7.3 Analytic functions and Cauchy-Riemann equations

Complex functions that are differentiable everywhere in their domain are called analytic. These functions have special properties, like being infinitely differentiable and satisfying the Cauchy-Riemann equations. They're key to understanding complex analysis.

Analytic functions include polynomials and exponentials. Their real and imaginary parts are harmonic functions, which pop up in physics. Some analytic functions, called entire functions, are differentiable on the whole complex plane. This stuff is crucial for complex integration and more advanced topics.

Complex Differentiability and Analytic Functions

Definition and Properties of Analytic Functions

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• An analytic function is a complex-valued function that is differentiable at every point in its domain
  • Differentiability in the complex plane requires the existence of the limit of the difference quotient approaching the same value regardless of the direction in which the limit is taken
• Analytic functions are infinitely differentiable, meaning all higher-order derivatives exist
• The real and imaginary parts of an analytic function satisfy the Cauchy-Riemann equations
  • If $f(z) = u(x, y) + iv(x, y)$ is analytic, then $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

Complex Differentiability and the Cauchy-Riemann Equations

• A complex function $f(z)$ is complex differentiable at a point $z_0$ if the limit $\lim_{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0}$ exists
  • This limit must be the same regardless of the path taken to approach $z_0$ in the complex plane
• The Cauchy-Riemann equations provide a necessary and sufficient condition for a complex function to be complex differentiable
  • For $f(z) = u(x, y) + iv(x, y)$, the partial derivatives of $u$ and $v$ must satisfy $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
• If a function is complex differentiable at every point in an open set, it is said to be holomorphic on that set

Examples of Analytic Functions

• The function $f(z) = z^2$ is analytic everywhere in the complex plane
  • Its real and imaginary parts, $u(x, y) = x^2 - y^2$ and $v(x, y) = 2xy$, satisfy the Cauchy-Riemann equations
• The exponential function $e^z = e^x(\cos y + i \sin y)$ is analytic everywhere
  • Its real and imaginary parts, $u(x, y) = e^x \cos y$ and $v(x, y) = e^x \sin y$, satisfy the Cauchy-Riemann equations

Special Classes of Analytic Functions

Harmonic Functions

• A harmonic function is a real-valued function that satisfies Laplace's equation, $\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$
• The real and imaginary parts of an analytic function are harmonic functions
  • If $f(z) = u(x, y) + iv(x, y)$ is analytic, then both $u(x, y)$ and $v(x, y)$ are harmonic functions
• Harmonic functions have important applications in physics, such as modeling electrostatic potentials and steady-state temperature distributions

Entire Functions

• An entire function is a complex-valued function that is analytic on the entire complex plane
  • Entire functions have no singularities or points of non-differentiability
• Polynomials and the exponential function are examples of entire functions
  • The function $f(z) = z^3 - 2z + 1$ is an entire function, as it is a polynomial in $z$
  • The exponential function $e^z$ is entire, as it is analytic everywhere in the complex plane
• Entire functions have important properties, such as being bounded on any compact subset of the complex plane (Liouville's theorem)

Singularities and Residues

Types of Singularities

• A singularity is a point in the complex plane where a function is not analytic
• Isolated singularities can be classified into three types: removable singularities, poles, and essential singularities
  • A removable singularity is a point where the function is not defined, but the limit of the function exists as the point is approached (e.g., $f(z) = \frac{\sin z}{z}$ at $z = 0$)
  • A pole is a point where the function approaches infinity as the point is approached (e.g., $f(z) = \frac{1}{z}$ at $z = 0$)
  • An essential singularity is a point where the function behaves erratically as the point is approached (e.g., $f(z) = e^{\frac{1}{z}}$ at $z = 0$)

Residues and Their Applications

• The residue of a function at an isolated singularity is the coefficient of the $\frac{1}{z - z_0}$ term in the Laurent series expansion of the function around the singularity
  • For a function $f(z)$ with a pole of order $n$ at $z_0$, the residue is given by $\frac{1}{(n-1)!} \lim_{z \to z_0} \frac{d^{n-1}}{dz^{n-1}} [(z - z_0)^n f(z)]$
• Residues have important applications in complex integration, particularly in evaluating integrals using the residue theorem
  • The residue theorem states that the integral of a meromorphic function along a closed contour is equal to $2\pi i$ times the sum of the residues enclosed by the contour
  • For example, the integral $\int_{-\infty}^{\infty} \frac{1}{1 + x^2} dx$ can be evaluated using the residue theorem by considering the contour integral around a semicircle in the upper half-plane