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##### Department of Mathematics, University of California San Diego

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## The Art of Subtracting Infinity

##### Abstract:

We begin this lecture by reviewing the concept of the trace of a matrix. Then we move on to study the vibrations of a guitar string of length $\pi$. The noise that the string produces contains a base tone and higher overtones. In the idealized mathematical model, the list of frequencies of all these tones are $$1,\ 2^2,\ 3^2,\ 4^2,\ \dots,$$ and so the wavelengths are $$1,\ \frac1{2^2},\ \frac1{3^2},\dots\dots$$ We explain why the {\it total wavelength}, $1+\frac1{2^2}+\frac1{3^2}+\dots$ can be computed by integrating the {\it Green's function} $x(1-x/\pi)$ along the guitar string, giving an answer of $\pi^2/6$. The trouble starts when we try to replace the guitar string by a sphere and carry out the same process. The resulting formula $$\sum_{n=1}^\infty \frac{2n+1}{n(n+1)}\ =\ \int_{S^2} G(x,x)\,dS(x)$$ simply boils down to $\infty=\infty$. However, by subtracting infinity very carefully from each side, we obtain an interesting formula which works for any surface, not just a sphere. Although this mathematics may never be used to help us design musical instruments, it is related to the vortex theory of fluids, and Einstein's relativity.

### AP&M B412

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