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

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

\[
f(z) = \frac{az + b}{cz + d}
\]

where \( a \), \( b \), \( c \), and \( d \) are complex numbers, and \( ad - bc \neq 0 \).

To find a specific bilinear transformation with one fixed point at \( z = a \) and another at \( z = \infty \), we use the fact that a fixed point \( z_0 \) satisfies the equation:

\[
z_0 = \frac{az_0 + b}{cz_0 + d}
\]

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

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

Now, for \( z = a \) to be a fixed point, we require:

\[
a = \frac{aa + b}{d}
\]

Simplifying this:

\[
ad = a^2 + b
\]

Thus, \( b = ad - a^2 \), and the bilinear transformation becomes:

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

This can also be written as:

\[
f(z) = \frac{az + a(d - a)}{d} = \frac{a(z + d - a)}{d}
\]

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

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

where \( a \) and \( d \) 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.

Explore the concept of bilinear (Möbius) transformations with fixed points 'a' and '∞'. Learn how to derive the specific form of such transformations using complex analysis techniques. Suitable for advanced mathematics enthusiasts.

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