3194 Fourier and Laplace
What we know is but a little thing;
what we are ignorant of is immense.
Pierre Simon de Laplace4.1 Introduction ...............................................................................................................
This chapter is devoted to the Fourier series (FS) and the Fourier and Laplace transforms
named after the French mathematicians Jean Baptiste Joseph Fourier (1768–1830)* and
Pierre Simon de Laplace (1749–1829).†
The FS expansion (presented in 1807), called “the great mathematical poem” by Lord
Kelvin, deals with waves that are periodic in nature, such as sound, light, radio, thermal,
ocean, pressure, or force waves.
Fourier (1822) while studying the problem associated with the fl ow, propagation, and
conduction of heat, showed that any arbitrary wave can be expressed as a linear infi nite
series of sinusoidal (sines and cosines) functions of harmonic-related frequencies. The pur-
pose of representing a function in terms of an FS is that the function can be analyzed in
terms of its frequency components, making it much easier to understand and calculate its
power and energy distribution content.
This model is, in many cases, the preferred form to analyze a function. The FS can be used
to evaluate the amount of power concentrated at each harmonic; hence, the relative impor-
tance of each harmonic of a given wave. The theory developed by Fourier is used for solving
many engineering problems in diverse areas such as communications, signal processing,
*^ Baron Jean Baptist Joseph Fourier—distinguished scientist, professor of mathematics and physicist, politi-
cian, and diplomat—served Napoleon and King Louis XVIII. He played an important role in Napoleon’s expe-
ditions, in particular the one to Egypt. He is credited (1807) with the idea that any arbitrary function, defi ned
by different analytical expressions over adjacent segments of its range can be expressed by a single analytical
expression.
† Pierre Simon Laplace (a count and years later a marquis)—distinguished scientist, mathematical astronomer,
and mathematician, professor at the Paris École Militaire, a member of the Académic Royale des Sciences,
astronomer, and an accomplished diplomat and politician—served Napoleon and King Louis XVIII govern-
ments, as president of the senate, and as a member of the Chamber of Peers. As a scientist he was second only
to Newton; but as a person his qualities are mixed. Laplace was born at Beaumont-en-Auge, Normandy, on
March 23, 1747. Facts of his life were destroyed by a fi re in the year 1929, and lost during the bombardments
of Caen, in WWII. Laplace adapted easily to the social and political changes of his time. He prospered fi nan-
cially, scientifi cally, socially, and politically. In 1784, Laplace was appointed examiner to the royal artillery,
a lucrative and prominent post. There he had the good fortune to examine a young ambitious 16-year-old
sublieutenant named Napoleon Bonaparte, who years later became the ruler of continental Europe. This rela-
tionship gave Laplace a number of opportunities that he took advantage of.