Complex Analysis: Cauchy-Riemann Equations

The Cauchy-Riemann equations ensure a function $f(z) = u(x, y) + i v(x, y)$ is holomorphic (complex differentiable). These equations in Cartesian and polar coordinates are:

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1. In Cartesian Coordinates:

Given $f(z) = u(x, y) + i v(x, y)$, the Cauchy-Riemann equations are: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

2. In Polar Coordinates:

Let $z = re^{i\theta}$, where $r = |z|$ and $\theta = \text{arg}(z)$. The function becomes: $f(z) = u(r, \theta) + i v(r, \theta)$ The Cauchy-Riemann equations in polar form are: $\frac{\partial u}{\partial r} = \frac{1}{r} \frac{\partial v}{\partial \theta}, \quad \frac{\partial v}{\partial r} = -\frac{1}{r} \frac{\partial u}{\partial \theta}$

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Solving Problems:

1. Given $u(x, y)$ or $v(x, y)$, find the other component:

   • Use the Cartesian Cauchy-Riemann equations to solve for the missing component $u(x, y)$ or $v(x, y)$.

2. Find the complex function $f(z)$:

   • Combine $u(x, y)$ and $v(x, y)$ to express $f(z) = u + iv$.
   • Identify any functional dependence (e.g., polynomial, exponential).

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Linear Fractional Transformations (LFTs):

A Linear Fractional Transformation (LFT) is a mapping of the form: $w = \frac{az + b}{cz + d}, \quad ad - bc \neq 0$

Properties:

1. Preservation of Circles and Lines: LFTs map generalized circles (including straight lines) to generalized circles.
2. Computation of Parameters: Use boundary conditions or fixed points to solve for $a, b, c, d$.
3. Finding Images of Regions: Break regions into boundary curves, find their images, then deduce the region's overall transformation.

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Examples:

1. Given $u(x, y) = x^2 - y^2$, find $v(x, y)$: Using $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$: $\frac{\partial u}{\partial x} = 2x, \quad \frac{\partial u}{\partial y} = -2y$ $\frac{\partial v}{\partial y} = 2x, \quad \frac{\partial v}{\partial x} = 2y$ Integrating, $v(x, y) = 2xy$.

Thus, $f(z) = z^2 = (x^2 - y^2) + i(2xy)$.

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2. Example LFT Problem: Find the LFT that maps $1, i, -1$ to $0, 1, \infty$. The general form is: $w = \frac{az + b}{cz + d}$ Substitute boundary conditions:

• $w(1) = 0$, $w(i) = 1$, $w(-1) = \infty$ Solve the resulting system for $a, b, c, d$.

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Questions for Expansion:

1. How are the Cauchy-Riemann equations derived in polar coordinates?
2. What is the significance of satisfying the Cauchy-Riemann equations in complex analysis?
3. How does an LFT preserve angles, and why is this property useful?
4. Can we use Cauchy-Riemann equations to determine whether a function is analytic? Why?
5. What is the relationship between LFTs and Möbius transformations?

Tip: Always check if the Cauchy-Riemann equations hold before concluding that a function is analytic or holomorphic!