Math Problem Statement

  1. What is the form of a bilinear transformation which has one fixed point ‘a’ and the other fixed point at ’∞’.

Solution

A bilinear (or Möbius) transformation is typically given by the formula:

f(z)=az+bcz+df(z) = \frac{az + b}{cz + d}

where aa, bb, cc, and dd are complex numbers, and adbc0ad - bc \neq 0.

To find a specific bilinear transformation with one fixed point at z=az = a and another at z=z = \infty, we use the fact that a fixed point z0z_0 satisfies the equation:

z0=az0+bcz0+dz_0 = \frac{az_0 + b}{cz_0 + d}

For z=z = \infty to be a fixed point, the transformation must map \infty to \infty. This implies that the coefficient of zz in the denominator must be zero, so c=0c = 0. The transformation thus simplifies to:

f(z)=az+bdf(z) = \frac{az + b}{d}

Now, for z=az = a to be a fixed point, we require:

a=aa+bda = \frac{aa + b}{d}

Simplifying this:

ad=a2+bad = a^2 + b

Thus, b=ada2b = ad - a^2, and the bilinear transformation becomes:

f(z)=az+(ada2)df(z) = \frac{az + (ad - a^2)}{d}

This can also be written as:

f(z)=az+a(da)d=a(z+da)df(z) = \frac{az + a(d - a)}{d} = \frac{a(z + d - a)}{d}

Simplifying further, we can express the transformation in the form:

f(z)=ad(z+(da))f(z) = \frac{a}{d} \cdot \left( z + (d - a) \right)

where aa and dd are constants determined by the specific transformation.

Would you like more details or have any questions? Here are some related topics you might find interesting:

  1. How to determine the coefficients of a general bilinear transformation?
  2. What are the geometric interpretations of fixed points in a bilinear transformation?
  3. How to find the fixed points of a given bilinear transformation?
  4. How are bilinear transformations related to conformal mappings?
  5. What is the significance of the cross-ratio in bilinear transformations?

Tip: When working with bilinear transformations, it’s useful to remember that they preserve the cross-ratio of four points, which is a key property in projective geometry.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Complex Analysis
Möbius Transformations

Formulas

Bilinear (Möbius) transformation: f(z) = (az + b) / (cz + d)

Theorems

-

Suitable Grade Level

Advanced Mathematics