Complex trigonometric functions

> what do the complex trigonometric functions correspond to?

e^(i*z)=cos(z)+i*sin(z)
cos(z)=(e^(i*z)+e^(-i*z))/2
sin(z)=(e^(i*z)-e^(-i*z))/2

The real trigonometric functions are just the real and imaginary components of e^(i*z) for real z.

e^z converts log-polar coordinates to Cartesian coordinates. e^(a+b*i)=e^a*e^(b*i) = e^a*(cos(b)+i*sin(b)). I.e. ln(|e^z|) is Re(z) and arg(e^z) is Im(z).

Does it give any additional insight whatsoever to inspect the actual geometry of complex functions in 4-dimensional space? I see this isn't usually done?
Is the geometric aspect of complex functions somehow trivial?

please dont call that a unit circle, because it is not a circle.

it's some algebraic variety, maybe of interest, but let me tell you what it is not:
a circle

Can someone please post a picture representing the manifold that the complex trigonometric functions correspond to?

Interesting! If x and y are complex, such that x^2 + y^2 = 1, is there a z such that x=cos(z) and y=sin(z)? Is z unique?

How do you prove that?

If you multiply x^2+y^2=1 out, where x = a + i b and y = c + i d, you get

a^2-b^2+c^2-d^2=1
and
a b+ c d = 0
tough to graph

Can someone please render a projection of the Riemann zeta function rotating in 4d space?
Surely this is simplistic for some of you to implement in Mathematica?

The problem with your question is the logic.

You think that trig functions correspond to circles so complex trig functions must correspond to something equally fundamental. They don't.

Suppose
x = cos(z) = (exp(i z ) + exp( -i z))/(2 )
y = sin(z) = (exp(i z ) - exp( -i z))/(2 i ). Then
x+i y = exp( i z). Let
z = u + i v, so
exp(iz)=exp(-v) exp(iu), and
|x+iy|=|exp( i z)| = exp(-v). Then
v=-log(|x+iy|) and
exp(i u) = (x+iy)/|x+iy|. Since the right
hand side is a point on the unit circle,
there is a u, unique up to the addition of 2 pi k
for integer k that satisfies this.

You could also use
x-i y = exp( - i z) = exp(v) exp(-i u), but you get the same answer, because x^2+y^2=1 implies
(x+i y)=1/(x-i y).

There is no logical problem with my question. I didn't assume the complex geometry to be fundamental like the circle is, I just wanted to know what it is. Now that I know, I think it's interesting and would like to know more about it despite it not being "fundamental".

Does complex (algebraic) geometry deal a lot with the actual geometry of complex functions as surfaces in 4d space?

I've been interested in this too and I haven´t found good information on it yet. What I would like is to have some kind of "polar coordinate system" for [math]\mathbb{C}^2[/math] which specifies points by giving a point on [math]x^2+y^2=1[/math], which would have until length in an appropriate sense (which may not be an actual metric) along with a complex-valued direction. Then the trigonometric functions would behave as they are supposed to when looking at triangles, etc., with complex-valued side lengths.

>complex-valued direction

I meant length here, the point on [math]x^2+y^2=1[/math] is the direction.

So does this surface have real dimension 2? Four variables a,b,c and d, and two constraints
a^2-b^2+c^2-d^2=1
and
a b+ c d = 0.

Is there a nice parameterization of the surface by the plane, i.e. C itself, or some nice subset thereof?

Isn't this a fairly interesting thread?

no, geometry stops at 3 dimensions.

Does it?

Complex polynomials, yes.

Affine varieties are literally just the zero locus of some of polynomials. i.e. Take [math]\left\{ {{f_1},..,{f_k}} \right\} \subseteq \mathbb{C}\left[ {{x_1},...,{x_n}} \right][/math] then you can define the affine variety [math]V\left( {{f_1},..,{f_k}} \right) = \left\{ {x \in {\mathbb{C}^n}|{f_1}\left( x \right) = ... = {f_k}\left( x \right) = 0} \right\}[/math].

Even more general algebraic varieties still must look like the zero locus of a set of polynomials locally.

Well, what field would generally deal with the geometry of complex functions as surfaces in 4-dimensional space?

>what field would generally deal with the geometry of complex functions as surfaces in 4-dimensional space?
Surely this question is simplistic to answer in terms of what field of mathematics deals most with the actual geometry of complex functions as they are defined?

We don't just study the graphs of arbitrary functions in geometry. Because the ""surfaces"" these graphs create are not always well behaved.

The behavior of these complex trigonometric functions is just a topic of complex analysis.

>We don't just study the graphs of arbitrary functions in geometry. Because the ""surfaces"" these graphs create are not always well behaved.
There should be a geometric description for the surfaces corresponding to non-well-behaving functions, shouldn't there, at least when the behaviour isn't pathological over some sensible limit, no? Surely the geometric description of such surfaces is far from impossible?

The thing is they aren't always actually surfaces (from either the differential or algebraic point of view).

What are they in those cases?

Are they not surfaces because they aren't manifolds? What are they?

I'm sure such objects are defined somehow in mathematics?

>Is there a nice parameterization of the surface by the plane, i.e. C itself, or some nice subset thereof?

z |---> (cos(z),sin(z)) obviously.

Sometimes things don't have nice structure. I don't know what you want to hear.

What I want to hear is what such not-quite-surfaces can be defined as. What is a suitable generalization that allows them to be defined? They might not have nice structure but they don't have an impossible structure either.

Is that really it, though?

Yes. You have to limit it to a strip -pi=x

Along the imaginary line they turn into their hyperbolical counterparts cos(bi) = cosh(b) and sin(bi) = sinh(b). Other than that I don't know how to interpret them intuitively.

>What is a suitable generalization that allows them to be defined?
"topological space"
You're welcome.

what's the metric and curvatures?

z=x+i y

F=[
Re(cos(z))
Im(cos(z))
Re(sin(z))
Im(sin(z))]

F = [
cos(x) cosh(y)
-sin(x) sinh(y)
sin(x) cosh(y)
cos(x) sinh(y) ]

dF = [
-sin(x) cosh(y), cos(x) sinh(y)
-cos(x) sinh(y), -sin(x) cosh(y)
cos(x) cosh(y), sin(x) sinh(y)
-sin(x) sinh(y), cos(x) cosh(y)]

The metric is conformal!

dF'*dF = cosh(2 y) * I

where I is the 2x2 identity matrix.

Conformal metric:
p(x,y) = (cosh(y))^(1/2)
Gaussian curvature:
K = -(\Delta log p)/p^2

log p(x,y) = 1/2 log cosh(y)
d^2/dy^2 log p(x,y) = 1/2 cosh^(-2)
p(x,y)^2 = cosh(y)

K = -1/2 (cosh(y))^(-3)

Does this mean that for large y the surface gets nearly developable?