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Section 21.1 The need for rigor
Objectives
See how infinite sums of continuous functions can produce discontinuous behavior.
Recognize that informal rules may fail when dealing with infinite processes.
Motivate real analysis as a rigorous framework that explains and controls these phenomena.
Recall the basic square wave with amplitude \(1\) and period \(2\pii\text{:}\)
\begin{equation*}
S(x)=\frac{4}{\pii}\qty(\sin x+\frac{1}{3}\sin 3x+\frac{1}{5}\sin 5x+\frac{1}{7}\sin 7x+\cdots)
\end{equation*}
Every term in this series is a continuous function — it’s just a sinusoid — but the resulting function is discontinuous!
The Gibbs phenomenon is the overshooting and undershooting that occurs around the points of discontinuity for any finite sum. It never goes away even with more and more times, because the series is not uniformly convergent.
From Wikipedia: “It is impossible for a discontinuous function to have absolutely convergent Fourier coefficients, since the function would thus be the uniform limit of continuous functions and therefore be continuous, a contradiction.”
If you integrate and then differentiate the infinite series, you get points where the derivative is undefined.
Derivative of the square wave term by term goes nuts!!
Fourier’s series caused a lot of mathematicians to slam on the brakes and try to figure out what was really going on. The result was real analysis.
