The tidiness of elementary mathematics: Unterschied zwischen den Versionen

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Version vom 9. November 2011, 00:34 Uhr

We all expect a certain tidiness to mathematics, and for many this provides a gratifying aesthetic pleasure. However, there are some wrinkles. One of these arises in the integral of x^k. Calculus provides us with the formula \int x^k\mathrm{d}x = \frac{x^{k+1}}{k+1}. However, this equation is only correct for k\neq -1. If we try to take k=-1 the right hand side is meaningless because we have a zero on the denominator of the fraction, i.e. \frac{1}{0}. But in this case a separate argument gives the answer \int x^{-1}\mathrm{d}x = \int \frac{1}{x}\mathrm{d}x = \ln(x).

Our expectation is that these two formulae should be reconciled. Indeed, if we let k approach -1 in the first we should end up with the second, but that fails to happen. For each x, \lim_{k\rightarrow -1} \frac{x^{k+1}}{k+1} is undefined.

This vingette explores this issue.

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