- REAL ANALYSIS 499
Cauchy's definition still does not specify what ways of expressing one variable
in terms of another are legitimate, but this definition was a step toward the basic
idea that the value of the independent variable determines (uniquely) the value of
the dependent variable or variables.
Fourier series. Daniel Bernoulli's work introduced trigonometric series as an al-
ternative to power series. In his classic work of 1811, a revised version of which
was published in 1821,^7 Theorie analytique de chaleur (Analytic Theory of Heat),
Fourier established the standard formulas for the Fourier coefficients of a function.
For an even function of period 2ð, these formulas are1 °° 1 Ã2ð
f(x) = -an + on cosnx; a„ — — I f(x)cosnxdx, ç = 0,1,....A trigonometric series whose coefficients are obtained from an integrable function
f(x) in this way is called a Fourier series.
After trigonometric series had become a familiar technique, mathematicians
were encouraged to look for other simple functions in terms of which solutions
of more general differential equations than Laplace's equation could be expressed.
Between 1836 and 1838 this problem was attacked by Charles Sturm (1803-1855)
and Joseph Liouville, who considered general second-order differential equations of
the form
[p(x)y'(x)}' + \\r(x) + q(x)}y(x) = 0.
When a solution of Laplace's equation is sought in the form of a product of functions
of one variable, the result is often an equation of this type for the one-variable
functions. It often happens that only isolated values of ë yield solutions satisfying
given boundary conditions. Sturm and Liouville found that in general there will
be an infinite set of values ë = ëç, ç = 1,2,..., satisfying the equation and a pair
of conditions at the endpoints of an interval [a, b], and that these values increase
to infinity. The values can be arranged so that the corresponding solutions yn(x)
have exactly ç zeros in [a,b], and any solution of the differential equation can be
expressed as a series
ooy(x) = X]cnyn(x).
n=l
The sense in which such series converge was still not clear, but it continued to
be studied by other mathematicians. It required some decades for all these ideas
to be sorted out clearly.
Proving that a Fourier series actually did converge to the function that gener-
ated it was one of the first places where real analysis encountered greater difficulties
than complex analysis. In 1829 Dirichlet proved that the Fourier series of f(x) con-
verged to f(x) for a bounded periodic function f(x) having only a finite number
of discontinuities and a finite number of maxima and minima in each period.^8
Dirichlet tried to get necessary and sufficient conditions for convergence, but that
is a problem that has never been solved. He showed that some kind of continuity
would be required by giving the famous example of the function whose value at ÷
is one of two different values according as ÷ is rational or irrational. This function
(^7) The original version remained unpublished until 1972, when Grattan-Guiimess published an
annotated version of it.
(^8) We would call such a function piecewise monotonic.