Thursday, 10 May 2018

complex analysis - Where does this equation come from?




Since I study 3 years i ask myself very often where does this equation come from?

eiθ=cos(θ)+isin(θ)
Is it found by series expansion?


Answer



This result is commonly shown via Taylor series, as explained in the comments, and is well-known. I'd like to offer a different sort of proof, for those who are interested, that I believe is easier yet less well-known.



Consider the second order linear differential equation
y
We know the most general solution is:
y = A\cos{x}+B\sin{x}
But y = e^{ix} is also a solution, and by existence and uniqueness theorems, that means e^{ix} = A\cos{x}+B\sin{x}

for some A,B. Plugging in x=0 for the expression and its first derivative, we see that A = 1, B = i.



Thus, e^{ix} = \cos{x}+i\sin{x}


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