Learn more. Asked 5 years, 5 months ago. Active 5 years, 4 months ago. Viewed 9k times. In Euler's book on complex functions he used the following proof. But was this his first proof? Improve this question.
Martin 10 10 bronze badges. Is this how he wrote the proof? Add a comment. Active Oldest Votes. Improve this answer. Could you provide a link to Cotes proof? It is likely included in his posthumously published Harmonia Mensuraum, but no online PDF appears to be available.
Cotes was also the first to derive the decimal expansion of e, also misattributed to Euler. An issue with Cotes' statement of the Euler identity is that, as we now understand, the ln function is multi-valued over C. Sign up or log in Sign up using Google. Sign up using Facebook.
Sign up using Email and Password. It certainly didn't to me when I first saw it. What does it really mean to raise a number to an imaginary power? I think our instinct when reasoning about exponents is to imagine multiplying the base by itself "exponent" number of times. But this line of reasoning leads to a dead end when we have an imaginary exponent, so we can't use that definition of exponentiation.
We'll have to use another definition to prove that this is true. So I will go ahead and prove it two different ways to convince you that it makes sense, and then I'll show how it actually makes life easier in spite of how strange it may seem.
Proving it with a differential equation I'm going to start with an easy proof of this using some basic calculus. That was easy! Visualizing Euler's Formula So hopefully now you can buy Euler's formula; if not at a concrete level, at least at some abstract level by noting that everything "works" mathematically if you accept that i is the square root of Now it's difficult to try to visualize it unless we come up with some conventions.
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