0
votes
1answer
24 views

Stereographic projection is conformal — from the line element

I'm looking over some fairly basic stuff on complex methods and the book I'm using takes the formula for the stereographic projection: $$z = \cot(\beta/2)e^{i\phi} $$ as well as the line element on ...
1
vote
0answers
35 views

Weird conformal map problem

Construct a conformal map from the region $\omega$ = open disk of radius 1 centered at 0 minus the closed disk of radius 0.5 centered at 0.5 to $\mathbb{D}$ = disk radius 1 centered at 0. I really ...
3
votes
1answer
56 views

Let $f(z) = z + z^2$ and let $V = \displaystyle \{z \in \mathbb{C} : |z| < \frac{1}{2}, \frac{3\pi}{4} < arg\{z\} < \frac{5\pi}{4}\}$.

Let $f(z) = z + z^2$ and let $V = \displaystyle \{z \in \mathbb{C} : |z| < \frac{1}{2}, \frac{3\pi}{4} < arg\{z\} < \frac{5\pi}{4}\}$. $(a)$ Show that $f(V) \subset V.$ $(b)$ Let $f_n$ be ...
1
vote
1answer
23 views

Find sup$\{|f′(3)| : f$ maps $Ω$ analytically into the unit disk $\}.$

Let $Ω=\{z=x+iy∈C : |y|<x\}.$ Find sup$\{|f′(3)| : f$ maps $Ω$ analytically into the unit disk $\}.$ Okay. So I can find a conformal map from $Ω\rightarrow \mathbb{D}$. I used the map $f(z) = ...
0
votes
1answer
113 views

Finding the transfinite diameter of the level sets of complex logarithm

Given a simply-connected domain $|g(z)|\ge C$ how can I find the analytic conformal mapping guaranteed by the Riemann mapping theorem? In particular I'm interested in finding the transfinite diameter ...
1
vote
1answer
40 views

Conformal mapping between symmetric region and unit disc

Exercise 3 of VII.4 of Conway's Complex Analysis states Let $G$ be a simply connected region which is not the whole plane, and suppose that $\bar{z}\in G$ whenever $z\in G$. Let $a\in ...
1
vote
1answer
27 views

holomorphic functions with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
5
votes
2answers
154 views

Find a conformal map from semi-disc onto unit disc

This comes straight from Conway's Complex Analysis, VII.4, exercise 4. Find an analytic function $f$ which maps $G:=$ {${z: |z| < 1, Re(z) > 0}$} onto $B(0; 1)$ in a one-one fashion. ...
3
votes
1answer
53 views

Conformal mapping of the domain bounded by a line segment and a circular arc

I am trying to construct a conformal map from the region $R$ which is the set of points in the complex plane bounded by the segment connecting $i$ and $1$ and the part of the unit circle in the first ...
2
votes
0answers
59 views

A hard Conformal Mapping problem

I am trying to construct a conformal map from $R = \{z \in \mathbb{C} : -1 < Re(z) < 1$ and $Im{(z)} > 0\} \cap \{z \in \mathbb{C} : |z| > 1\}$ to the unit disk $\mathbb{D}$. I am really ...
2
votes
1answer
56 views

an inequality derived from conformal automorphisms of unit disk

Let $f$ be a holomorphic function on $D(0,1)$ such that $|f(z)|<1$ for all $z\in D(0,1)$. I have obtained $$ \frac{|f(0)|-|z|}{1+|f(0)||z|}\leq |f(z)|\leq \frac{|f(0)|+|z|}{1-|f(0)||z|}. $$ Is it ...
0
votes
2answers
32 views

The proof of the Area Theorem for Conformal Maps

The Area Theorem: Suppose $f(z)$ is one-to-one and analytic on the punctured unit disk, and is given by $f(z) = 1/z + \sum_0^\infty a_nz^n$ Then $\sum_0^\infty n|a_n|^2 \le 1$ I'm reading the ...
0
votes
0answers
20 views

