Some comments from Gene, 7/16:
A Geometric Rep: "But before that, let's take a step back and recall that the parametric representation in terms of the sine and cosine functions of a unit circle is sin(x)^2 + cos(x)^2 = 1"
- I think it's confusing to have x here, since we have the x-axis to worry about. Why not use theta so you can leave out "random point A"?
Next paragraph: how about starting with something about (cos theta, i sin theta) is not a point in the real plane R2, since it consists of pairs of real numbers and i sin theta ain't a real number. Then say something about introducing the complex plane C, where everything is cool.
Moving Particle Argument: You might warn folks at first that this makes modest use of calculus--the derivative anyway.
"Let's now "extend" the action of the ordinary exponential function from real values of x to imaginary ones, so that:"
- What you're doing is extending the action of differentiation, not of the exponential function (what is its action?)
"For the purpose of this page, we will only come across derivative of complex numbers that are constant (i), and we will simply assume such derivatives can be manipulated just like real numbers."
"but how are we to understand the derivation in the above function?"
The rest of the Moving Particle Argument needs to be cleared up a bit.
Power Series Argument:
"Recall that the Taylor series of a real of complex function..."
Argument seems pretty good thereafter.
Will finish tomorrow morning. Though I should point this out to you now! G
Swu2 11:44, 21 May 2012 (EDT) I got this
Smaurer1 16:48, 7 June 2012 (EDT)
- Bill, let's talk about the logic of your proofs Friday morning.
- Four technical matters for now.
- \Theta and \theta are different. The standard Greek letter in math is \theta.
- Never write <math>sin x</math>; always write <math>\sin x</math>.
- Look in the history, at the bottom, to see how I have changed your code to illustrate the use of forced blank space in TeX, namely, "\ " and \quad
- Notice how equation numbering is done, with EquationRef2. Look in WikiTricks for the details.
Talk to you soon.
Jorin 09:44, 17 July 2012 (EDT) Great page!
- At the beginning of the moving particle argument section, maybe make the x in the exponent italicized
- In the dot product, maybe have z(t) dotted with i*z(t) to show that it really is zero from the rules of dot products.
- I think in the cos(3theta) section, just show how the sin^2 + cos^2 = 1 can make that transformation.