6.2 Analytic Functions and Cauchy-Riemann Equations

Complex functions are a fascinating realm in mathematical physics. Analytic functions, with their unique properties and the Cauchy-Riemann equations, form the foundation of this field. They offer powerful tools for solving problems in physics and engineering.

Singularities add depth to our understanding of complex functions. By studying poles, essential singularities, and removable singularities, we gain insights into function behavior. This knowledge is crucial for tackling advanced problems in mathematical physics.

Analytic Functions

Properties of analytic functions

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• Analytic functions are complex functions differentiable at every point in their domain
  • Differentiability implies the function has a unique derivative at each point
  • The derivative of an analytic function is also analytic
• Analytic functions are infinitely differentiable, having derivatives of all orders
• Single-valuedness is a property of analytic functions
  • For any input value in the domain, an analytic function yields a single, unique output value ($f(z) = z^2$ is single-valued, while $f(z) = \sqrt{z}$ is multi-valued)
  • Contrasts with multi-valued functions, which may have multiple output values for a single input

Cauchy-Riemann equations for analyticity

• The Cauchy-Riemann equations are partial differential equations providing a necessary and sufficient condition for a complex function to be analytic
• For a complex function $f(z) = u(x, y) + iv(x, y)$, where $z = x + iy$, the Cauchy-Riemann equations are:
  • $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$
  • $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
• To determine if a complex function is analytic:
  1. Compute its partial derivatives
  2. Check if they satisfy the Cauchy-Riemann equations
• If the Cauchy-Riemann equations are satisfied and the partial derivatives are continuous, the function is analytic ($f(z) = z^2$ satisfies the equations, while $f(z) = |z|$ does not)

Geometry of Cauchy-Riemann equations

• The Cauchy-Riemann equations have a geometric interpretation related to conformal mapping
• Conformal mapping preserves angles between curves in the complex plane
• The Cauchy-Riemann equations ensure the mapping between the real and imaginary parts of an analytic function is conformal
  • The partial derivatives of the real and imaginary parts are related such that the angle between any two curves in the domain is preserved in the image of the mapping
• This geometric interpretation helps understand the behavior of analytic functions and their mappings (the mapping $f(z) = e^z$ preserves angles, while $f(z) = z^2$ doubles angles)

Singularities of Complex Functions

Types of complex function singularities

• Singularities are points in the complex plane where a function is not analytic
• Poles are isolated singularities where the function approaches infinity as the input approaches the singularity
  • The order of a pole is the power to which the denominator of the function vanishes at the singularity
  • A simple pole has order 1 ($f(z) = \frac{1}{z}$ has a simple pole at $z = 0$), while higher-order poles have orders greater than 1 ($f(z) = \frac{1}{z^2}$ has a pole of order 2 at $z = 0$)
• Essential singularities are isolated singularities where the function exhibits complex, oscillatory behavior near the singularity
  • The function does not approach any specific value or infinity as the input approaches an essential singularity ($f(z) = e^{\frac{1}{z}}$ has an essential singularity at $z = 0$)
• Removable singularities are points where the function is undefined, but the limit of the function exists as the input approaches the singularity
  • Removable singularities can be eliminated by redefining the function value at the singularity to be equal to the limit ($f(z) = \frac{z^2 - 1}{z - 1}$ has a removable singularity at $z = 1$)
• To classify singularities, examine the function's behavior near the point and determine if it matches the characteristics of a pole, essential singularity, or removable singularity