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How do you think Euler felt when he discovered/invented $e^{ix}=\cos(x)+i\sin(x)$. Do you think he doubted 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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He proved it; see:

And compare with: William Dunham, Euler: The Master of Us All (1999), page 93, for a "modern" formulation.

Why do you think that:

Was it Readily accepted by the mathematical community quickly or did they object at first? If so when was it finally accepted and why?

The result has been proved by the leading mathematician of his times and it is contained into the fundamental mathematical analysis textbook of his time.

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