![](https://csdnimg.cn/release/download_crawler_static/10270543/bg13.jpg)
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
π
4
15
] 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
7
ð
π=2
0
cos
1
cos xðÞ
1 þ 2 cos xðÞ
dx ¼
5π
2
24
:
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
7
MATLAB’s quad says this integral is 2.0561677 ..., which agrees quite nicely with
5π
2
24
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
6
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