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Main question: Calculating the coefficients of a Fourier series can be difficult and time-consuming. How might a student be motivated/convinced to go through these (potentially tedious) details? Are there recognizable benefits that we, as teachers and/or mathematicians, can point to in the hopes of motivating students?

Example: Consider a simple periodic function like a square wave. This function can be easily written (e.g. defined piece-wise), but the Fourier series is an infinite series of sines and cosines, and we all know infinite series pose notorious conceptual challenges for students at all levels.

A student might ask, "Why bother with this format when we can quite easily deal with the given periodic function?" In answering such a query we might want to address similar questions: Does this format tell us something useful? Does it reveal deeper behavior of this function and similar ones? etc.

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    $\begingroup$ See also math.stackexchange.com/questions/660170/…. $\endgroup$ – J W Feb 22 '15 at 16:02
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    $\begingroup$ What students do you have in mind: Undergrad or graduate? Fluent with computational software or not? Specializing in chemistry, data science, electrical engineering, finance, math, mechanical engineering, music, physics, statistics....? I'd expect slightly different answers for all of these. $\endgroup$ – user173 Feb 24 '15 at 3:25
  • $\begingroup$ I removed almost all comments now that the question got rephrased and reopened. In case somebody should really miss one just let me know. $\endgroup$ – quid Feb 25 '15 at 21:43
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    $\begingroup$ Mathematics and physics undergraduate students. $\endgroup$ – matqkks Feb 27 '15 at 10:01
  • $\begingroup$ www-personal.umich.edu/~ojwalch/tina.png from www-personal.umich.edu/~ojwalch/drawanything.html is a striking demonstration that knowledge about Fourier series can be used to win app-making contests $\endgroup$ – Mark S. Feb 28 '15 at 21:02
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The Fourier coefficients contain useful information that is not at all apparent just from the shape of the periodic function itself. For example, when Fourier series are used for musical vibrations we may remove the terms in the Fourier series with coefficients having magnitude below a certain cutoff (kind of like taking out fundamental tones in a musical note at a frequency beyond the range of human hearing) and the resulting finite Fourier series may be indistinguishable from the original but require a lot less information to store it. That's the whole idea behind data compression. If your students ever listened to an MP3 file then they know why this is important without even realizing it.

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    $\begingroup$ Another good example is the sampling theorem, and the transmission of audio data over telephone wires. $\endgroup$ – Steven Gubkin Feb 25 '15 at 2:40
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There are tons of reasons:

  • Signal theory: The key phrase here is bandwidth. If you want to transmit a signal over an analog channel you could modulate it onto a carrier frequency. To ensure that you do not have inference with other signals transmitted simultaneously, you should ensure that the signal you want to transmit does not have too large frequency components. In other words: For a periodic signal you need that Fourier coefficients for high frequencies are zero. Check the Wikipedia pages for Bandwidth and Frequency response.

  • Vibrations: The key phrase is natural frequency. If you have a physical object that is subject to excitation and start to vibrate, you could model this by finite elements and end up with certain eigenfrequencies. In a linear model the different eigenfrequencies all have their own damping coefficient i.e. oscillations at different frequencies are damped away independent of each other, each with its own speed. Coming back to the square wave: If the excitation is a square wave, then what matters for the resulting vibration is the expansion of the excitation in eigenfrequencies. A square wave has contributions from all frequencies, i.e. it excites all eigenfrequencies.

  • Electrical circuits: Circuits consisting of resistors, inductors and capacitators behave similarly, i.e. the behavior of the circuit is described by its eigenfrequencies. Again, exciting the circuit with a square wave (that is, switching a current on and off) excites all eigenfrequencies which is good to know since this may cause undesired behaviour. Check RLC circuits.

  • Regularity of functions: More on the mathematical side, the order of differentiability can be seen in the decay of the Fourier coefficients. If $f$ is $n$-times continuously differentiable, then the Fourier coefficients decay like $1/k^n$ and conversely, if the coefficients decay fast enough, the function is indeed differentiable… Check, e.g., these notes for details and exact statements.

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You cited the square wave, saying that it can be written as a piecewise function. True, but when given a box of resistors, capacitors, and inductors, you can't build a piecewise filter. This series is taught in engineering school and is one of the most beautiful examples of a Fourier series being used in real life. A filter creating a square wave that can get as clean as you need it depending on the number of components used to build it, i.e the number of terms of the series.