Conformal map two slits to circles

I am trying to find a conformal mapping that maps a double slitted plane onto a plane with two circles. The two slits are both located along the real axis with similar lengths. For a single slit ...
2
votes
1answer
56 views

Schwarz-Christoffel mapping onto infinite L-shaped region

I'm trying to map the upper half plane onto the infinite L-shaped region $$ \Omega = \{z = x+iy; \ x > 0, \ y > 0, \ \min(x,y) < 1 \} $$ My first try is a Schwarz-Christoffel function $$ F(w) ...
1
vote
1answer
20 views

Find a conformal mapping from lens to first quadradrant

Consider the disks of radius 1 centered at 0 and 1 in the complex plane. Their intersection forms a lens shape. I want a complex function which is a conformal map from this lens to the first quadrant. ...
0
votes
1answer
33 views

Injective polynomial on the unit disc

Let $P(z)=\sum_{k=0}^{n}{a_kz^k}$ be polynomial that is injective in the open unit disc. Show that $|a_n|\le |a_1|/n$. I know that if $P$ is injective function than $P$ is conformal map and therefore ...
1
vote
1answer
24 views

arctan maps the unit disk onto a band around the imaginary axis

Let $D\subseteq\mathbb{C}$ be the unit disk; that is, $D=\{z\in\mathbb{C}:\ |z|<1\}$. Let $B\subseteq \mathbb{C}$ be some band around the imaginary axis: $B=\{z\in\mathbb{C}:\ ...
0
votes
1answer
26 views

Conformal Mapping Question Relating Solution of Laplace Equation on Different Domains

Above is my question. The issue that I'm having is that I find that the given mapping, call it $f$, maps $C_1$ to itself, and $C_2$ maps to a circle of radius $5$ and centre $-2$. As much as this in ...
1
vote
0answers
29 views

Complex Variables Conformal Mapping in Complex Plane of harmonic Functions

Consider the harmonic function $u(x,y) = 1 - y + x/(x^2+y^2)$ on the upper half plane $y > 0$. What is the corresponding harmonic function on the first quadrant $x>0$, $y>0$, under the ...
0
votes
2answers
60 views

Map a half sliced unit disk to upper half plane

"half sliced unit disk" Can somebody tell me how to map this conformally to the upper half plane? I think the symmetry principle should be applied here but stuck on that for hours. Pardon my hasty ...
1
vote
1answer
71 views

Conformal map entire domain to a strip with specific branchcuts

I am looking for a conformal mapping function that maps the entire z-plane to an infinite strip. (e.g. T=f(z) & -b < Real(T) > b ) I hope to find a function that cuts open to original domain ...
6
votes
1answer
101 views

Conformal equivalence of resistance

Link to the question in the physics portal. I'm currently working on a system that uses a logarithmic and a Schwarz-Christoffel transformation to calculate the resistance of a specific area. With ...
0
votes
1answer
64 views

Find a conformal map from the exterior of the closed unit disk to the unit disk

Question: Find a conformal map from the exterior of the closed unit disk to the unit disk. Also, prove that it is indeed a conformal map (bijective and holomorphic along with its inverse). I missed ...
1
vote
1answer
28 views

Conformal Mapping with homeomorphic extension

Suppose that $$D=\{z:0<x<a,0<y<b\}$$ and that $$D'=\{w:0<u<c,0,v<d\}$$ Then there is a conformal mapping $f$ of $D$ onto $D'$ whose homeomorphic extension $\tilde{f}$ to ...
1
vote
2answers
59 views

Conformal Mapping Between Two Domains (log)

Does anyone have a recommendation as how to go about solving this problem? I want a conformal from G to H where $$ G = \{ z \in \Bbb C \ | \ |z|<1, |z+i|>\sqrt{2} \}, S = \{ z \in \Bbb C \ | \ ...
0
votes
1answer
54 views

Prove that a pseudo-hyperbolic ball is a Euclidean ball. Find the radius and center of the Euclidean ball.

