Next page : Euler's identity
What could eiθ mean? Assuming it means anything, and that the result will be a complex number, we could write
Where u and v must be functions of θ. Now let us, of some obscure reason, take the derivative of the above with respect to θ. If we treat i as any constant, we get
If we take the derivative again, we get
Comparing this to the original function we get that
So, what functions would, if we take the derivative twice, give the function back but with a negative sign?
The obvious candidates are the sine and cosine functions. How to find which is which? We know that e0=1, so if θ=0 then u must be cosine, and v must be sine, since cos(0)=1 and Asin(0)=0. We thus have
But what value would A have? If we square both sides we get
Looking at the imaginary part we must thus have that
But the double angle formula gives us that A=±1. (If we instead look at the imaginary part we have A in both sides, so we cannot use that to determine the value of A. ) We select 1, and that will finally give us
This is a very important formula, and arguably one of the most beautiful in mathematics, and it will be used extensively in these pages.
How about is we would have chosen A= –1? That would actually work pretty fine. What is the difference between i and –i? In a sense nothing but the sign. All the maths would essentially remain the same.Up a level : Euler's identity
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