Fourier series is used in mathematics to decompose a periodic function into a sum of functions considered to be simple. These functions may either be cosines or sines. Fourier series study is one of Fourier analysis branch. Joseph Fourier is the person who contributed in the trigonometric series studies and led to the naming of the Fourier series. According to Folland (2009), one of the reasons for introduction of the series by Joseph Fourier was to assist in cracking heat equations. This heat equation is an equation said to be partial different. There was no answer for the heat equation prior to the work done by Joseph Fourier. There were specific solutions attained incase the heat equation performed in a modest way like to be a cosine or sine wave.
Sporadically, these solutions acquired are named eigensolutions. One of the ideas of Joseph Fourier was to perfect heat equations that were complicated like the linear combinations of cosine and sine waves then to inscribe the solutions as the equivalent eigensolutions superposition. The superposition and the linear combinations given are now known as the Fourier series. Later, the results achieved by Joseph Fourier were familiar because of the absence of a function notion that was precise (Folland, 2009).
Even though the main purpose of the Fourier series was to crack the heat equations, there were realizations that the similar methods could be used to solve physical and mathematical problems. ƒ(x) will stand for real variable x function which is periodic. These functions will be written as sums that are infinite. Braun (1993) asserts that this is by using cosines and sines on [−π, π] intervals. In a case whereby f(x) is able to be integrated on [−π, π], an = 1/n (x) cos(nx) dx, n ≥ 0 and bn = 1/n (x) sin(nx) dx, n ≥ 1(Braun, 1993). The figures accrued are referred to as figures of f. The introduction of Fourier series partial sums of f are symbolized by , N ≥ 0(Braun, 1993). These trigonometric equations may be expanded by using formulas of multiple angles.
This series does not come together and when it does for a precise value like x0 of x, the series x0 sum might be different from the function’s f(x0) value. According to the harmonic analysis, this is one of the questions used in deciding when conversions is to be done in the Fourier series and when the new function accrued is the same as the sum. In a case whereby the function is square integrable on [−π, π], this series converges to that function repeatedly. Mainly, converges of Fourier series are to f(x).
On the other hand, Fourier series has many uses in various applications. An example of a differential equation that is common is provided by: x (t) + ax (t) + b = f (t) (Braun, 1993). In this equation, description of a restrained harmonic oscillator motion which is driven by f (t) function is showed. Folland (2009) states that the Fourier series may be used in modeling wide physical portents like analog circuits that have capacitors, inductors, resistors or strings which are vibrated at given frequencies. The equation’s solution contains two fragments. The first fragment is a transient which disappears fairly rapidly.
After the transient has vanished, a steady state solution remains. Incase f (t) is said to be sinusoid, the solution to be accrued will be sinusoid and not problematic to find. One of the problems is that the driver is not a sinusoid that is simple but it is periodic functions. The oscillating systems physical properties which make the analysis of Fourier series to be useful are the superposition’s properties. What if f1 (t), with other conditions that are initial produce a steady x1 (t) and f2 (t) considered to be a driving force produces a steady x2 (t), then f3 (t) = f1 (t) + f2 (t) will produce x3 (t) = x1 (t) + x2 (t) (Braun, 1993).
In conclusion, the Fourier series elementary equations led to Fourier transform development that can decompose a function considered to be non periodic. These advanced techniques of the Fourier transforms and series have being used as essential parts of scientists and mathematicians toolboxes. Currently, they are being used in applications such as processing of signals in astronomy, optics and compressing of files (Folland, 2009).