Analytic Mappings - Conformality, Arcs and Closed Curves, Conformal Mapping, Linear Fractional Transformations, Cross Ratio and Symmetry

1. Arcs and Closed Curves

In complex analysis, we often deal with curves and regions in the complex plane. These are fundamental to understanding functions of a complex variable.

1.1 Arcs

An arc in the complex plane is a continuous curve. It can be represented parametrically by a function $z(t) = x(t) + iy(t)$, where $t$ is a real parameter, typically in an interval $[a, b]$.

For an arc to be considered "smooth," its derivative $z'(t)$ must exist and be non-zero for all $t$ in the interval. If $z'(t)$ exists and is continuous, the arc is called a "C¹" arc.

A simple arc is one that does not intersect itself. This means that if $t_1 \neq t_2$, then $z(t_1) \neq z(t_2)$ for any $t_1, t_2$ in the domain, except possibly at the endpoints if the arc is closed.

1.2 Closed Curves

A closed curve is an arc where the starting point and the ending point are the same. Mathematically, for an arc $z(t)$ defined on $[a, b]$, it is closed if $z(a) = z(b)$.

A simple closed curve is a closed curve that does not intersect itself at any other point. The classic example is a circle or an ellipse. Jordan Curve Theorem states that a simple closed curve divides the complex plane into exactly two disjoint regions: an "interior" and an "exterior."

The interior of a simple closed curve is the bounded region, and the exterior is the unbounded region. Both the interior and exterior are open connected sets.

2. Conformality

Conformality is a crucial property of analytic functions. An analytic function preserves angles between intersecting curves at a point, provided the derivative at that point is non-zero. This property is what makes analytic mappings "conformal."

2.1 Angle Preservation

Consider two smooth curves, $C_1$ and $C_2$, intersecting at a point $z_0$. Let the angle between $C_1$ and $C_2$ at $z_0$ be $\theta$. If a function $f(z)$ is analytic in a neighborhood of $z_0$ and $f'(z_0) \neq 0$, then the images of these curves, $f(C_1)$ and $f(C_2)$, will intersect at $f(z_0)$ with the same angle $\theta$. This includes preserving the orientation of the angle.

2.2 Conditions for Conformality

A function $f(z)$ is conformal at a point $z_0$ if it is analytic in a neighborhood of $z_0$ and $f'(z_0) \neq 0$.

If $f'(z_0) = 0$, the function is not conformal at $z_0$. In this case, the angle between intersecting curves can be magnified. The magnification factor depends on the order of the zero of $f'(z)$ at $z_0$. If $f'(z_0) = 0$ and $f''(z_0) \neq 0$, angles are multiplied by 2.

2.3 Analytic vs. Conformal

All analytic functions are conformal at points where their derivative is non-zero. However, not all conformal functions are analytic. For example, the function $f(z) = \bar{z}$ (complex conjugation) is conformal everywhere because it preserves angles and orientation, but it is not analytic anywhere.

The key insight is that analyticity implies conformality (at points with non-zero derivative), but conformality does not necessarily imply analyticity. However, in the context of analytic mappings, we are primarily concerned with functions that are both analytic and conformal.

3. Conformal Mapping

A conformal mapping is a transformation that preserves angles locally. When we talk about conformal mapping in complex analysis, we are usually referring to mappings defined by analytic functions.

3.1 Properties of Conformal Mappings

A non-constant analytic function $f(z)$ maps open connected sets to open connected sets. This means that if $D$ is an open connected set in the $z$-plane, then $f(D)$ is also an open connected set in the $w$-plane.

If $f(z)$ is analytic in a domain $D$ and $f'(z) \neq 0$ for all $z \in D$, then $f$ is a conformal mapping of $D$ onto $f(D)$. This mapping is also one-to-one (injective) if $f'(z) \neq 0$ throughout $D$.

Conformal mappings have significant applications in various fields, including fluid dynamics, heat transfer, and electrostatics, because they preserve the geometric properties of angles and shapes locally.

3.2 Examples of Conformal Mappings

Example 1: $f(z) = z^2$

The derivative is $f'(z) = 2z$. This is zero only at $z=0$. So, $f(z) = z^2$ is conformal everywhere except at the origin.

At $z=0$, $f'(0)=0$. The angle between the positive real axis and the positive imaginary axis is $\pi/2$. Their images under $f(z)=z^2$ are the positive real axis ($x$-axis) and the negative real axis ($x$-axis). The angle between them is $\pi$, which is twice the original angle.

Consider two curves $y=x$ and $y=-x$ intersecting at $z=0$. The angle is $\pi/2$. Their images are $v = \pm 2u$, which are the same line. The angle becomes $\pi$.

Example 2: $f(z) = e^z$

The derivative is $f'(z) = e^z$. Since $e^z$ is never zero, $f(z) = e^z$ is conformal everywhere in the complex plane.

