探索趣味积分：方法与未来应用

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"《Inside Interesting Integrals》是一本专为完成大学一年级或高中AP微积分课程，并对微分方程概念有一定了解的学生编写的轻松读物。作者保罗·J·纳欣通过这本书，旨在深入探讨和教授计算定积分的方法，而非仅仅关注具体的解。尽管书中提供的结果都经过了详尽的推导，其价值在于它所传授的技巧，这些技巧对于解决未来可能会遇到的积分问题至关重要。该书属于《Undergraduate Lecture Notes in Physics》系列的一部分，这个系列由Springer出版社出版，专注于提供纯物理学和应用物理学领域的权威教材，适合用作本科教学的基础材料。每本书通常包含练习题、工作示例、章节总结以及进一步阅读的建议，强调了新颖、原创和独特教学方法在本科物理教育中的应用。《Inside Interesting Integrals》作为系列的一部分，可能满足以下标准：一是对标准本科主题进行了清晰且精炼的阐述；二是为研究生或高级主题提供坚实的入门；三是提供了一个学科的新视角或非传统的教学方法。这本书特别鼓励创新的教学方式，以保持读者在整个学术生涯中对其内容的兴趣和依赖。《Inside Interesting Integrals》不仅提供了丰富的积分技巧，还激发学生对数学理论的热情，使之成为学生在求学道路上的一个长期参考资源。无论是为了深化对已学知识的理解，还是为了掌握处理复杂问题的新策略，这本书都是一个不可或缺的工具。"

ln 1 þ xðÞ

1 þ x

dx ¼

πln 2ðÞ

which equals 1.088793 .... One can only wonder at how many students struggled

(and for how long) to do this, as the given answer is incorrect. Late r in this book, in

(5.1.3), we’ll do this integral correctly, but a use of quad (not available to Spiegel

in 1964) quickly shows the numerical value is actually the signiﬁcantly greater

1.4603 .... At the end of this Preface I’ll show you two examples (including

Spiegel’s integral) of this helpful use of quad.

Our use of quad does prompt the question of why, if we can always calculate the

value of any deﬁnite integral to as many decimal digits we wish, do we even care

about ﬁnding exact expressions for these integrals? This is really a philosophical

issue, and I think it gets to the mysterious interconnectedness of mathematics—how

seemingly unrelated concepts can turnout to actually be intimately related. The

expressions we’ll ﬁnd for many of the deﬁnite integrals evaluated in this book will

involve such familiar numbers as ln(2) and π, and other numbers that are not so well

known, like Catalan’s constant (usually written as G) after the French mathemati-

cian Euge

ne Catalan (1814–1894). The common thread that stitches these and other

numbers together is that all can be written as inﬁnite series that can, in turn, be

written as deﬁnite integrals:

ln 2ðÞ¼1 



 ... ¼

1 þ x

dx ¼ 0:693147 ...

¼ 1 



 ... ¼

1 þ x

dx ¼ 0:785398 ...

G ¼



 ... ¼

ln xðÞ

1 þ x

dx ¼ 0:9159655 ...:

And surely it is understanding at a far deeper level to know that the famous

Fresnel integr als

cos(x

)dx and

sin(x

)dx are exactly equal to

ﬃﬃ

, com-

pared to knowing only that they are ‘pretty near’ 0.6267.

In 2004 a wonderful book, very much like this one in spirit, was published by

two mathematicians, and so I hope my cavalier words will appear to be appalling

ones only to the most rigid of hard-core purists. That book, Irresistible Integrals

(Cambridge University Press) by the late George Boras, and Victor Moll at Tulane

University, is not quite as willing as this one is to return to the devil-may-care,

eighteenth century mathematics of Euler’s day, but I strongly suspect the authors

were often tempted. Their subtitle gave them away: Symbolics, Analysis and

Preface xv

Experiments [particularly notice this word!] in the Evaluation of Integrals. Being

mathematicians, their technical will-power was stronger than is my puny electrical

engineer’s dedication to rigor, but every now and then even they could not totally

suppress their sheer pleasure at doing deﬁnite integrals.

