Next page : Euler's identity

What could *e ^{i}*

*mean? Assuming it means anything, and that the result will be a complex number, we could write*

^{θ}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

or

and thus

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 *e*^{0}=1, so if *θ*=0 then *u* must be cosine, and *v* must be sine, since cos(0)=1 and *A*sin(0)=0. We thus have

But what value would *A *have? If we square both sides we get

and

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.

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