It maps horizontal lines $y=c$ to rays starting from the origin in the $w$-plane. It maps vertical lines $x=c$ to circles centered at the origin.

4. Linear Fractional Transformations (LFTs)

Linear Fractional Transformations, also known as Möbius transformations or bilinear transformations, are a class of functions of the form:

$$w = f(z) = \frac{az + b}{cz + d}$$

where $a, b, c, d$ are complex constants and $ad - bc \neq 0$.

The condition $ad - bc \neq 0$ is crucial. If $ad - bc = 0$, then $ad = bc$, which implies $a/c = b/d$ (assuming $c, d \neq 0$). This means the numerator is a constant multiple of the denominator, resulting in a constant function, which is not a transformation in the usual sense.

4.1 Properties of LFTs

1. Analyticity: LFTs are analytic everywhere except at the pole $z = -d/c$ (if $c \neq 0$).

2. Conformality: LFTs are conformal everywhere except at their pole.

3. Mapping of Circles and Lines: LFTs map circles and lines to circles and lines. A "generalized circle" is either a circle or a line.

4. One-to-One Mapping: Every LFT is a one-to-one mapping of the extended complex plane ($\mathbb{C}_\infty$) onto itself. The extended complex plane includes a point at infinity, $\infty$.

5. Fixed Points: An LFT can have one or two fixed points, which are solutions to $z = \frac{az+b}{cz+d}$.

4.2 The Extended Complex Plane ($\mathbb{C}_\infty$)

To handle the pole $z = -d/c$ and to ensure LFTs map $\mathbb{C}$ onto $\mathbb{C}$, we introduce the point at infinity. We define:

• If $c \neq 0$:
  • $f(-d/c) = \infty$
  • $f(\infty) = a/c$
• If $c = 0$:
  • $f(z) = \frac{az+b}{d}$ (a simple linear transformation)
  • $f(\infty) = \infty$

4.3 Special Cases of LFTs

• Translations: $w = z + b$ (here $a=1, c=0, d=1$). Corresponds to $a=1, b, c=0, d=1$. $ad-bc = 1 \neq 0$.
• Rotations: $w = az$ where $|a|=1$ (here $b=0, c=0, d=1$). Corresponds to $a, b=0, c=0, d=1$. $ad-bc = a \neq 0$.
• Magnifications: $w = az$ where $a$ is a positive real number.
• Inversions: $w = 1/z$ (here $a=0, b=1, c=1, d=0$). $ad-bc = -1 \neq 0$.

4.4 Determining an LFT

An LFT is uniquely determined by its action on any three distinct points. If we know that $f(z_1) = w_1$, $f(z_2) = w_2$, and $f(z_3) = w_3$, where $z_1, z_2, z_3$ are distinct, then the LFT $f(z)$ is unique.

5. Cross Ratio

The cross-ratio is an invariant under Linear Fractional Transformations. It is a fundamental concept for studying LFTs.

The cross-ratio of four distinct complex numbers $z_1, z_2, z_3, z_4$ is denoted by $(z_1, z_2; z_3, z_4)$ and is defined as:

$$ (z_1, z_2; z_3, z_4) = \frac{(z_1 - z_3)(z_2 - z_4)}{(z_1 - z_4)(z_2 - z_3)} $$

Note: There are other conventions for the order of points, but this is a common one. The key is consistency.

5.1 Invariance Property

Let $f(z)$ be a Linear Fractional Transformation. If $z_1, z_2, z_3, z_4$ are four distinct points, and $w_1 = f(z_1), w_2 = f(z_2), w_3 = f(z_3), w_4 = f(z_4)$ are their images under $f$, then the cross-ratio is preserved:

$$ (w_1, w_2; w_3, w_4) = (z_1, z_2; z_3, z_4) $$

5.2 Cross Ratio with Infinity

If one of the points is $\infty$, the cross-ratio is defined by taking appropriate limits. For example, if $z_4 = \infty$:

$$ (z_1, z_2; z_3, \infty) = \lim_{z_4 \to \infty} \frac{(z_1 - z_3)(z_2 - z_4)}{(z_1 - z_4)(z_2 - z_3)} $$

To evaluate this limit, we can rewrite the expression:

$$ \frac{(z_1 - z_3)}{(z_2 - z_3)} \cdot \frac{(z_2 - z_4)}{(z_1 - z_4)} $$

As $z_4 \to \infty$, the term $\frac{(z_2 - z_4)}{(z_1 - z_4)}$ approaches $\frac{-z_4}{-z_4} = 1$.

