Cauchy-Riemann Equations

The Cauchy–Riemann equations are a set of two partial differential equations that are a fundamental part of complex analysis, particularly in the study of functions of a complex variable. They provide a necessary and sufficient condition for a complex function to be holomorphic, meaning that it is differentiable at every point in its domain.

Understanding complex numbers

Before we dive into the Cauchy-Riemann equations, let's review the basics of complex numbers. A complex number is of the form z = x + iy, where x and y are real numbers, and i is an imaginary unit with the property i 2 = -1.

In this form, x is called the real part of z, denoted by Re(z), and y is the imaginary part, denoted by Im(z).

Functions of a complex variable

A function of a complex variable is represented as f(z), where z is a complex number. Such a function can also be expressed in terms of its real and imaginary components. Let us express f(z) as follows:

f(z) = u(x, y) + iv(x, y)

Here, u(x, y) and v(x, y) are real-valued functions that represent the real and imaginary parts of f(z), respectively.

Cauchy–Riemann equations

For a function f(z) to be differentiable at a point z_0 in its domain, the Cauchy-Riemann equations must be satisfied at that point. These equations are given as:

∂u/∂x = ∂v/∂y
∂u/∂y = -∂v/∂x

Visual representation

Derivation of the Cauchy–Riemann equations

Let's derive the Cauchy-Riemann equations in a simple way. Consider the derivative of f(z) at the point z_0:

f'(z_0) = lim(Δz→0) [(f(z_0 + Δz) - f(z_0)) / Δz]

Express Δz as Δz = Δx + iΔy. Then, we can write:

f(z_0 + Δz) = u(x + Δx, y + Δy) + iv(x + Δx, y + Δy)

Expanding this in terms of Δx and Δy:

f(z_0 + Δz) ≈ f(z_0) + (∂u/∂x + i∂v/∂x)Δx + (∂u/∂y + i∂v/∂y)Δy

By solving the above for holomorphicity, you get the Cauchy-Riemann equations.

Geometrical interpretation

Geometrically, the Cauchy–Riemann equations imply that mappings defined by holomorphic functions locally preserve angles and shapes. In other words, such functions exhibit conformality. For example, if a smaller shape were deformed while being moved by such a function, the new shape would still be the same as the original in terms of angles and ratios.

Example: identity function

Consider the identity function f(z) = z. Here, u(x, y) = x and v(x, y) = y. Check the Cauchy-Riemann equations:

∂u/∂x = 1, ∂v/∂y = 1
∂u/∂y = 0, ∂v/∂x = 0

satisfying the Cauchy–Riemann equations, so f(z) = z is holomorphic.

Application

The Cauchy–Riemann equations are fundamental in making a function analytic, allowing the use of powerful theorems in complex analysis such as Cauchy's integral theorem and the Taylor series for complex functions. They are used in a variety of fields such as fluid dynamics, electromagnetism, and quantum mechanics.

Example: complex powers

Consider f(z) = z n, where n is a positive integer. Express f(z) = (x + iy) n Using the binomial theorem, we can expand and separate into real and imaginary parts. Show that the expanded expression indeed satisfies the Cauchy-Riemann equations.

Conclusion

The Cauchy–Riemann equations serve as an essential bridge between real and complex analysis, containing essential criteria for the differentiability of complex functions. Understanding these equations helps in a deeper understanding of the nature of complex functions and their wide applicability in various scientific fields.

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