41–50 of 62 posts
Here is w = e^z:
https://i.imgur.com/pAALOh2.png
Just look at the above picture and every detail until it starts to makes sense.
3blue1brown also has some nice visual explanations: https://www.youtube.com/watch?v=F_0yfvm0UoU
exp(iy) = 1 + iy + (iy)^2/2! + ...
(the Taylor expansion of cos) + i(the Taylor expansion of sin)I once had an interesting thought about the function e^x. I think this is a key idea in the theory of Lie groups. If the x in e^x = (1+x/N)^N is understood as some transformation, then e^x is essentially repeating an infinitesimal transformation lots of times. So it's like a for-loop where the body of the loop is some infinitesimal transformation. I tried to define the integration operator in terms of e^x. The 1 + x/…
You're close! This can indeed be done properly and is then called the Euler-Maclaurin formula. For this, you define the "shift to the left by n operator" e^(nD) where D is the differentiation operator d/dx.
You then always take the current value of f(x), multiply it by the small shift n to get the first rectangle. Then you shift to the left by n, i.e. to e^(nD)*f(x) = f(x+n), multiply that by the small shift n to get the next rectangle etc.
The book "street-fighting mathematics" [1][pdf] has a very hands-on and playful derivation of this in chapter 6.3.
[1] https://mitpress.mit.edu/books/street-fighting-mathematics
[pdf]https://www.dropbox.com/s/722rlvrwy9l9w73/7728.pdf?dl=1
Earlier quoted context omitted.
I'll be honest, I've never understood how humanity didn't invent Cartesian coordinates until 1637, with all the other engineering we had. Once we had linear equations, for example with the ancient Greeks, not one person ever thought to plot a line with it? Or to use it to calculate the necessary building materials for something like a pediment or cathedral?
I think one of the keys to understanding math, and to a lesser but still significant extent, physics history is to remember what you believed as a child, and how you struggled with the concepts taught to you. And that's even with a math curriculum designed to lead you to modern math. (One can debate how effective it is at that, but that's a separate topic.) Those misconceptions we had as children are pretty fundament…
A sense of the phrase "knowledge is power" aligns with this.
I read Alan Kay's "User Interface - A Personal View" [1] recently, wherein he discusses Seymour Papert's [2] ideas on learning, specifically the 3 stages of learning. I found Papert's conception (with only mild exaggeration) to be illuminating. For example, I now have a explanatory model as to why certain inventions that did not require the du jour technology of the industrial age were developed so late in the game.
[1]: http://www.vpri.org/pdf/hc_user_interface.pdf [2]: https://en.wikipedia.org/wiki/Seymour_Papert
The article is quite right that multiplying by i gives a rotation. But it doesn't quite explain the reason for this: it's because that's the whole point of defining imaginary numbers in the first place! Remember you start off wanting to find a solution for the equation: i^2 = -1 This is actually easier to think about if you multiply it by a general real number r: r i^2 = -r In other words you want i such that if you…
It gets even more interesting when you add additional degrees of freedom so that such rotations can happen via more than one path. For example, quaternions add two extra degrees of freedom, and this lets you have an infinite number of square roots of -1. Any imaginary unit quaternion (i.e. ip+jq+kr where p^2+q^2+r^2=1) is a square root of -1.
The proof of the formula is beautiful. It's common to defined the complex exponential as the extension of the Taylor expansion of exp(x) to the complex plane. Thus, exp(iy) = 1 + iy + (iy)^2/2! + ... Now just group the even-numbered terms together and group the odd-numbered terms together, the i's multiply to become 1 in the even numbered terms, and what you get is (the Taylor expansion of cos) + i(the Taylor expansi…
https://www.mathsisfun.com/algebra/eulers-formula.html
as a reference template, e^-ix would seem to involve:
(taylor cosine series) - i * (taylor sine series)
or e^-ix = cos x - i sin x = -1
which suggests another twist to the familiar identity:
e^ix * e^-ix = (cos x + i sin x) * (cos x - i sin x) = (cos x)^2 + (sin x)^2 = (-1) ^ 2 = 1 = e^0
The proof of the formula is beautiful. It's common to defined the complex exponential as the extension of the Taylor expansion of exp(x) to the complex plane. Thus, exp(iy) = 1 + iy + (iy)^2/2! + ... Now just group the even-numbered terms together and group the odd-numbered terms together, the i's multiply to become 1 in the even numbered terms, and what you get is (the Taylor expansion of cos) + i(the Taylor expansi…
https://www.circuitlab.com/textbook/complex-numbers/
(Thanks to someone who found a small equation typo and emailed me!)
Earlier quoted context omitted.
> it's because that's the whole point of defining imaginary numbers in the first place! People used imaginary numbers for a long time before Cartesian coordinates even existed.
Indeed. There were 18th century mathematicians like Gauss who realized the importance of the geometric nature of complex numbers, but it didn't become central to the subject until the 19th century. The appearance of imaginary numbers as formal square roots of negative numbers goes back to the mid 16th century. As for Cartesian coordinates, i^2 = -1 has an intrinsic, coordinate-free interpretation in terms that would…
Do you know of any resource treating that subject explicitly? I assume you mean the same kind of operator as in 'differential operator'—is that right? I can kinda see it maybe... but would definitely be interested in hearing the idea expanded on :)
Earlier quoted context omitted.
There is certainly a connection between Euler's formula and the matrix exponential, but I think you have confused some details about how e^x is defined. The connection is to consider C as a 2-dimensional real vector space with basis 1,i. Multiplication by i is a linear transformation of this vector space. In more detail: The exponential of a matrix X is an infinite sum just like that of the normal exponential functio…
OK, and where was I confused? e^X could also be defined by \lim_{N \to \infty}(I + X/N)^N for X a linear map. For my intuition, I find that better than your definition.