4
votes
0answers
53 views
+50

Conformal equivalence of resistance

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 resistance I mean the resistance that would ...
0
votes
1answer
41 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
12 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 ...
0
votes
0answers
7 views

Conceptual explanation for what happens to euclidean center and radius of pseudohyperbolic disk as pseudo hyperbolic center tends to unit circle

If $P(w, r)=\{ z \in \mathbb{D} | \rho(z, w)<r\}$ where $\mathbb{D}$ is the unit disk, and $\rho$ is the pseudohyperbolic metric, can someone please explain on a conceptual level how the Euclidean ...
1
vote
2answers
19 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
38 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
36 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
23 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
23 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
16 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 ...
0
votes
2answers
53 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
31 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
63 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
28 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
42 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
151 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
50 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
57 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
65 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
86 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
46 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
42 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
162 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
51 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
93 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
34 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
35 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
96 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
38 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
81 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
2
votes
1answer
64 views

conformal mappings of Riemann sphere

We know that Möbius transformations are the conformal selfmaps of the Riemann sphere. But what does being conformal at infinity correspond to?what does it mean?
1
vote
1answer
23 views

Can there exist an FLT mapping UHD to UHP?

Can there exist a fractional linear transformation mapping the upper half-disc conformally onto the upper half-plane? I think there cannot be, because I am looking for where the unit circle and the ...
4
votes
1answer
85 views

Conformal mapping of disk, surjective, not injective

Is there an example of a conformal mapping of the disk onto itself which is not injective? If not, how may we prove there does not exist such a map? This came up in the answer to this question.
4
votes
1answer
71 views

Which conformal maps UHP$\to$UHP extend continuously to the closure?

Does every conformal map of the upper half-plane $\{\text{Im }z >0\}$onto itself extend continuously to a map from its closure $\{\text{Im }z \geq 0\}$ to itself? If not, which ones do? In ...
1
vote
1answer
53 views

Dirichlet problem on a ring

For the domain \begin{equation} E_r=\{z: r < |z| < 1\}, \end{equation} find a harmonic function $ u $ in $ E_r $ so that \begin{equation} u(re^{i\theta})=1 \text{ and } u(e^{i\theta})=0 ...
3
votes
0answers
202 views

A conformal mapping onto a region bounded by convex contours (Ahlfors)

I want to solve the following exercise (from Ahlfors' text, page 261) *3. Using Ex. 2, show that $p + q$ maps $\Omega$ in a one-to-one manner onto a region bounded by convex contours. Comments: ...
1
vote
1answer
31 views

Why do these properties determine this map up to an additive constant?

In Ahlfors' text, he considers a region $\Omega$ bounded by analytic contours $C_1, \dots, C_n$. On page 259 he states: We have thus established the existence of a function $p(z)$ which is ...
2
votes
1answer
229 views

Is my proof correct? (Conformal equivalence of two circular annuli)

I want to show that the two annuli $$A=\{r<|z-z_0|<R\} $$ $$A'=\{r'<|z-z_0'|<R'\} $$ are conformally equivalent (i.e. there exists a biholomorphic map between the two) iff ...
1
vote
1answer
96 views

Find explicitly a conformal mapping $f$ of $R$ onto the unit disc $U$ such that $f(1)=0$ and $f'(1)\gt 0$

Question Let $R$ be the complex plane with the non positive real axis taken out. Find explicitly a conformal mapping $f$ of $R$ onto the unit disc $U$ such that $f(1)=0$ and $f'(1)\gt 0$ ...
4
votes
1answer
109 views

Construct a conformal mapping from $\Bbb C$ Onto $R$ if such a map exists. And explain why if does not exist.

Let $R$ be the domain obtained by removing the non negative real numbers from $\Bbb C$. Construct a conformal mapping from $\Bbb C$ Onto $R$ if such a map exists. And explain why if does not exist. ...
6
votes
1answer
66 views

Prove or disprove that $f$ is Möbius transformation.

The value of analytic function $f(z)$ is defined to be the value of $f(1/t)$ at $t=0$ as an element of $\Bbb C\cup \{\infty\}$. We can consider a meromorphic function as a function from $\Bbb C\cup ...
0
votes
1answer
44 views

Conformal mappings of multiply-connected regions

I'm reading about conformal mappings in Ahlfors' text, and I got to this section: In the following $\Omega$ denotes a plane region of connectivity $n > 1$. The components of the complement are ...
2
votes
1answer
284 views

Finding a conformal map from semi disk to upper half plane.

Find a conformal mapping $f$ of semi-disk$S=\{z: \vert z\vert \lt 1, Im z\gt 0\}$ onto the upper plane. Again I used composition of conformal map. First of all, let's define a conformal map ...
4
votes
0answers
217 views

Find a conformal map from the disc to the first quadrant.

Find a conformal mapping of the disk $x^2+(y-1)^2\lt 1$ onto the first quadrant $x, y \gt 0$ I did something, which may be false or not, I cannot exactly say anything. I used the composition of ...
2
votes
1answer
400 views

How to find a conformal mapping of the first quadrant.

Find a conformal mapping of the first quadrant onto the unit disc mapping the points $1+i$ and $0$ onto the points $0$ and $i$ respectively. I think that i need to use "the change of variables ...
2
votes
2answers
204 views

Conformal map example $ f(z)=e^z$

I an studying the example-1. I understand $f(z)=e^z$ has a nonzero derivative at all points, hence it is everywhere conformal and locally $1-1$. But I dont understand th part I underlined with ...
3
votes
2answers
120 views

Why not $f(z)=z^2$ conformal at $z=0$?

$$f(z)=z^2$$ is not conformal at $z=0$ Why? Conformal definition: $f$ is conformal at z if f preserves angles there.
1
vote
1answer
77 views

Book suggestion- complex analysis -conformal mapping.

I am studying complex analysis. And I am using J. Bak and D.J. Newman's book.(springer) And now my studying topic is conformal map. In addition to this book, I want to learn other book names which ...
1
vote
0answers
39 views

Conformal mapping of non-convex finite domain

Can a non-convex finite domain be conformally mapped to the unit circle? I am interested in finding such map for this one: a square of length 1 with quarter circles of radius 1/3 removed from each ...
0
votes
1answer
74 views

Is my proof correct? (Invariance of subharmonicity under a conformal map)

I want to prove the following (exercise from Ahlfors' text): Prove that a subharmonic function remains subharmonic if the independent variable is subjected to a conformal mapping. Here is my ...