# conformal

1. ### Complex Analysis: Conformal Mappings

I am looking for conformal transformations to map: 1. Disk of radius R to equilateral triangular region with side A. 2. Disk of radius R to rectangular region with length L and width W. 3. Disk of radius R to elliptic disk with semi-major axis a and semi-minor axis b. Thanks!
2. ### Conformal and non-conformal transformations

It is well known that from a two-dimensional solution of Laplace equation for a particular geometry, other solutions for other geometries can be obtained by making conformal transformations. Now, I have a function defined on a disc centered at the origin and is given by f(r) = a r where a is...
3. ### Conformal and non-conformal mappings

My understanding is that conformal mapping is restricted to analytic functions. What sort of mapping (if any) that can be used for non-analytic functions?
4. ### Conformal mapping

Is there a conformal mapping that transforms regular polygons (e.g. triangle and square) to circle?
5. ### Show f is a conformal mapping

My working: A basis for the tangent space (at any point) is (1,0), (0,1) so I need only show it for this . df(1,0)=df/dx and df(0,1)=df/dy Is this the right approach? Thanks
6. ### Conformal map of this region

Hi everyone, I have the following region S = { z : 0 < Im(z) < PI } in C and I'm looking for its image under z --> w = (1+ie^z)/(1-ie^z). I'm aware of a technique of finding images by looking at what happens to the boundaries of the region being transformed. In this case, it looks like I have...
7. ### Proof, bijective aplication and conformal mapping

Hi there. I have to prove that if f:A \rightarrow B it's a bijective and analytic function with analyitic inverse, then f is conformal. I think I should prove the angle preserving using the analyticity, but I'm not sure how.
8. ### Conformal equivalence between rectangles

For 0<a<1, let R_a be the rectangle whose side lengths are a and 1. Show that every rectangle is conformally equivalent to one and only one of this rectangles. Hint: use the schwarz reflection principle and the fact that every ring in C (subset whose complement has two components) is...

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10. ### Conformal mapping of an infinite strip

Can anyone explain to me why the exponential function maps the strip { z \epsilon \mathbb{C} : - \Pi / 2 < Imz < \Pi / 2 } onto the RHP? Thanks!!
11. ### Conformal mapping Q - find minor and major axis length of the ellipse on semi circle

Hi, i really dont know where to start with this q. there are 3 parts. Help on any part of the q is really appreciated. Thankyou for any help in advance. :) The question is attached below.
12. ### Complex open mapping & conformal mapping problems.

1. Let f:D(0,2) -> C be a holomorphic function. Suppose that f is real-valued on the set {|z| = 1}. Show that f is a constant. 2. Can you find a 1-1 conformal map from {z complex number | 0 < |z| < 1} onto the annulus X = {z complex | 1 < |z| < 2}. Thanks for helping.
13. ### Mathematical methods - Conformal mapping Q

Hi, im really stuck on this question, Thankyou for any help in advance. :) 4) i) For the line segment Lz and the semi-circle Cz Lz = {x, y : x = ln r, 0 < y < pi} Cz = {r, θ : r ∈ R+, 0 < θ < pi}, show that the function f1(z) = ez maps Lz onto Cz. ii) Determine the length of the...
14. ### Conformal Mapping to Unit Disc

Problem: Find a conformal map from the set A = { z = x + iy in C : x^2 + y^2 > 1, x > 0} to the unit disc {z in C : |z| < 1}. In class we've been through some examples of this form where we construct a conformal map as a composition of mobius maps and other analytic functions. (exp(z), z^n...
15. ### Conformal Proof

Here's an interesting question I can't seem to get my head around. A function f is called a diffeomorphism if J_f \neq 0, where J_f = u_xv_y - u_yv_x. Let \alpha_1(t) and \alpha_2(t) denote contours C_1 and C_2 in a domain D, where \alpha_1(a)= \alpha_2(b)=z_0. Let \Theta(C_1,C_2) =...
16. ### Conformal equivalences (biholomprphic self-maps)

The question is Find all biholomorphic self-maps (conformal equivalences) of \mathbb{C}\setminus \{ 0\}. Also find all biholomorphic self-maps of \mathbb{C} \setminus F where F=\{ z_1,z_2,...z_n\} is some finite subset. I was thinking about the linear fractional transformations from...
17. ### Conformal preserves harmonicity

Mobius transform (Apologies for the incongruous title: I solved a problem in the middle of typing it, but have a different one now on Mobius transforms.) [math Let f(z) = \frac{az+b}{cz+d} be the unique mobius transformation
18. ### Conformal Reflection across Real Axis

I have an exercise and an approach to it's solution with a gap in it. The exercise comes from Ahlfors Complex Analysis text in Ch. 6 on Conformal mappings. Suppose that f is a Riemann map (conformal) from a simply connected domain \Omega \varsubsetneqq \mathbb{C} to the open unit disk...
19. ### conformal map, upper half-plane

Find a conformal map w(z) of the right half-disk \{ \text{Re}(z), |z|<1 \} onto the upper half-plane that maps -i to 0, +i to \infty, and 0 to -1. What is w(1)? The back of the book says that w=\frac{-(z+i)^2}{(z-i)^2}, w(1)=1. However, I still do not see how they get...
20. ### conformal map, strip

Find a conformal map w(z) of the strip \{ \text{Im}(z)<\text{Re}(z)<\text{Im}(z)+2 \} onto the upper half-plane such that w(0)=0, w(z) \rightarrow +1, as \text{Re}(z) \rightarrow -\infty, and w(z) \rightarrow -1 as \text{Re}(z) \rightarrow +\infty. Sketch the images of the straight lines \{...