### Fourier Series Term Paper:

**Fourier series** in mathematics is the way of presentation of the certain wave-like function with the help of the sum of the simple sine functions.

Generally, the number of simple functions can be unlimited; moreover, the more simple functions one represents during the calculation, the more chances he has to receive the most precise result. In the majority of cases the role of the simple functions are performed by sines and cosines. In this case Fourier series is called the trigonometric one. The Fourier series has got its name after the outstanding French physician and mathematician Jean-Baptiste Joseph Fourier who is known to make the considerable contribution into the development of trigonometric series. He decided to work out the series in order to solve the heat equation in the plate made from metal.

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The idea of presentation of a single complex function by the complex of the simple ones was noticed about 2300 years ago and the ancient astronomers suggested the empiric system of the motion of planets relying ondeferents and epicycles. Fourier was the inventor of the solution of the problem on heat equation, because no one before him could do it in the effective way. The heat equation belongs to the partial differential equation based on sine and cosine waves. It is obvious that the mathematician planned just to apply his invention for the solution of heat equation, but this technique can be also used in other spheres – electrical engineering, acoustics, quantum mechanics, vibration analysis and many others.

Fourier series is a useful topic for the research, because it touches upon the important issues based on the complex wave-like functions and their calculations. The student is expected to observe the history of the origin of Fourier series in order to understand the logic of the inventor and to see the connections with other processes and equations. The young person is expected to dwell on the structure of the Fourier series, their variety, their multiple application on practice and the principles of their calculation. The student’s aim is to demonstrate the effectiveness of the equations and support his point of view with the definite examples and cases. In the end, one is supposed to conclude the paper wisely.

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