Recent content by JungleJesus

Thanks a lot. This was a big help.

I used this result earlier to move z along the arc to the perpendicular bisector of \stackrel{\rightarrow}{ab}, yielding an isosceles triangle inscribed in the three point circle. That didn't seem to help me. Is there a more useful approach?

I am reading Visual Complex Analysis by Dr. Tristan Needham and am hung up on some of the geometrical concepts. In particular, I am having trouble with ideas involving the geometric properties of numbers like: \frac{z-a}{z-b} Note: I am still in the first and second chapters, which deal...

I have the book as a pdf. The process of composing rotations is conceptually simple. Recall that a single rotation about an arbitrary point is a translation to the origin, a rotation about the origin, and then the inverse of the translation to the origin. Composing two rotations means you simply...

I like the idea of series, but I don't think they would be applied in that way exactly. Integrating (3x^2 + 2x +1)^-1 would not give the inverse of f(x) = x^3 + x^3 + x.The existing formula requires that the inverse be known to find the derivative. Once you have the derivative, you can integrate...

Yes. That's exactly what I'm talking about. I want to investigate a general method of finding inverse functions using calculus. For the trig functions, this is easy due to right-triangle trigonometry and the inverse function rule for derivatives. Unfortunately, in its present form, the inverse...

We have a formula for the derivative of an inverse function: dy/dx = 1/(dx/dy). Just how useful is it? Say we want to find the inverse of a complicated function, f(x), on an interval (a,b) on which f(x) is one-to-one. Can we use integration to find such a function? Example: Say we didn't...

I'm thinking more on the abstract side of things. I would like to arrive at a complex ordering that yields the real ordering as a special case. I'm also beginning to look at complex analysis as a major area of study for me. The function G() which I described earlier would surely have some...

Yikes. This is not the way to go. Drop the order axioms. I just want a way to say that any two complex numbers satisfy trichotomy (<, >, =). I was hoping to do this by mapping C onto R via a one-to-one function and then comparing the real numbers produced by that function. Strictly speaking...

Sorry about that. I know exponentiation will not preserve order, but the lexicographic ordering that I am familiar with only allows order to be preserved if a constant is added to both sides. If a > b and c > 0, then I would like ac > bc. The lexicographic ordering fails here due to the...

If I can have a bijection, I would like it to be continuous. I need to be able to use it to define an ordering of the complex numbers that satisfies the properties of order on the real number line (the order axioms).

Can a function be defined such that for a complex argument z = x + iy, the function will uniquely map z onto the real number line? I have a hunch that this would not be possible, but if such a function existed, it could be used to define a unique ordering of the complex numbers without the need...

Cool. I'll look into this process in more detail. Who knows, I might even write the book I'm looking for. Thanks for the attention.

A standard complex number can be conceptualized as a pair of real numbers (x,y) of the form "x + iy" where i is the square-root of -1. I am wondering if/how the non-standard complex numbers can be constructed. By my intuition alone, I would construct the non-standard complex number with a...

Ahh. So the Transfer principle allows the complex structures to be defined for the non-standard model. Very interesting. Thank you, Hurkyl. Does anyone know where I could find an online book or something along those lines? I know Robinson briefly mentioned the complex extension in his...