The Cauchy-Riemann equations are key to understanding complex differentiability. They give us a way to check if a function is analytic, which is super important in complex analysis. These equations link the real and imaginary parts of a complex function.

By using these equations, we can figure out if a function is analytic and find harmonic conjugates. This helps us solve all sorts of problems in math and physics, like fluid dynamics and electrostatics. It's a powerful tool for working with complex functions.

Cauchy-Riemann Equations

Cartesian and Polar Forms

• The Cauchy-Riemann equations provide a necessary and sufficient condition for a complex function to be complex differentiable (analytic) at a point
• In Cartesian form, for a complex function $f(z) = u(x, y) + iv(x, y)$, the Cauchy-Riemann equations are:
  • $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$
  • $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
• In polar form, for a complex function $f(z) = u(r, \theta) + iv(r, \theta)$, the Cauchy-Riemann equations are:
  • $\frac{\partial u}{\partial r} = \frac{1}{r}\frac{\partial v}{\partial \theta}$
  • $\frac{\partial v}{\partial r} = -\frac{1}{r}\frac{\partial u}{\partial \theta}$
• The Cauchy-Riemann equations relate the partial derivatives of the real and imaginary parts of a complex function

Derivation from Differentiability

• A complex function $f(z) = u(x, y) + iv(x, y)$ is differentiable at a point $z_0$ if the limit of $\frac{f(z) - f(z_0)}{z - z_0}$ exists as $z$ approaches $z_0$, independent of the path along which $z$ approaches $z_0$
• Applying the limit definition of the derivative to the real and imaginary parts of $f(z)$ separately leads to the Cauchy-Riemann equations
• The derivation involves considering the limit along the real and imaginary axes and equating the corresponding components, resulting in the partial derivative relations
• The derivation can also be performed using the polar form of the complex function, leading to the polar form of the Cauchy-Riemann equations
• The existence of the complex derivative implies the existence and continuity of the partial derivatives satisfying the Cauchy-Riemann equations

Differentiability and Cauchy-Riemann Equations

Analyticity and Differentiability

• A complex function is analytic (complex differentiable) at a point if it is differentiable in a neighborhood of that point
• For a complex function to be analytic at a point or in a region, the Cauchy-Riemann equations must be satisfied at that point or throughout the region
• If the Cauchy-Riemann equations are satisfied and the partial derivatives are continuous at a point, then the function is analytic at that point
• If the Cauchy-Riemann equations are satisfied throughout a region and the partial derivatives are continuous in that region, then the function is analytic in that region
• Analyticity is a stronger condition than differentiability, as it requires the function to be differentiable in a neighborhood of a point

Checking Analyticity

• To determine if a complex function is analytic, compute the partial derivatives of the real and imaginary parts and check them against the Cauchy-Riemann equations
• Example: For $f(z) = z^2 = (x + iy)^2 = (x^2 - y^2) + i(2xy)$, we have:
  • $u(x, y) = x^2 - y^2$, $v(x, y) = 2xy$
  • $\frac{\partial u}{\partial x} = 2x$, $\frac{\partial v}{\partial y} = 2x$
  • $\frac{\partial u}{\partial y} = -2y$, $\frac{\partial v}{\partial x} = 2y$
  • The Cauchy-Riemann equations are satisfied, and the partial derivatives are continuous everywhere, so $f(z) = z^2$ is analytic in the entire complex plane
• Example: For $f(z) = \bar{z} = x - iy$, we have:
  • $u(x, y) = x$, $v(x, y) = -y$
  • $\frac{\partial u}{\partial x} = 1$, $\frac{\partial v}{\partial y} = -1$
  • $\frac{\partial u}{\partial y} = 0$, $\frac{\partial v}{\partial x} = 0$
  • The Cauchy-Riemann equations are not satisfied, so $f(z) = \bar{z}$ is not analytic anywhere

Analyticity Using Cauchy-Riemann Equations

Harmonic Functions and Conjugates

• A real-valued function $u(x, y)$ is called harmonic if it satisfies Laplace's equation: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$
• If $u(x, y)$ is a harmonic function, then there exists a harmonic conjugate $v(x, y)$ such that $f(z) = u(x, y) + iv(x, y)$ is analytic
• The harmonic conjugate can be found by integrating the Cauchy-Riemann equations:
  • $v(x, y) = \int \frac{\partial u}{\partial x} dy + C(x)$ or $v(x, y) = -\int \frac{\partial u}{\partial y} dx + C(y)$
• Example: If $u(x, y) = e^x \cos y$, then $v(x, y) = e^x \sin y$ is its harmonic conjugate, and $f(z) = e^x \cos y + i e^x \sin y = e^z$ is analytic

Laplace's Equation

• The Cauchy-Riemann equations can be used to derive Laplace's equation in two dimensions
• If $f(z) = u(x, y) + iv(x, y)$ is analytic, then both $u(x, y)$ and $v(x, y)$ satisfy Laplace's equation:
  • $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ and $\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0$
• This property is useful in solving boundary value problems in various fields, such as fluid dynamics and electrostatics
• Example: In electrostatics, the electric potential $\phi(x, y)$ satisfies Laplace's equation in charge-free regions, and the electric field components can be found using the Cauchy-Riemann equations:
  • $E_x = -\frac{\partial \phi}{\partial x}$ and $E_y = -\frac{\partial \phi}{\partial y}$

Applications of Cauchy-Riemann Equations

Analytic Function Properties

• The Cauchy-Riemann equations can be used to prove properties of analytic functions, such as the Cauchy-Riemann theorem and the Cauchy integral formula
• Cauchy-Riemann Theorem: If $f(z)$ is analytic in a simply connected domain $D$ and $C$ is a simple closed curve in $D$, then $\oint_C f(z) dz = 0$
• Cauchy Integral Formula: If $f(z)$ is analytic in a simply connected domain $D$ and $C$ is a simple closed curve in $D$ enclosing a point $z_0$, then $f(z_0) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z - z_0} dz$
• These theorems are fundamental in complex analysis and have numerous applications in mathematics and physics

Boundary Value Problems

• The Cauchy-Riemann equations can be used to solve boundary value problems in various fields, such as fluid dynamics, electrostatics, and heat transfer
• In these problems, the solution is often a complex function whose real and imaginary parts satisfy certain boundary conditions
• Example: In fluid dynamics, the complex potential $W(z) = \phi(x, y) + i\psi(x, y)$ is used to describe the flow of an ideal fluid, where $\phi(x, y)$ is the velocity potential and $\psi(x, y)$ is the stream function
  • The Cauchy-Riemann equations relate the velocity components to the potential and stream functions: $u_x = \frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y}$ and $u_y = \frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x}$
  • Boundary conditions on the velocity components or the stream function can be used to determine the complex potential and solve for the flow field
• Example: In electrostatics, the complex potential $W(z) = \phi(x, y) + i\psi(x, y)$ is used to describe the electric field in two dimensions, where $\phi(x, y)$ is the electric potential and $\psi(x, y)$ is the electric flux function
  • The Cauchy-Riemann equations relate the electric field components to the potential and flux functions: $E_x = -\frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y}$ and $E_y = -\frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x}$
  • Boundary conditions on the electric potential or the flux function can be used to determine the complex potential and solve for the electric field