Questions tagged [mobius-transformation]

For questions about the geometry, complex-analytic, and group-theoretic properties of the Möbius transformations (linear fractional transformations) $z \mapsto \frac{az + b}{cz + d}$ of the complex plane, which can be identified with the group $PGL(2, \mathbb{C})$, or certain subgroups thereof.

457 questions
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What is the radius and center of the image of $|z|=1$ under $f(z) = \frac{3z+2}{4z+3}$?

I would like to compute the image of the circle $|z|=1$ about the fractional linear transformation: $$f(z) = \frac{3z+2}{4z+3}$$ In particular, I'd like to compute the new center and radius. ...
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Finding bilinear transformation which maps $|z|=1$ on to $|w|=1$ [closed]

How can I show that every bilinear transformation which maps $|z|=1$ on to $|w|=1$ must be of the form $$w=K\frac{z-\alpha}{\overline{\alpha}z-1}$$ where $|K|=1$? Please help me. Thanks.
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Mobius transformation producing a curved triangle with 3 intersecting circles

Let $ABC$ be a curved triangle on a plane, whose side $AB$, $BC$ and $CD$ are arcs of circles $S_1$, $S_2$ and $S_3$ passing though a point $D$ (i.e. $S_1∩S_2∩S_3 = D$, $D≠A$, $D≠B$, $D≠C$). Assume ...
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Möbius transformation mapping

I would like to understand how to choose the right Mobius transformation. For example, the Mobius transformation that maps upper half plane onto the unit disk is: $z \rightarrow \frac{z-i}{z+i}$ ...
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Find a center and radius of circle that is an image of Mobius Transformation of real axis

I need to find a center and radius of a circle that is an image of real axis under homography $$h(z)= \frac{z-z_1}{z-z_2}$$ I found out that homography preserves symetric points, therefore ...
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Holomorphic bijection from intersection of two circles to a region between two rays

What is a holomorphic map from the nonempty intersection of two circles with "tip points" $a$ (below) and $b$ (neither of which are included in one another) to the region $A$ between two rays? Pictue ...
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Mapping using mobius transformations

I had a fundamental question regarding mobius transforms. Suppose I want to map the unit circle to the upper half plane ($Im$ $z \geq 0$) on the complex plane. I know mapping any three points on the ...
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Evaluate the image of complex function

Given the function $f:\mathbb{C}\setminus\{-i\}\rightarrow \mathbb{C}\setminus \{1\}$, defined by $f(z)=\frac{z-i}{z+i}$. I'm supposed to find the image for $f(\{z\mid\Im (z) > 0\})$. However I'm ...
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Prove this Möbius function maps unit disc to itself bijectively.

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 3.9 I got $(a)$ and $(b)$. My attempt for $(c)$: First, I interpret that $(c)$ is ...
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Suppose $f$ holomorphic and its image is a subset of the unit circle. Then show f is constant.

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 3.8 Suppose $f$ is holomorphic in region $G$, and $f(G) \subseteq \{ |z|=1 \}$. Prove $f$ ...
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Finding Mobius transformations that maps one set to another

I am having a hard time understanding how we find mobius maps from circles, discs to half planes etc. I know how to find maps that take a set of points to another but not sets. I know about cross ...
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Find a conformal map 2

$\textbf{Exercise}$ Find a conformal mapping which maps the domain $D$ onto the open unit disc, where $D$ is the intersection of $\vert z \vert <1$ and $\vert z-1 \vert <1$. I knew that ...
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Need help with $\frac{w-a}{1-\overline{a}w} = \mu\frac{z-a}{1-\overline{a}z}$

I have one exercises that i need help. Let the Möbius transformation $w=T(z)$ defined by $\frac{w-a}{1-\overline{a}w} = \mu\frac{z-a}{1-\overline{a}z},\mu=e^i\alpha, \lvert a\rvert<1$ 1.Put the ...
Let $D$ denote the unit disc (|z| < 1). Let $a \in \mathbb{C}, B \in \mathbb{R}$ and $r > 0.$ I want to find a Mobius map $f$ such that (1) $f(D) = \{z = x+iy : |z-a| < r\}$ $\textbf{Sol}$ ...
Find the Mobius Transformation that maps the right half plane to the unit disc carrying the point $z=15$ to the origin. Since the Mobius transformation takes the point $z=15$ to the origin, so I ...