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Life is an approximate art. If we consider every detail of every aspect of life, we will never be able to make new progress. Of course, we need to choose carefully which things to ignore, because if those details contain well-known demons, they may bite us in turn.

Mathematicians have suffered many times. A typical example is the Stokes' phenomenon phenomenon. It originated from a problem about rainbows nearly two hundred years ago and derived a subfield of mathematics. In fact, Cambridge this year brought together some of the smartest people in the field to launch a virtual research project on the topic. The problem involves a very small amount-- exponentially small. But after the passage of time and space, this small amount can grow to a very large exponentially. Understanding these potentially explosive quantities is crucial not only to mathematics but also to engineering and science from the manufacture of jet engines to theoretical physics.

The problem under the rainbow began in 1838, when astronomer George George Biddell Airy was interested in rainbows.

If you are lucky, if you look closely at the rainbow, you will find one or more less obvious arcs below the main body of the rainbow (the main rainbow), mainly green, pink and purple. Eli is interested in these extra stripes (attached rainbow, supernumerary fringes) not because of them, but because of a similar edge effect in optical lenses. As an astronomer who often uses binoculars, Eli wants to understand the reasons behind this phenomenon.

A rainbow with an attached rainbow. Photography: Johannes Bahrdt Eli function Ai (r) is a solution of the following differential equations:

It is given by this integral:

Along the axis that passes vertically through the rainbow, the intensity of light is related to the square of the Eli function.

At the beginning of the 17th century, Ren é Descartes explained the cause of the main rainbow using a theory that imagined light as consisting of rays. " But the ray theory of light can't predict the existence of accessory stripes, so we can't simulate what it is, "said Chris House, who is also a co-sponsor of the Newton Institute project." Eli uses the wave theory of light, which naturally leads to accessory stripes. "

Eli wrote down a mathematical formula, now known as the Airy function function, from which you can get the light intensity of the main rainbow and the accessory rainbow, and when describing the rainbow with a straight axis perpendicular to the rainbow, we can also get the position of the rainbow arc. "Eli wants to calculate where these extra stripes are, because it will help improve the optical performance of the telescope." House said.

The problem with the Eli function is that it is difficult to calculate. Given a specific x value, it is difficult to calculate the Eli function value Ai (x). At first, using the quadrature method (quadratures), Eli painstakingly calculated the value of the Eli function when x was from-4 to 4 at intervals of 0.2. Eleven years later, he improved the result using the method recommended by the mathematician Augustus Morgan (Augustus de Morgan): using the sum of infinitely many series to approximate the function.

Using modern methods, we can calculate the value of the Eli function and draw an image. The rightmost main bulge represents the main rainbow and the smaller bulge on the left represents the accessory rainbow. The square of the Eli function gives the light intensity. ) Source: the idea of summing the infinite series of House Index may seem strange at first glance. Let's take a look at an example.

Examine the exponential function:

Where e is the Euler constant Euler 2.718281...

This function is given by the Taylor series of the sum of infinitely many terms:

Every term of a series is a power function of the variable x.

Now to assign any specific value to the variable x, we can never add up every term of this series (because there is no infinite time), but we can sum the first n terms and get the so-called partial sum. The result we get is an approximation of ex: the larger the n (that is, the more terms are contained in the partial sum), the more accurate the approximation is. In fact, as long as n is large enough (that is, there are enough items in the partial sum), we can get an approximation of arbitrary precision. Mathematically, it is believed that this series can converge to the value f (x) for all x.

For example, now in order to estimate the value of ex at x2, we simply calculate the first few terms of Taylor series (also known as McLaughlin series), keep the first five terms, and we get:

The real value of the function f (x) is f (2) = e ~ 2 ≈ 7.4.

So in this case, even taking only the first five terms of the Taylor series can give a reasonable approximation of the value of the function at the time of x _ 2.

Taylor series exists in a whole class of functions. And Taylor theorem can tell us how big the difference is between the approximate value and the real value of the function.

Taylor's failed Taylor series is great in theory, and Eli can indeed calculate the value of the function when x is from-5.6 to 5.6 using the Taylor series corresponding to the Eli function. But there is still an obstacle. Although the Taylor series of the Eli function can converge to the function itself, it converges too slowly. "We even need to calculate 13 to 14 terms before we get the first accessory stripe," House said. "it was very difficult in 1838 because scientists had to do it by hand, which was impractical."

The blue curve is the Eli function, and the red curve retains the approximation obtained by the first three Taylor series, and you can see that the approximate value is only consistent with the first bulge on the right of the main rainbow. Figure source: House in order to find a simpler way to approximate the Eli function, mathematician George Gabriel Stokes (George Gabriel Stokes) decided in 1850 to risk using an unconvergent series as an approximation.

Satan series is easy to imagine, not all series converge to a finite value. A simple example is the following series:

As the partial sum contains more and more terms, the results become larger and larger and eventually exceed all boundaries-they do not approach a finite value. This series will diverge to infinity.

The divergence series is like a wild animal in a circus, dangerous but can be controlled with a variety of techniques. In 1828, shortly before Stokes began to study the Eli function, the Norwegian mathematician Nils Henryk Abel (Niels Henrik Abel) described the divergence series as "the invention of the devil" and claimed that "any proof based on the divergence series is shameful".

But Stokes was not intimidated when he sought to approximate the Eli function. Out of an in-depth analysis of the mathematical nature of Eli function, he began to consider the use of divergent series. In fact, the divergence series gives a good approximation to the Eli function.

The trick of "taming animals" is to know where to stop. Because the series used by Stokes diverges to infinity, if there are too many terms in the partial sum, the approximation will become huge and far deviate from the corresponding finite size Eli function value. However, if the number of terms of the partial sum is just right, then the approximate value will be very close to the actual function value.

When we add up more and more terms of the divergence series, we get a larger and larger result, which eventually diverges to infinity. But Stokes knows that for the divergence series he uses, taking an appropriate number of terms can get a good approximation of the Eli function.

Stokes's ingenious method made it "very convenient" for him to approximate the value of the Eli function at the value of x, so he basically solved the problem of calculating the accessory rainbow. The blue curve below represents the actual Eli function, and the red curve represents the Stokes approximation. You can see that the red line is very close to the blue line. The only discrepancy occurs near x zero, in the middle of the red curve diverging toward infinity.

As far as rainbows are concerned, this difference is not important because we are interested in the behavior that the Eli function represents the attached rainbow on the left side of xrain0.

The blue curve is the actual Eli function, and the red curve is Stokes's asymptotic approximation. The formula gives the approximation in different parts. Image source: House here, the word "progressive" means that the approximation is valid only when x is large enough and negative enough. It is similar to the straight asymptote we learned at school. A strict definition of gradual progress is given here. )

Despite his success in solving the problem, Stokes was not satisfied. The two parts of his approximation are described by two very different mathematical formulas (shown above), which bothers Stokes. "what Stokes wants to know is how to transition from one expression to another." "the problem haunted him from 1850 to 1902," House said. " Stokes's final answer shows that when it comes to asymptotic approximation, tiny exponential terms can suddenly appear and grow to dominate. For details, please listen to the next decomposition.

Original text link:

Https://plus.maths.org/content/stokes-phenomenon-asymptotic-adventure

The content of the translation only represents the author's point of view, not the position of the Institute of Physics of the Chinese Academy of Sciences.

This article comes from the official account of Wechat: Institute of Physics, Chinese Academy of Sciences (ID:cas-iop), author: Marianne Freiberger, translator: Tibetan idiot, revision: zhenni

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