We have that the pseudo-hyperbolic metric in the open unit disk $\mathbb D$ is defined by $$ \rho(z,w) = |\phi_w(z)|, \qquad \phi_w(z) = \frac{w - z}{1 - \overline w z}$$ where $z,w \in \mathbb D.$ ...
2
votes
1answer
159 views

Conformal map from exterior of unit circle to upper half plane

I'm trying to find a conformal map from the space $\Omega = \mathbb{H}\setminus\{z : |z-\frac{i}{2}|\leq\frac{1}{2}\}$ to the upper half plane. I think I'm most of the way there, but I wanted to check ...
1
vote
1answer
43 views

Schwarz-Christoffel transformation understanding

I've been reading this explanation (with pic and formula) about the Schwarz-Christoffel mapping. I'm not really used to this sort of argument. My question is why are all terms constant in $(21.3)$ ...
0
votes
1answer
41 views

Does this conformal map from a rectangle exist?

It is well known (by Schwarz-Christoffel) that if $k \in (0,1)$, then the Jacobi elliptic function $\mathrm{sn}(\cdot,k)$ provides a biholomorphic map from the rectangle $(-K(k),K(k)) \times ...
2
votes
1answer
40 views

The fixed points of analytic self-maps of $\mathbb{D}$

So far, I have assumed that $z_1$ is a fixed point of an analytic self map of $\mathbb{D}$. Then, I summoned the conformal self map of $\mathbb{D}$, $\phi$ to take $z_1\to 0$. It follows from Schwarz ...
1
vote
2answers
162 views

conformal map/Mobius transformation from annulus to $\mathbb{C}\setminus \overline{D(0,1)}$

Does there exist a conformal bijection/Mobius transformation from the open unit disk to the whole complex plane? Does there exist a conformal bijection/Mobius transformation from the annulus $\{z\in ...
0
votes
1answer
62 views

Conformal mapping from exterior of semi disk onto exterior of unit disk

Can you construct a conformal mapping from exterior of upper semi-disk onto exterior of unit disk and fixes infinity?
0
votes
1answer
115 views

Conformal maps from the upper half-plane to the unit disc has the form

Prove that the conformal maps from the upper half-plane $\mathbb{H}$ to the unit disc $\mathbb{D}$ has the form $$e^{i\theta}\dfrac{z-\beta}{z-\overline{\beta}},\quad\theta \in \mathbb{R} \text { and ...
0
votes
1answer
39 views

Conformal Mapping and Relating Solutions (of Laplace) of Domains (via the Mapping)

Find a conformal equivalence between the following domains: the strip $ S = \{ z \in \Bbb C \ | \ 0 < \Bbb Im(z) < 1 \} $ and the quadrant $ Q = \{z \in \Bbb C \ | \ \Bbb Re(z) > 0, \Bbb ...
0
votes
1answer
75 views

Conformal maps from the left half plane to a sector

I'm trying to do a conformal mapping from the left half plane to a sector symmetrical around -X axis. would it be $Z=W^m$ while $m= (\frac{2}{\pi})(\pi-\theta)$. theta is the angle of the sector ...
2
votes
2answers
220 views

Mapping circles via inversion in the complex plane

Consider two unshaded circles $C_r$ and $C_s$ with radii $r>s$ that touch at the origin of the complex plane. The shaded circles $C_1,C_2...C_7$ (labeled in counterclockwise direction sequentially) ...
1
vote
0answers
66 views

Why are Mobius map conformal at infinity?

Why are Mobius map conformal at infinity? I think I'm missing a subtlety! So we know an analytic function, $f$, is conformal at $z$ iff $f'(z) \neq 0$ But we see that the derivative of Mobius ...
1
vote
1answer
97 views

Conformal mapping on unit disk

Let $\mathbb{C}_{-}^r:=\mathbb{C}\setminus(]-\infty,-\frac{1}{r}])$, $~\mathbb{C}_{-}:=\mathbb{C}\setminus(]-\infty,0])$, $~\mathbb{D}:= \{z\in \mathbb{C}\mid |z|<1\}$ and $\mathbb{G}=\{z\in ...
0
votes
1answer
86 views

Conformal map from unit disc $D\setminus\lbrace0\rbrace$ to $\mathbb{C}\setminus\bar D$

Question: If $D$ is the unit disc, find a conformal equivalence from $D\setminus\lbrace0\rbrace$ to $\mathbb{C}\setminus \bar D$ My Attempt: I have no idea how to start this problem... I don't think ...
1
vote
2answers
196 views

What is the conformal equivalence from the half-plane to the unit disc?

