# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $f(D)$?
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $z,w \in \mathbb D.$ ...
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### 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 ...
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### 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 $(21.3)$ ...
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### 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?
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### 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 ...
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### 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 ...
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### 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: $U = \{x + iy : 0\le y\le1\}; u(x, 0) = 0, u(x, 1) = 1$. ...
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### 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 ...
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### 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 ...
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### 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 $$\mathcal{S} = \{f:H\to V, \,\,f \text{ is holomorphic and } f(i)=2\}.$$ ...
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### 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 ...
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### 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 $f(z)$ ...
### 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 ...