Cauchy-Riemann Equations & Analytic Functions Quiz

Cauchy-Riemann Equations & Analytic Functions

Understanding Complex Functions

The Cauchy-Riemann equations are a crucial concept in understanding analytic functions, which involve complex numbers and their derivatives. These equations provide an alternative way to comprehend complex functions by showing how the real and imaginary parts of a function are related. They were introduced in the context of complex analysis.

Real Partial Differential Equations

The Cauchy-Riemann equations can be rewritten as a pair of real partial differential equations, which are given by:

[\dfrac{\partial u}{\partial x} = \dfrac{\partial v}{\partial y}] (CR1)

and

[\dfrac{\partial u}{\partial y} = -\dfrac{\partial v}{\partial x}.] (CR2)

These equations link together the real parts ((u)) and imaginary parts ((v)) of a complex function, (f(z)), where (z = x + iy).

Proving Analyticity with Cauchy-Riemann Equations

If a function, (f(z)), satisfies these two conditions, then it is analytic within its domain, meaning that every point within that region is differentiable. This ensures that the real and imaginary parts of the input (z) do not independently affect the output value.

For example, consider the function, (f(z) = e^z). Writing this in exponential form, we have:

[e^z = e^{x+iy} = e^x\cos(y) + ie^x \sin(y),]

where (u = e^x\cos(y)) and (v = e^x\sin(y)). Applying the Cauchy-Riemann equations, we find that:

[\dfrac{\partial u}{\partial x} = -\dfrac{\partial v}{\partial y} = e^x \sin(y)]

and

[\dfrac{\partial u}{\partial y} = -\dfrac{\partial v}{\partial x} = -e^x \cos(y),]

which indeed satisfy the equations, demonstrating that (e^z) is an analytic function.

Implications & Usage in Complex Analysis

The Cauchy-Riemann equations have numerous implications and are used extensively in complex analysis. Some of these include:

• Compositions: If a function, (f(z)), is analytic within a domain (D\subset \mathbb{C}), and another function, (g(z)), is analytic over the image set of (f(z)), then their composition, (g(f(z))), is also analytic within (D).
• Reciprocals: If a function, (f(z)), is analytic in a domain (D\subset \mathbb{C}), then the reciprocal of that function (excluding singular points), (1/f(z)), is also analytic everywhere in (D).

These properties allow analysts to apply operations and functions to complex variables, expanding the scope of mathematical analysis.

Analyticity Conditions

A function (f(z) = u(x, y) + iv(x, y)) is analytic within an open region (D) if and only if at every point in (D):

• The real partial derivatives of (u) and (v) exist and are continuous.
• Equations (CR1) and (CR2) are satisfied.

By checking these conditions, we can determine whether a given function is analytic within a specified domain.

Derivatives & Jacobians

Using the Cauchy-Riemann equations, we can derive the derivative of a complex function, represented as the matrix representation of its Jacobian. For a function (f(z) = f(x, y)), its derivative is given by:

[f'(z) = \begin{bmatrix} u_x & u_y \ v_x & v_y \end{bmatrix},]

where (u_x = \dfrac{\partial u}{\partial x}), (u_y = \dfrac{\partial u}{\partial y}), and similarly for (v).

The Cauchy-Riemann equations provide a powerful tool for working with complex functions, enabling a deeper understanding of the relationship between real and imaginary parts, and allowing us to perform various calculations and manipulations within the realm of complex analysis.