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The well-known physicist A. A. Michelson started quite an interesting correspondence in the journal Nature in 1898. He complained about the convergence of continuous Fourier series approximations to a discontinuous function as being 'utterly at variance with the physicist' s notions of quantity' . J. W. Gibbs essentially settled matters in 1899 and this situation has become to be called the Gibbs' phenomenon. It is discussed in many texts but appears to be always focused on the discontinuity of a simple step function. The details for an arbitrary jump discontinuity are illuminating and provide an interesting example to discuss the diVerence between pointwise convergence and uniform convergence. Two proofs are given that the Gibbs' phenomenon only depends on the size of the jump and is a multiple of the integral " º 0 …sin x=x † dx. The demonstration and calculations are suitable for an advanced calculus class and provide very nice applications of Riemann sums and uniform convergence.

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