So, the cross-ratio becomes:

$$ (z_1, z_2; z_3, \infty) = \frac{z_1 - z_3}{z_2 - z_3} $$

Similarly:

• $(z_1, z_2; \infty, z_4) = \frac{z_1 - z_2}{z_1 - z_4}$
• $(z_1, \infty; z_3, z_4) = \frac{z_1 - z_3}{z_1 - z_4}$
• $(\infty, z_2; z_3, z_4) = \frac{z_2 - z_3}{z_2 - z_4}$

And if two points are infinity:

• $(z_1, z_2; \infty, \infty)$ is not defined in this form.
• $(\infty, \infty; z_3, z_4)$ is not defined.

However, the cross-ratio is defined for any four distinct points in the extended complex plane $\mathbb{C}_\infty$. For example, $(\infty, 1; 0, -1) = \frac{(\infty - 0)(1 - (-1))}{(\infty - (-1))(1 - 0)} = \frac{\infty \cdot 2}{\infty \cdot 1} = 2$. Using the limit definition: $(\infty, 1; 0, -1) = \frac{1-0}{1-(-1)} = \frac{1}{2}$. This shows care must be taken with definitions. A more robust way is to use the invariant property.

5.3 Properties of Cross Ratio

1. The cross-ratio is real if and only if the four points lie on a circle or a line.

2. The cross-ratio is preserved under LFTs.

6. Symmetry

Symmetry plays a vital role in complex analysis, particularly in relation to conformal mappings and the Schwarz reflection principle.

6.1 Reflection Across a Line

In the complex plane, reflection across the real axis is given by the complex conjugate: $z \mapsto \bar{z}$.

Reflection across a line passing through the origin with angle $\alpha$ with the positive real axis is given by $z \mapsto e^{2i\alpha} \bar{z}$.

Reflection across an arbitrary line $L$ can be understood by considering a coordinate system where $L$ is the real axis. If $z_0$ is a point on the line $L$ and $\vec{u}$ is a unit vector along $L$, then any point $z$ can be written as $z = z_0 + r \vec{u} + i s \vec{v}$, where $\vec{v}$ is a unit vector perpendicular to $\vec{u}$. The reflection $z'$ across $L$ is $z' = z_0 + r \vec{u} - i s \vec{v}$.

Alternatively, let the line $L$ be represented by the equation $|z-a| = |z-b|$. The reflection of $z$ across $L$ is $z'$.

6.2 Reflection Across a Circle (Inversion)

Reflection of a point $z$ across a circle $|z - c| = R$ (center $c$, radius $R$) maps $z$ to a point $z'$ such that $c, z, z'$ are collinear, and the distances satisfy $|z - c| |z' - c| = R^2$.

The formula for inversion with respect to the circle $|z|=R$ is $z' = R^2 / \bar{z}$.

For a circle $|z - c| = R$, the inversion formula is $z' - c = \frac{R^2}{\overline{z - c}}$.

6.3 Schwarz Reflection Principle

This principle provides a powerful method for extending analytic functions. It states:

Suppose $f(z)$ is analytic in a domain $D$ that includes a segment $L$ of a straight line or a circular arc on its boundary. Let $D'$ be the reflection of $D$ across the boundary curve (line or circle).

If $f(z)$ maps the boundary segment $L$ onto a curve $\Gamma$ in the $w$-plane, and if the reflection of $\Gamma$ across the corresponding boundary curve in the $w$-plane is $\Gamma'$, then there exists an analytic function $g(z)$ defined in $D'$ such that $g(z) = f(z)$ for $z \in D$ and $g(z)$ maps $D'$ onto the reflection of the image of $D$ under $f$.

More simply:

Let $L$ be a line or a circle. Let $D$ be a domain on one side of $L$, and let $D'$ be its reflection across $L$. Suppose $f(z)$ is analytic in $D \cup L$ and maps $L$ onto a line or circle $\Gamma$. Let $f(z) = w$. Then there exists an analytic function $g(z)$ in $D' \cup L$ such that $g(z) = \overline{f(\bar{z})}$ for $z \in D'$ (if $L$ is the real axis) and $g(z) = f(z)$ for $z \in D$. The function $g(z)$ is the analytic continuation of $f(z)$ across $L$.

6.4 Symmetry and LFTs

Linear Fractional Transformations have a special relationship with symmetry. They map generalized circles (circles and lines) to generalized circles.

Consider the unit circle $|z|=1$. The inversion $w = 1/z$ maps this circle to itself (since $|1/\bar{z}| = 1$ if $|z|=1$).

Consider a general LFT $w = \frac{az+b}{cz+d}$. If this transformation maps a circle or line $C_1$ to another circle or line $C_2$, it preserves the property of being a generalized circle.

The cross-ratio property $(w_1, w_2; w_3, w_4) = (z_1, z_2; z_3, z_4)$ implies that if four points $z_1, z_2, z_3, z_4$ lie on a generalized circle, their images $w_1, w_2, w_3, w_4$ under an LFT will also lie on a generalized circle.