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The statement $e^{ix}=\cos x+i\sin x$ can only be proven if the operation $e^z,z\in\mathbb C$ is already defined, otherwise $e^{ix}$ makes no sense. Students are likely to see $e^{ix}=\cos x+i\sin x$ before they see any other definition of exponentiating complex numbers, so the short answer to your question is yes, this identity should be treated as a ...

8

In my view Euler's Formula $e^{ix} = \cos(x)+ i \sin(x)$ and the resulting extension ( this is (5.) in Steven Gubkin's excellent complex answer ) of $e^{x+iy} = e^x( \cos y + i \sin y)$ is not a good definition for the exponential because it is special to the context of $\mathbb{C}$. In other algebras the interplay between the exponential function and ...

1

First, we start with those equations for sine and cosine with an infinite number of terms: \begin{align} \sin x &= \left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} - \cdots \right) \\[8pt] \cos x &= \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \cdots \right) \\ \end{align} We also take ...

4

A guiding principle for operations extension is the principle of permanence: a definition of an operation should be extended from a restricted domain to a wider one in such a way as to conserve the crucial properties of the operation. For example, the crucial algebraic properties of exponentiation: a#(b+c) = a#b a#c, (a#b)#c = a#(bc) and (ab)#c = a#c b#c ...

5

I'll put out there that it might not be helpful to have this as a definition because it wasn't discovered in that way. That is to say, Euler discovered this formula by clever manipulation of $\sin$ and $\cos$ in ways that might not quite pass muster today in terms of rigor, but were still pretty cool. According to the linked article, it used 3. in Steven's ...

14

I think it is worth acknowledging that there is a huge list of equivalent definitions of the complex exponential. Any presentation should be explicit about the fact that choosing one definition over others is a choice of convenience, or taste, rather than being canonical. Showing that all of these definitions are equivalent is the real work. Here are some ...

2

I am grateful to Mark Conger for finding a video of this presentation and getting the University of Michigan to digitize it. It isn't produced in the style of math youtube content, because that was presumably not the goal of QED TV, which I do not know the history of. But it is found! I finally have a link to the video! Here it is: The video spends a lot of ...

1

Chris. I know you are really excited about this idea, but I'm very worried about a critical omission from your list of considerations. I notice that you teach at a college with a lot of faculty members. One of the biggest blind spots that many faculty (and apparently, you) have is which publisher resources their fellow faculty use. Some use online ...

2

Books about middle school algebra and trigonometry written by someone like Gelfand are rare (I don't know other examples; for example V.I. Arnold's books nomially for children are aimed at very special children, their author's claims to the contrary), but there are many undergraduate textbooks written by superb mathematicians. I list below some that occur to ...

0

This may not qualify, as it is a stretch to classify this as a "textbook." It arose from a series of lecture Klein gave to high-school teachers. Klein, Felix. Elementary Mathematics from an Advanced Standpoint: Geometry. Transl. from the Third German Ed. by ER Hedrick and CA Noble. Dover, 1939.           I love this (...

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