Life is an approximation of 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 be careful about what we choose to ignore, because if those details contain the well-known devil, they may bite us in turn.
Mathematicians have suffered many times. A typical example is the Stokes' phenomenon. It originated nearly two hundred years ago with a problem about the rainbow and gave rise to a subfield of mathematics. In fact, this year Cambridge brought together some of the brightest minds in the field to launch a virtual research project on the topic. The problem involves very small amounts — exponentially small. But over time and space, this small amount can grow exponentially to a large size. Understanding these potentially explosively growing quantities is critical not only for mathematics, but also for all areas of engineering and science, from making jet engines to theoretical physics.
Under the rainbow
The problem began in 1838, when astronomer George Biddell Airy was interested in rainbows.
If you're lucky, a closer look at the rainbow reveals that there are one or more less obvious arcs under the rainbow body (the main rainbow), mainly green, pink, and purple. Airy was interested in these supernumerary fringes, not because of themselves, but because similar edge effects appeared in optical lenses. As an astronomer who needs to use telescopes regularly, Airy wants to understand the reasons behind this phenomenon.
There is a rainbow of affiliated rainbows. Photo by Johannes Bahrdt
Airy function
The Airy function Ai(r) is a solution to the following differential equations:
It is given by this integral:
Along the axis of coordinates that run vertically through the rainbow, the intensity of the light is related to the square of airy's function.
In the early 17th century, René Descartes used a theory that imagined light as composed of rays to explain the cause of the main rainbow. "But the ray theory of light doesn't predict the existence of a secondary stripe, so we can't simulate what it is," said Chris House, who is also a co-sponsor of the Newton Institute project. "Airy used the wave theory of light, which naturally derives the secondary stripes."
Airy wrote down a mathematical formula, now known as the Airy function, from which the light intensity of the main and secondary rainbows can be obtained, and when a rainbow is described by a linear coordinate axis perpendicular to the rainbow, we can also get the position of the rainbow arc. "Airy wanted to calculate where these extra fringes were, because it would help improve the optical performance of the telescope." House said.
The problem with airy's function is that it is difficult to calculate, given a specific x value, it is difficult to calculate the value of Airy's function Ai(x). At first, Using quadratures, Airy painstakingly calculated the value of the Airy function x at intervals of -4 to 4 at 0.2 intervals. Eleven years later, he improved the result using the method recommended by the mathematician Augustus de Morgan: using the sum of infinite multinomial series to approximate the function.
Using modern methods we can calculate the value of the Airy function and draw an image. The rightmost main bump represents the main rainbow, and the smaller bulge on the left represents the secondary rainbow. (The square of Airy's function gives the intensity of light.) Image source: House
The power of exponential
The idea of summing infinite series may seem strange at first glance, so let's look at an example.
Consider the exponential function:
where e is Euler's constant e= 2.718281...
This function is given by the Taylor series summed by the following infinite multinomial sum:
Each item of the series is a power function of the variable x.
Now that we assign any particular 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 to get the so-called partial sum. The result we get is an approximation of e: the larger n (i.e. the more terms contained in the partial sum), the more precise this approximation becomes. In fact, as long as n is large enough (i.e. the partial sum contains enough terms), we can get an approximation of arbitrary precision. Mathematically , this series is considered to converge to the value f ( x ) for all x.
For example, now in order to estimate the value of e at x=2, we take x=2 and simply calculate the first few terms of the Taylor series (also called the McLaughlin series), leaving the first five terms, and we get:
The true value of the function f(x) at x=2 is f(2) = e ≈ 7.4.
So in this example, even taking only the first five terms of the Taylor series gives a reasonable approximation of the value of the function at x=2.
Taylor series exist in a whole class of functions. And Taylor's theorem can tell us how far the approximate value differs from the true value of the function.
Taylor's failure
Taylor series are great in theory, and Airy can indeed calculate the value of the x function when it takes -5.6 to 5.6 using the Taylor series corresponding to the Airy function. But there is still a hurdle. Although the Taylor series of the Airy function can converge to the function itself, it converges too slowly. "Before we got the first subordinate stripe, we even needed to calculate 13 to 14 items," House said, "and in 1838 it was very difficult because scientists at the time had to do it by hand, which was impractical." ”
The blue curve is an Airy function, and the red curve retains the approximation obtained by the first three Taylor series, and it can be seen that the approximation only matches the first bump on the right side of the main rainbow. Source: House
To find a simpler way to approximate Airy's function, the mathematician George Gabriel Stokes decided in 1850 to venture out with an unconvergential series for approximation.
Satanic progression
It is easy to imagine that not all series converge at finite values. A simple example is the following series:
As parts and sums contain more and more terms, the results are also getting larger and larger, eventually exceeding all boundaries— they don't approach a finite value. This series will diverge to infinity.
Divergent progressions are like wild beasts in the circus, dangerous but controllable with a variety of tricks. In 1828, shortly before Stokes began studying the Airy function, the Norwegian mathematician Niels Henrik Abel described divergent series as "the invention of the devil" and claimed that "any proof based on divergent series is shameful."
But Stokes wasn't intimidated when he sought to approximate the Airy function. Out of an in-depth analysis of the mathematical nature of Airy's functions, he began to consider the use of divergent series. In fact , divergent series gives a good approximation to airy's function.
The trick of "taming the beast" is to know where to stop. Since stokes uses series that diverges to infinity, if too many terms in the partial sum are taken, the approximation becomes large and deviates far from the corresponding finite size of the Airy function value. But if the number of terms of the partial sum is just right, then the approximation will be very close to the actual function value.
When we add up more and more terms of divergent progression, we get a larger and larger result, and eventually divergence to infinity. But Stokes knew that for the divergent series he used, taking the appropriate number of terms would give a good approximation to airy's function.
Stokes' ingenious method allowed him to approximate the value of airy's function "very conveniently" at the value of the x value he had obtained, so he basically solved the problem of calculating the secondary rainbow. The blue curve below represents the actual Airy function, and the red curve represents the approximation of Stokes. You can see that the red line fits the blue line very closely. The only discrepancies appear near x=0, in the middle of the red curve diverging toward infinity.
In the case of rainbows, this difference is not important, since we are interested in the behavior of the Airy function representing the secondary rainbow on the left side of x=0.
The blue curve is the actual Airy function, and the red curve is Stokes' progressive approximation. The formula gives approximations in the different parts. Source: House
Here, the word "progressive" stands for approximation only if x is a sufficiently large positive number and a small enough negative number. (Similar to the straight-line asymptote we learned in school.) A strict definition of progressiveness is given here. ）
Despite successfully 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." House said, "This problem haunted him from 1850 to 1902. Stokes's final answer showed that when it comes to progressive approximations, tiny exponential terms can pop up and then grow to dominance. For details on each, please listen to the next breakdown.
By Marianne Freiberger
Translation: Tibetan idiot
Reviewer: zhenni
Original link:
https://plus.maths.org/content/stokes-phenomenon-asymptotic-adventure