and vice versa.
Thus, in the immediate vicinity of an anti-Stokes line neither
series is dominant, so it is mathematically consistent to include
both series in the asymptotic expansion.

The hypergeometric function
is a perfectly well-behaved,
single-valued, analytic function in the complex plane. However, our
two asymptotic series are, in general, multi-valued functions in the
complex plane [see Equation (1162)]. Can a single-valued function
be represented asymptotically by a multi-valued function? The short answer
is no. We have to employ different combinations of
the two series in different
regions of the complex plane in order to ensure that
remains
single-valued. Equations (1162)-(1164) show how this is achieved.
Basically, the coefficient in front of the subdominant series
changes discontinuously at certain values of
. This
is perfectly consistent with
being an analytic function
because the subdominant series is ``invisible'': in other words, the contribution
of the subdominant series to the asymptotic solution falls below the
intrinsic error associated with the dominant series, so that it does not really
matter if the coefficient in front of the former series
changes discontinuously. Imagine tracing a large circle, centered on the
origin, in the complex plane. Close to an anti-Stokes line, neither
series is dominant, so we must include both series in the asymptotic
expansion. As we move away from the anti-Stokes line, one series
becomes dominant, which means that the other series becomes
subdominant, and, therefore, drops out of our asymptotic expansion.
Eventually, we approach a second anti-Stokes line, and the subdominant
series reappears in our asymptotic expansion. However, the
coefficient in front of the subdominant series, when it
reappears, is different to that when the series disappeared. This new
coefficient is carried across the second anti-Stokes line into the
region where the subdominant series becomes dominant. In this new
region, the dominant series becomes subdominant, and disappears
from our asymptotic expansion. Eventually, a third anti-Stokes line
is approached, and the series reappears, but, again, with a different
coefficient in front. The jumps in the coefficients of the subdominant series
are chosen in such a manner that if we perform a complete circuit in the complex
plane then the value of the asymptotic expansion is the same at the beginning
and the
end points. In other words, the asymptotic expansion is single-valued,
despite the fact that it is built up out of two asymptotic
series that are not single-valued. The jumps in the coefficient of the
subdominant series, which are needed to keep the asymptotic expansion
single-valued, are called *Stokes phenomena*, after the
celebrated nineteenth century British mathematician
Sir George Gabriel Stokes, who first drew attention to this effect.

Where exactly does the jump in the coefficient of the subdominant
series occur? All we can really say is ``somewhere in the
region between two anti-Stokes lines where the series in question
is subdominant.'' The problem is that we only retained the
first term in each asymptotic series. Consequently, the intrinsic
error in
the dominant series is relatively large, and we lose track of
the subdominant series very quickly after moving away from
an anti-Stokes line. However, we could include more terms in each
asymptotic series. This would enable us to reduce the intrinsic error in
the dominant series, and, thereby, expand the region of the complex
plane in the vicinity of the anti-Stokes lines where
we can see both the dominant
and subdominant series. If we were to keep adding terms to our
asymptotic series, so as to minimize the error in the dominant
solution, we would eventually be forced to conclude that a jump in the
coefficient of the subdominant series can only take place on
those lines
in the complex plane on which
: these are
called *Stokes lines*. This result was first proved by Stokes in 1857.^{}On a Stokes line, the magnitude of the dominant
series achieves its maximum value with respect to that of
the subdominant series. Once we know that a jump in the coefficient
of the subdominant series can only take place at a Stokes line,
we can retain the subdominant series in our asymptotic expansion
in all regions of the complex plane. What we are basically saying is that,
although, in practice,
we cannot actually see the subdominant series very far away
from an anti-Stokes line, because we are only retaining the
first term in each asymptotic series, we could, in principle, see the
subdominant series at all values of
provided that
we retained a sufficient number of terms in our asymptotic series.

Figure 784 shows the location in the complex plane of the Stokes and anti-Stokes lines for the asymptotic expansion of the hypergeometric function. Also shown is a branch cut, which is needed to make single-valued. The branch cut is chosen such that on the positive real axis. Every time we cross an anti-Stokes line, the dominant series becomes subdominant, and vice versa. Every time we cross a Stokes line, the coefficient in front of the dominant series stays the same, but that in front of the subdominant series jumps discontinuously [see Equations (1162)-(1164)]. Finally, the jumps in the coefficient of the subdominant series are such as to ensure that the asymptotic expansion is single-valued.