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Euler's Theorem (1 Viewer)

Arrowshaft

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This approach would have to establish that the differential operator applies to a complex function e^(ix) in the same way as a real function e^(kx)...which is not trivial.
I don’t remember it exactly as I haven’t covered complex analysis yet, but he’s fairly credible, I may just be recalling it bad
 

Trebla

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I would rather call it a definition.

I don't think there exists a rigirous "proof" unless you make various assumptions that complex analysis behaves similarly to real analysis.
The syllabus treats Euler’s formula as if it is a definition rather than an actual result. That means the only benefit of introducing it is simply notation.

But doing so does not explain why you can suddenly exponentiate a complex number and why this suddenly relates to the polar form of complex numbers.
 

stupid_girl

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I don’t remember it exactly as I haven’t covered complex analysis yet, but he’s fairly credible, I may just be recalling it bad
I'm not saying anyone is not credible. However, any attempt to "prove" this formula will require some "hidden" assumptions that are definitely out of reach for 4U students. I have seen various versions of sloppy "proof" that omits those assumptions.
 

Arrowshaft

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I'm not saying anyone is not credible. However, any attempt to "prove" this formula will require some "hidden" assumptions that are definitely out of reach for 4U students. I have seen various versions of sloppy "proof" that omits those assumptions.
ok boomer.
jk I agree
 

stupid_girl

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The syllabus treats Euler’s formula as if it is a definition rather than an actual result. That means the only benefit of introducing it is simply notation.

But doing so does not explain why you can suddenly exponentiate a complex number and why this suddenly relates to the polar form of complex numbers.
You need to define e^x in the complex plane before you can exponentiate a complex number. There are different ways to do it.

One approach is to define e^x as a complex power series, which would lead to a question why it converges in the same way as the real counterpart.

Another approach is to define y=e^x as the solution to the complex differential equation dy/dx=y, ie. assume e^x can be differentiated in the same manner as a real function. This is not trivial for 4U students.

Yet another approach is to define e^x as a complex limit, which would lead to a question why the limit exists in the same way as the real counterpart.

Taking this formula as the definition is a cleaner approach for me because it defines real part and imaginary part of e^x separately, which makes the derivation of other results straight forward.
 
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HeroWise

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"e^x is a shaky definition, only works by power series "
- Some math guru
 

stupid_girl

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"e^x is a shaky definition, only works by power series "
- Some math guru
All the mentioned definitions are equivalent. My prof preferred to take this formula as the definition because it avoids introducing complex power series at the very beginning of the course.
 

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