Euler's formula - properHoc

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What could eⁱ^θ mean? Assuming it means anything, and that the result will be a complex number, we could write

${e^{i\theta }} = u(\theta ) + iv(\theta ) = u + iv$

Where u and v must be functions of θ. Now let us, for some obscure reason, take the derivative of the above with respect to θ. If we treat i as any constant, we get

$i{e^{i\theta }} = u' + iv'$

If we take the derivative again, we get

${i^2}{e^{i\theta }} = u'' + iv''$

$- {e^{i\theta }} = u'' + iv''$

and thus

${e^{i\theta }} = - u'' - iv''$

Comparing this to the original function we get that

$u = - u'',\quad v = - v''$

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

${e^{i\theta }} = \cos \theta + iA\sin \theta$

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

${({e^{i\theta }})^2} = {e^{i2\theta }} = \cos (2\theta ) + iA\sin (2\theta )$

and

${(\cos \theta + iA\sin \theta )^2} = {\cos ^2}\theta - A^2{\sin ^2}\theta + 2iA\cos \theta \sin \theta$

Looking at the real part we must thus have that

$\cos (2\theta ) = {\cos ^2}\theta - A^2{\sin ^2}\theta$

The double angle formula gives us that A=1 and that will finally give us

${e^{i\theta }} = \cos \theta + i\sin \theta$

This is a very important formula, and arguably one of the most beautiful in mathematics, and it will be used extensively in these pages.

(If we instead look at the imaginary part we have A on both sides, so we cannot use that to determine the value of A. )

How about if we would have chosen A= –1? That would 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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Last modified: Dec 28, 2023 @ 14:03