History
It was Johann Bernoulli who noted that
And since
the above equation tells us something about complex logarithms. Bernoulli, however, did not evaluate the integral. Bernoulli's correspondence with Euler (who also knew the above equation) shows that Bernoulli did not fully understand logarithms. Euler also suggested that the complex logarithms can have infinitely many values.
Meanwhile, Roger Cotes, in 1714, discovered that
(where "ln" means natural logarithm, i.e. log with base e). We now know that the above equation is true modulo integer multiples of, but Cotes missed the fact that a complex logarithm can have infinitely many values due to the periodicity of the trigonometric functions.
It was Euler (presumably around 1740) who turned his attention to the exponential function instead of logarithms, and obtained the correct formula now named after him. It was published in 1748, and his proof was based on the infinite series of both sides being equal. Neither of these mathematicians saw the geometrical interpretation of the formula: the view of complex numbers as points in the complex plane arose only some 50 years later (see Caspar Wessel).
Read more about this topic: Euler's Formula
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