Could you provide a proof of Euler’s formula: eiφ=cos(φ)+isin(φ)?
Answer
Proof:
Consider the function f(t)=e−it(cost+isint) for t∈R. By the product rule
f′(t)=e−it(icost−sint)−ie−it(cost+isint)=0
identically for all t∈R. Hence, f is constant everywhere. Since f(0)=1, it follows that f(t)=1 identically. Therefore, eit=cost+isint for all t∈R, as claimed.
Attribution
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