Abstract
The short history of the implicit function theorem displays a number of characteristic features of the history of mathematics in the period we are considering. For many years the result was taken for granted, and attention paid to the task of finding explicitly functions defined only implicitly. Then it was realised that there was something to prove, but the task was assigned to complex analysis—which says something about the state of real variable theory. Only much later was the theorem proved in the context of real analysis.
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Notes
- 1.
To control what happens when the choice of initial value \(y_0\) is varied Cauchy added the condition that the partial derivative \(\frac{\partial f}{\partial y}\) is also continuous and bounded as a function of x and y on the interval \(x_0 \le x \le X\), but we need not pursue that here.
- 2.
This account follows the version that he published in 1841. See also Bottazzini and Gray (2013, 150–155).
- 3.
The opaque notation is Cauchy’s.
- 4.
Mentioned without further discussion in Bottazzini and Gray (2013, 378).
- 5.
He used almost the same words in the edition of his lecture notes at Bologna in 1901.
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Gray, J. (2015). Implicit Functions. In: The Real and the Complex: A History of Analysis in the 19th Century. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-23715-2_26
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DOI: https://doi.org/10.1007/978-3-319-23715-2_26
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