And then 3 years later, in another book coauthored by Moll, we ﬁnd a statement

of philosophy that exactly mirrors my own (and that of this book): “Given an

interesting identity buried in a long and complicated paper on an unfamiliar subject,

which would give you more conﬁdence in its correctness: staring at the proof, or

conﬁrming computationally that it is correct to 10,000 decimal places?”

That book

and Irresi stible Integrals are really fun math books to read.

Irresistible Integrals is different from this one, though, in that Boras and Moll

wrote for a more mathematically sophisticated audience than I have, assuming a

level of knowledge equivalent to that of a junior/senior college math major. They

also use Mathematica much more than I use MATLAB. I, on the other hand, have

assumed far less, just what a good student would know—with one BIG exception—

after the ﬁrst year of high school AP calculus, plus just a bit of exposure to the

concept of a differential equation. That big exception is contour integration, which

Boras and Moll avoided in their book because “not all [math majors] (we fear, few)

study complex analysis.”

Now that, I have to say, caught me by surprise. For a modern undergraduate

math major not to have ever had a course in complex analysis seems to me to be

shocking. As an electrical engineering major, 50 years ago, I took complex analysis

up through contour integr ation (from Stanford’s math department) at the start of my

junior year using R.V. Churchill’s famous book Complex Variables and Applica-

tions. (I still have my beat-up, coffee-stained copy.) I think contour integration is

just too beautiful and powerful to be left out of this book but, recognizing that my

assumed reader may not have prior knowledge of complex analysis, all the integrals

done in this book by contour integration are gathered together in their own chapter

at the end of the book. Further, in that chapter I’ve included a ‘crash mini-cour se’ in

the theoretical complex analysis required to understand the technique (assuming

only that the reader has already encountered complex numbers and their

manipulation).

Irresistible Integrals contains many beautiful results, but a signiﬁcant fraction of

them is presented mostly as ‘sketches,’ with the derivation details (often presenting

substantial challenges) left up to the reader. In this book every result is fully derived.

Indeed, there are results here that are not in the Boras and Moll book, such as the

famous integral ﬁrst worked out in 1697 by the Swiss mathematician John Bernoulli

(1667–1748), a result that so fascinate d him he called it his “series mirabili”

(“marvelous series”):

Experimental Mathematics in Action, A. K. Peters 2007, pp. 4–5. Our calculations here with quad

won’t be to 10,000 decimal places, but the idea is the same.

xvi Preface

done at Princeton by Morris] that the integral should have a value between 6.3 and

6.9. Thus the value above [6.4939 ...¼

] furnishes a theoretical veriﬁcation of

experimental results.”

So here we have an important deﬁnite integral apparently ‘discovered’ by a

physicist and solved by a mathematician. In fact, as you’ll learn in Chap. 5,

Buchanan was not the ﬁrst to do this integral; it had been evaluated by Riemann

in 1859, long before 1936. Nonetheless, this is a nice illus tration of the fruitful

coexistence and positive interaction of experiment and theory, and it is perfectly

aligned with the approach I took while writing this book.

There is one more way this book differs from Irresistible Integrals, that reﬂects

my background as an engineer rather than as a professional mathematician. I have,

all through the book, made an effort to bring into many of the discussions a variety

of physical applications, from such diverse ﬁelds as radio theory and theoretical

mechanics. In all such cases, however, math plays a central role. So, for exam ple,

when the topic of elliptic integrals comes up (at the end of Chap. 6), I do so in the

context of a famous physics problem. The origin of that problem is due, however,

not to a physicist but to a nineteenth century mathematician.

