# Image under a mobius transformation

• jimmycricket
In summary, the problem is to find a Mobius transformation that maps the points 0, 1, and -i to -1, 0, and infinity, respectively. After solving for the coefficients of the transformation, the next step is to find the image of the given domain under this transformation. This can be done by determining where the boundary points of the domain are mapped to, which can help visualize the image.
jimmycricket

## Homework Statement

Find the Mobius transformation which carries the points $$0,1,-i$$ to the points $$-1,0,\infty$$ respectively. Find the image of the domain $$\{z:x<0,-x+y<t\}$$ under this mobius transformation.

## The Attempt at a Solution

Let $$T(z)=\frac{az+b}{cz+d}$$.
Then $$T(0)=-1\Longrightarrow \frac{b}{d} \iff b=-d$$

$$T(1)=0\Longrightarrow \frac{a+b}{c+d}=0\Longrightarrow a+b=0 \iff a=-b=d$$

$$T(-i)=\infty\Longrightarrow \frac{-ia+b}{-ci+d}\iff d-ci=0\iff c=\frac{d}{i}=bi=-ai$$

Now we have $$T(z)=\frac{az-a}{a-aiz}=\frac{z-1}{1-zi}$$

So I now have to find the image under this map which is where I'm a bit stumped. Would it help to find where the intersections of the boundary of the domain with the axes are mapped to?

Yes, the first part seems to be correct.

Last edited:
Now let z= x+iy where x> 0 and x+ y< t for some fixed real number, t. For example, if t= 1, that would be the part of the complex plane to the right of the imaginary axis and below x+ y= 1. Some points on the boundary of that set would be 1, i, 2- i, etc. Determining what the transformation does to boundary points should help you see what the image is.

## 1. What is a Mobius transformation?

A Mobius transformation is a type of function that maps points from one complex plane to another. It takes the form of f(z) = (az + b) / (cz + d), where a, b, c, and d are complex numbers and z is a complex variable. It is also known as a linear fractional transformation or a conformal mapping.

## 2. How does a Mobius transformation affect an image?

A Mobius transformation can stretch, rotate, or translate an image in the complex plane. It can also flip an image over and change its orientation. The specific effect on an image will depend on the values of a, b, c, and d in the transformation function.

## 3. What is the significance of the Mobius transformation in mathematics?

The Mobius transformation is significant in mathematics because it is a powerful tool for studying and understanding complex functions and their properties. It has applications in geometry, topology, and complex analysis, and is also used in fields such as physics, engineering, and computer graphics.

## 4. Can a Mobius transformation preserve certain properties of an image?

Yes, a Mobius transformation can preserve certain properties of an image, such as angles, circles, and angles between intersecting curves. This is because it is a conformal mapping, meaning it preserves the local angles and shapes of objects in the image.

## 5. Are there any limitations or restrictions to using a Mobius transformation on an image?

Yes, there are some limitations and restrictions when using a Mobius transformation on an image. For example, the transformation is only defined for points in the complex plane, so it cannot be applied to images with points outside of this domain. Additionally, certain values of a, b, c, and d may cause the transformation to be singular or undefined.

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