- REAL ANALYSIS 501
[They] seem in general to be limited to a relatively narrow condi-
tion, one which is insufficient for even the simplest type of applica-
tion, namely that of absolute integrability of f{x) over an infinite
interval. There are, as far as I know, only a few exceptions.Thus, to the question as to whether physics could get by with sufficiently
smooth functions f(x) that decay sufficiently rapidly, the answer turned out to
be, in general, no. Physics needs to deal with discontinuous integrable functions
f(y), and for these f(z) cannot decay rapidly enough at infinity to make its integral
converge, at least not absolutely. What was to be done?
One solution involved the introduction of convergence factors, leading to a more
general sense of convergence, called Abel-Poisson convergence. In a paper on wave
motion published in 1818 Poisson used the representation
f(x) = - / f(a) cos a{x - a)e~ka da da.
ð J0 J-ooThe exponential factor provided enough decrease at infinity to make the integral
converge. Poisson claimed that the resulting integral tended toward f{x) as k
decreased to 0.
Abel used a similar technique to justify the natural value assigned to nonabso-
lutely convergent series such as
.. Ill , ir , 1 1 1
1ç(2) = 1-2 + 3-4+··· and 4 = 1-3 + 5-7 + -·
which can be obtained by expanding the integrands of the following integrals as
geometric series and integrating termwise:
/ , dr; / ,^1 » dr.
J 0 1 + r Jo 1+r^2
In Abel's case, the motive for making a careful study of continuity was his having
noticed that a trigonometric series could represent a discontinuous function. From
Paris in 1826 he wrote to a friend that the expansion
X ·^1 · Ï^1 · Ï^1 · ,
— = sm ð — — sin 2x + - sin 3x — — sin 4x + • • •
Α Ä ü Q
was provable for 0 < ÷ < ð, although obviously it could not hold at ÷ = ð.
Thus, while the representation might be a good thing, it meant, on the other hand,
that the sum of a series of continuous functions could be discontinuous. Abel
also believed that many of the difficulties mathematicians were encountering were
traceable to the use of divergent series. He gave, accordingly, a thorough discussion
of the convergence of the binomial series, the most difficult of the elementary Taylor
series to analyze.^10
For the two conditionally convergent series shown above and the general Fourier
integral, continuity of the sum was needed. In both cases, what appeared to be a
necessary evil—the introduction of the convergence factor e~ka or r—turned out to
have positive value. For the functions rn cos çè and r™ sin çè are harmonic functions(^10) Unknown to Abel, Bolzano had discussed the binomial series in 1816, considering integer,
rational, and irrational (real) exponents, admitting that he could not cover all possible cases,
due to the incomplete state of the theory of complex numbers at the time (Bottazzini, 1986, pp.
96-97). He performed a further analysis of series in general in 1817, with a view to proving the
intermediate value property (see Section 4 of Chapter 12).