Problem: Find a conformal equivalence from the half-plane {z : Re(z) > 1} to the unit disc D. My Attempt: Just a FYI that I am completely new to conformal mapping. Okay, so I know that $$h(z) = ...
0
votes
1answer
95 views

A holomorphic function is conformal

I am trying to show that if a function $f = u+iv$ is holomorphic with $\partial_z f(z)$ always non zero, then $f$ is a conformal mapping, i.e. it preserves angles between smooth curves. If $f$ is ...
1
vote
1answer
55 views

Mirror point with respect to Riemann circle (Möbius transformation)

The problem is "Find a Möbius transformation $w(z)$ that maps the area $Re( z) > 0, |z-1| > 1$ to the strip $0<Re(w)<2$." I realize there are many ways to skin a cat, but what I wanted ...
1
vote
1answer
334 views

Conformal map from unit disk to strip

I have the following question: Write down the solution $u(x, y)$ to the Dirichlet problem for the following region and boundary conditions: $U = \{x + iy : 0\le y\le1\}; u(x, 0) = 0, u(x, 1) = 1$. ...
2
votes
1answer
70 views

Conformal map taking quadrants to half-planes

The function $f$ takes the values indicated in the picture. It has poles at $1$ and $-1$ and $f(i)=f(-i)=1$. Elsewhere it is analytic and maps quadrants one-to-one onto the indicated ...
3
votes
2answers
116 views

Map the area to the left and below $xy=1\quad (x,y>0)$ to the UHP

The region to the left and below the hyperbola $xy=1$ is to be mapped conformally and one-to-one onto the upper the half-plane. (See picture below.) Ideas: The only thing I've seen before in ...
2
votes
1answer
36 views

Bound on $f'(i)$ for $f:\text{UHP} \to \text{ region right of a hyperbola }$

Denote $$H=\{z=x+iy: x,y\in \mathbb{R}, y>0\}, \quad \text{and}\\[12pt]V=\{z=x+iy: x^2-y^2>1, x>0\}.$$ Let $$\mathcal{S} = \{f:H\to V, \,\,f \text{ is holomorphic and } f(i)=2\}.$$ ...
0
votes
1answer
37 views

Prove that real f(3)>3

Let $f$ be a conformal mapping of the domain $$ \Omega = \{z:\Re(z)>0\}-(0,1] $$ onto the domain $\{z:\Re(z)>0\}$ so that $f(2)=2$, $f^\prime (2)>0$. Prove that we have real $f(3)>3$. I ...
2
votes
3answers
113 views

Map $\{x+iy \mid x^2+y^2<1 \text{ and } x^2 + (y-1)^2<2\}$ conformally to UHP

From an old qualifying exam: Let $D$ be the domain $$D :=\{x+iy \mid x^2+y^2<1 \text{ and } x^2 + (y-1)^2<2\}.$$ Map the domain onto the upper half-plane. Obtain a function $f(z)$ ...
1
vote
1answer
42 views

Images of two arcs under $z+1/z$

Let $C$ be the circle $|z-ai|=\sqrt{1+a^2}$, where $a>0$. Let $f(z)=z+\frac{1}{z}$, where $z\in \mathbb{C}$. Let $C_1, C_2$ be the two arcs on $C$ determined by $-1$ and $1$. I am looking for an ...
-1
votes
1answer
135 views

The image under mapping $w=(z+i)/(z-i)$, of the third quadrant?

The title says it all. I am not sure how to approach this problem. The only related problems i have done is mapping a (unbounded)line /circle to a line/circle. Regards Exatic