$y=\sin x\space +\space \frac{\sin \left(3x\right)}{3}\space +\frac{\sin \left(5x\right)}{5}\space +\space \frac{\sin \left(7x\right)}{7}\space +\frac{\sin \left(9x\right)}{9}+\space \frac{\sin \left(11x\right)}{11}\space +\frac{\sin \left(13x\right)}{13}+\frac{\sin \left(15x\right)}{15}$..........

enter image description here

In sum, Fourier series are an important part of many branches of engineering, offering an answer to "when is this used in real life?"

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  • $\begingroup$ You mean "examples of a Fourier series" rather than of a Taylor series. $\endgroup$ – KCd Mar 13 '15 at 19:24
  • $\begingroup$ Yes! Edited. Thank you. $\endgroup$ – JTP - Apologise to Monica Mar 13 '15 at 19:32
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On one hand understanding Fourier series is very important in Engineering, Physics and Math. On the other hand, to actually calculate the coefficients by hand, instead of using software, do we really need that nowadays? We do not teach how to use an engineering sliding ruler or logarithmic tables to do operations anymore, because mastering those skills is not necessary today. I try to show my engineering students in Acoustics the meaning of the Fourier series using software (Mathematica), please see the PDF documents in the several "Acoustic Spectrum" sections of this link http://matecmaticaacustica.weebly.com, perhaps those documents can be useful for you, even if you insist (or need to insist) in calculating the coefficients by hand (I mean without software)

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  • $\begingroup$ Doing it in professional (post grad) work, can rely on software to help with the tedious algebra, but I recommend to still have new students do by hand until they are comfortable. The reason is pedagogical. We learn better what we break down, rather than rely on black box. Also, some general practice in tedious algebra is still useful for other things also (like Jackson E&M homework, reading complicated math, physics papers, etc.). So this is a side benefit. Build the muscles... $\endgroup$ – guest Jan 29 '18 at 2:25
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I often use the module "How to Tune a Radio--Trigonometric integrals explain tuning a radio." It is in Volume 3 (Applications of Calculus) of the MAA Resources for Calculus Collection (MAA Notes Number 29, edited by Philip Straffin).

The author of the module, Clark Benson, provides very good context to the problem of tuning a radio, connects the problem to integration, and introduces Fourier analysis. Even if you do not assign the series of problems in the module, having students read the module can help convince them of why one might want to represent a square wave as a Fourier series.

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on possibly infinite representations

I once had this same question, so the following is the way I came to understand it.

There are many ways of defining a function. A piece-wise definition (like for your square wave) or a single succinct formula that may include logarithms, polynomials, or anything else we might like to throw at it. Or we might only have some Differential Equations which can be used to generate the function.

However, just because we can specify functions like that doesn't mean we can do much with them. Different mathematical tools or manipulations require different representations. In particular integration is often intractable for many arbitrary functions.

Common formats that give us access to common "Toolkits" are an (often infinite) list of coordinates, an (often infinite degree) polynomial, and (often infinite) combinations of sines and cosines. That all of these representations may be infinite does not mean that they are not useful, as there are often shorter ways expressing them (Sigma notation is common for infinite Taylor series), or hopeful they "converge" so a finite estimate gives us enough information about the part of the function we are interested in.

The Fourier Transform

I'm not an expert on the Fourier Transform (I've only ever studied Discrete FT, so others feel free to change or add) but they have strengths and weaknesses just like other representations. An obvious strength is that differentiation of sine waves is easy: simply multiply the amplitude of each wave by its frequency.

In your example, the infinite series of sines and cosines that represent a square wave can be simply expressed as $\frac{sin(\pi t)}{\pi t}$ (think of it like that formula gives the amplitude of the sine wave at frequency $t$). It also converges (high frequencies contribute much less to the sum). Once you have it in this form, all of the tools developed for Fourier series are at your disposal.

You can do signal processing on it. What will your square wave look like after it has gone through filters, or after it has been realised as an analogue waveform? If it is a sound wave, what harmonics will people hear? How will it be encoded into an MP3?

Is your square wave part of a control system? How will it interact with other components in the system?

Is your square wave part of a Differential Equation? You can now use Fourier methods to solve it algebraically.

If you want an example of how insanely flexible Fourier transforms are, look up the "Ptolemy and Homer Simpson" YouTube video. Now we can integrate Homer!

Here is a page which goes into more detail about the Fourier transform with some interesting visualisations: http://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-homer-simpsons-face

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  • $\begingroup$ If there is something I have got wrong, please edit or add a comment. $\endgroup$ – Richard Feb 23 '15 at 22:57
  • $\begingroup$ Formula for square wave is for the non-periodic version, will fix when question is reopened $\endgroup$ – Richard Feb 24 '15 at 20:18

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