Let me close this Preface on the same note that opened it. Despite all the math in

it, this book has been written in the spirit of ‘let’s have fun.’ That’s the same attitude

Hardy had when, in 1926, he replied to a plea for help from a young undergraduate

at Trinity College, Cambridge. That year, while he was still a teenager,

H.S.M. Coxeter (1907–2003) had undertaken a study of various four-dimensional

shapes. His investigations had suggested to him (“by a geometrical consideration

and veriﬁed graphically”) several quite spectacular deﬁnite integrals, like

π=2

cos

1

cos xðÞ

1 þ 2 cos xðÞ



dx ¼

5π

In a letter to the Mathematical Gazette he asked if any reader of the journal could

show him how to derive such an integral (we’ll calculate the above so-called

Coxeter’s integral later, in the longest derivation in this book). Coxeter went on

to become one of the world’s great geometers and, as he wrote decades later in the

Preface to his 1968 book Twelve Geometric Essays, “I can still recall the thrill of

receiving [solutions from Hardy] during my second month as a freshman at

Cambridge.” Accompanying Hardy’s solutions was a note scribbled in a margin

MATLAB’s quad says this integral is 2.0561677 ..., which agrees quite nicely with

5π

¼ 2:0561675 .... The code syntax is: quad(@(x)acos( cos(x)./(1 + 2*cos(x))),0,pi/2).

For the integral I showed you earlier, from Spiegel’s book, the quad code is (I’ve used 1e6 ¼10

for the upper limit of inﬁnity): quad(@(x)log(1 + x)./(1 + x.^2),0,1e6). Most of the integrals in this

book are one-dimensional but, for those times that we will encounter higher dimensional integrals

there is dblquad and triplequad, and MATLAB’s companion, the Symbolic Math Toolbox and its

command int (for ‘integrate’), can do them, too. The syntax for those cases will be explained when

we ﬁrst encounter multidimensional integrals.

xviii Preface

declaring that “I tried very hard not to spend time on your integrals, but to me the

challenge of a deﬁnite integral is irresistible.”

If you share Hardy’s (and my) fascination for deﬁnite integrals, then this is a

book for you. Still, despite my admiration for Hardy’s near magical talent for

integrals, I don’t think he was always correct. I write that nearly blasphemous

statement because, in addition to Boras and Moll, another bountiful source of

integrals is A Treatise on the Integral Calculus, a massive two-volume work of

nearly 1,900 pages by the English educator Joseph Edwards (1854–1931). Although

now long out-of-print, both volumes are on the Web as Google scans and available

for free download. In an April 1922 review in Nature that stops just short of being a

sneer, Hardy made it quite clear that he did not like Edwards’ work (“Mr.

Edwards’s book may serve to remind us that the early nineteenth century is not

yet dead,” and “it cannot be treated as a serious contribution to analysis”). Finally

admitting that there is som e good in the book, even then Hardy couldn’t resist

tossing a cream pie in Edwards’ face with his last sentence: “The book, in short,

may be useful to a sufﬁciently sophisticated teacher, provided he is careful not to

allow it to pass into his p upil’s hands.” Well, I disagree. I found Edwards’ Treatise

to be a terriﬁc read, a treasure chest absolutely stuffed with mathematical gems.

You’ll ﬁnd some of them in this book. Also included are dozens of challenge

problems, with complete, detailed solutions at the back of the book if you get stuck.

Enjoy!

Durham, NH Paul J. Nahin

And so we see where Boras and Moll got the title of their book. Several years ago, in my book Dr.

Euler’s Fabulous Formula (Princeton 2006, 2011), I gave another example of Hardy’s fascination

with deﬁnite integrals: see that book’s Section 5.7, “Hardy and Schuster, and their optical

integral,” pp. 263–274. There I wrote “displaying an unevaluated deﬁnite integral to Hardy was

very much like waving a red ﬂag in front of a bull.” Later in this book I’ll show you a ‘ﬁrst

principles’ derivation of the optical integral (Hardy’s far more sophisticated derivation uses

Fourier transforms).

Preface xix

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探索趣味积分：方法与未来应用

inside interesting integrals

table of integrals, series, and products

[1] G. Eason, B. Noble, and I. N. Sneddon, “On certain integrals of Lipschitz-Hankel type involving products of Bessel functions,” Phil. Trans. Roy. Soc. London, vol. A247, pp. 529–551, April 1955. 生成上面格式的参考文献代码

tables of integrals

The Mathematics of Computerized Tomography

G. Eason, B. Noble, and I. N. Sneddon, ``On certain integrals of Lipschitz-Hankel type involving products of Bessel functions,'' Phil. Trans. Roy. Soc. London, vol. A247, pp. 529--551, April 1955.里面的A247是什么意思

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