Question

Study of Fourier series is important for Engineering. Justify this statement in
your own words.​​

Answer 1

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Fourier series it's a mathematics method to represent [ any function ] as a summation of sine and cosine.

• Why are sine and cosine are good represent functions ?

The answer is: sine and cosine are so special function his amplitude value are border between (1,-1)! and give you any value from - infinity to +infinity in easy way!; a lot of identities, integrable , differentiable that help engineer and mathematician to solve any problems.

Many problems in physics involve vibrations and oscillations. Often the oscillatory motion is simple (e.g. weights on springs, pendulums, harmonic waves etc.) and can be represented as single sine or cosine functions. However,in many cases, (electromagnetism, heat conduction, quantum theory,etc.) the wave forms are not simple and, unlike sines and cosines, can be difficult to treat analytically.Fourier methods give us a set of powerful tools for representing any periodic function as a sum of sines and cosines.

This problems you can found it when you design a system like Mobile Communication Systems that you see it today ; As Telecommunication engineer you need infinite Bandwidth (BW)* if you send Square Pulse Wave! and that not possible because we can't make a transfer media can hold infinity number of data...so we escape to Fourier series to represent this square wave as cosine and sine wave with contain same data [0,1].

*(Bandwidth refers to the data throughput capacity of any communication channel).

As you can see in this fig by increase number of n the Fourier series be more close to this rectangular pulse:

but you can ask

• Why digital signals (rectangular pulse , square pulse...) need infinity BW ?

The answer is: the digital signals (ex. rectangular pulse) it's go from Vmin to Vmax in zero second! practically that not possible! system need time to charge capacitor to store data [0,1]! so anything (except zero) divided by zero is infinity.

Did you see! why Joseph Fourier is genius! this application as one of thousand application help mathematician,control systems,electrical engineers, mechanical engineers, physics ...etc) in their life's.

Answer 2

Answer:

Many problems in physics involve vibrations and oscillations. Often the oscillatory motion is simple (e.g. weights on springs, pendulums, harmonic waves etc.) and can be represented as single sine or cosine functions. However,in many cases, (electromagnetism, heat conduction, quantum theory,etc.) the wave forms are not simple and, unlike sines and cosines, can be difficult to treat analytically.Fourier methods give us a set of powerful tools for representing any periodic function as a sum of sines and cosines.

That's why it is very important.....