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When Euler discovered/invented $e^{ix} = \cos(x)+i\sin(x)$. Did he doubt his calculations for a length of time? Was it Readily accepted by the mathematical community quickly or did they object at first? If so when was it finally accepted and why?

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Work with power series was innovated by James Gregory and Isaac Newton almost a century earlier so the kind of arguments Euler presented in his 1748 Introductio were easily digestible by his contemporary mathematicians. The Introductio is full of calculations with power series a lot more complicated than the one you mentioned. I have seen no evidence that anybody doubted Euler's formula. His infinite product formula for the sine function was far more innovative; that also seems to have elicited no objections. For a detailed study of the formula see for example this acticle.

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  • $\begingroup$ Other than the his work on the Basil problem are there any large contributions he made that were harshly doubted? $\endgroup$ – shai horowitz May 20 '16 at 8:08
  • $\begingroup$ In chapter 3 of the Introductio there is a lengthy discussion of the philosophical foundations of the infinitesimal method. This may have been a reaction to Georg Berkeley's criticism or to criticisms by other authors. However, Berkeley explicitly stated that he is not criticizing the results of infinitesimal calculus; Rolle tried that thirty years earlier, and was greatly embarrassed as a result when his own mistakes were exposed. Rather, Berkeley objected on philosophical grounds; the same kind of objections resurfaced in Cantor a century and a half later, and can still be seen on... $\endgroup$ – Mikhail Katz May 20 '16 at 8:12
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    $\begingroup$ ...SME today; see for example this answer and the comments following. $\endgroup$ – Mikhail Katz May 20 '16 at 8:13
  • $\begingroup$ Cant edit comment but I want it noted for math's history. Basil is an herb. Basel is a city with a great problem named after it. $\endgroup$ – shai horowitz May 20 '16 at 8:19
  • $\begingroup$ :-) The formula for sine as an infinite product also provides a proof of the Basel problem; see that article. @shai $\endgroup$ – Mikhail Katz May 20 '16 at 8:27

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