Comment by rathel | Hacker News Reader

Comment by rathel | Hacker News Reader

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True, but I don't agree with what he said here @ 8:00

https://youtu.be/ZxYOEwM6Wbk?t=481

Is this really the case? I checked Wikipedia and MathWorld and nobody makes a distinction between e^x and exp(x).

Even if it's a "white lie" for didactic reasons, I don't buy it, it will be much more confusing down the line for students.

Math is about finding structures governed by some rules and then generalizing them. To me the e^x defined as repeated multiplication conceptually is the same thing as the exponential function, just over broader domain. Why? You can interpolate between integer-valued X's using geometric mean. e^3 = sqrt(e^2 * e^4). What stops you from interpolating this recursively to achieve in the limit the exponential function over real numbers?

I think the way he said it is a bit weird (and definitely nobody would think of e^x and exp(x) ss being two different things). But I would agree that it's probably best to define e^x through its power series, especially because it directly generalises to complex arguments. (Another good definition is as the function f whose derivative equals itself, and which satisfies f(0)=1, but that's not very constructive.)

Of course, you can also go another route: you can define the number e (e.g. as the limit of (1+1/n)^n), then you can straightforwardly define natural number powers of e by repeated multiplication, then integer and rational powers through reciprocals and roots (you have to prove n-th roots exist, but that's doable), and then you can define real number exponents via limits (maybe this is what you mean by "interpolating recursively"). Now, when we come to complex numbers, you can use Euler's formula as a definition instead of a theorem, and define exp(a+bi):=e^a * (cos b + i sin b). Of course, at the end of this whole exercise you can prove that this definition is exactly equal to the power series definition.

Another benefit of the power series definition is that it also generalises e.g. to the matrix exponential (exp(A), when A is a matrix).

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