∂f/∂z Vs Limit Definition Understanding Complex Derivatives For Non-Holomorphic Functions

In complex analysis, the derivative of a complex function f(z), denoted as f'(z) or df/dz, plays a crucial role in determining the function's holomorphic nature. A function is holomorphic in a region if it is complex differentiable at every point in that region. The limit definition of the derivative, a cornerstone of calculus, extends to complex functions, but its implications become particularly nuanced when dealing with non-holomorphic functions. This article delves into the intricacies of the partial derivative ∂f/∂z for complex, non-holomorphic functions, contrasting it with the limit definition of a derivative and exploring the implications through a detailed example. Understanding these distinctions is essential for grasping the behavior of complex functions and their applications in various fields such as physics, engineering, and mathematics.

The limit definition of a complex derivative closely mirrors its real-variable counterpart. For a complex function f(z), where z is a complex variable, the derivative at a point z₀ is defined as:

f'(z₀) = lim (z→z₀) [f(z) - f(z₀)] / (z - z₀)

This limit, if it exists, must be the same regardless of the path taken as z approaches z₀ in the complex plane. This path independence is a critical distinction between complex and real differentiability. In real analysis, we only consider the function's behavior as we approach a point from the left or the right. In the complex plane, there are infinitely many directions from which z can approach z₀. For a complex function to be differentiable, the limit must exist and be identical along all possible paths.

If the limit exists, the function is said to be complex differentiable at z₀. A function is holomorphic in a region if it is complex differentiable at every point in that region. Holomorphic functions possess several remarkable properties, including infinite differentiability and analyticity, meaning they can be locally represented by a power series. These properties make holomorphic functions indispensable tools in complex analysis and its applications.

However, if the limit does not exist or depends on the path of approach, the function is not complex differentiable at that point. This leads to the fascinating world of non-holomorphic functions, which, despite not satisfying the stringent conditions of complex differentiability, can still exhibit interesting behaviors and have practical applications. The key takeaway here is that the limit definition embodies a strict criterion for differentiability in the complex plane, demanding path independence.

To understand the partial derivative ∂f/∂z, we first need to express a complex function f(z) in terms of its real and imaginary parts. Let z = x + iy, where x and y are real variables and i is the imaginary unit (i² = -1). Similarly, let f(z) = u(x, y) + iv(x, y), where u(x, y) and v(x, y) are real-valued functions representing the real and imaginary parts of f, respectively. The partial derivatives ∂f/∂z and ∂f/∂z̄ (where z̄ is the complex conjugate of z) are defined using the chain rule as follows:

∂f/∂z = (1/2) (∂f/∂x - i ∂f/∂y)

∂f/∂z̄ = (1/2) (∂f/∂x + i ∂f/∂y)

Here, ∂f/∂x and ∂f/∂y represent the usual partial derivatives of f with respect to the real variables x and y. These partial derivatives are fundamental in complex analysis, particularly in the context of the Cauchy-Riemann equations, which provide a crucial link between the partial derivatives and the complex differentiability of a function.

The expressions for ∂f/∂z and ∂f/∂z̄ might seem a bit abstract at first, but they provide a powerful way to analyze complex functions. They essentially decompose the complex derivative into components that relate to changes in the real and imaginary parts of the input. This decomposition is not just a mathematical trick; it reflects the fundamental structure of complex functions and their behavior in the complex plane. By examining these partial derivatives, we can gain insights into whether a function is holomorphic and, if not, how it deviates from holomorphicity.

The Cauchy-Riemann equations are a pair of partial differential equations that provide a necessary (but not sufficient) condition for a complex function to be holomorphic. These equations relate the partial derivatives of the real and imaginary parts of a complex function f(z) = u(x, y) + iv(x, y) and are expressed as:

∂u/∂x = ∂v/∂y

∂u/∂y = -∂v/∂x

These equations arise directly from the requirement that the limit definition of the derivative must be path-independent. If a function f(z) is complex differentiable at a point, then these equations must hold at that point. Conversely, if the Cauchy-Riemann equations are satisfied and the partial derivatives are continuous in a neighborhood of a point, then the function is complex differentiable at that point.

The connection between the Cauchy-Riemann equations and the partial derivatives ∂f/∂z and ∂f/∂z̄ is particularly insightful. Using the definitions of these partial derivatives, we can rewrite the condition for holomorphicity in a concise form:

∂f/∂z̄ = 0

This equation states that for a function to be holomorphic, its partial derivative with respect to z̄ (the complex conjugate of z) must be zero. This is a powerful criterion because it encapsulates the Cauchy-Riemann equations in a single equation. If ∂f/∂z̄ is not zero, the function is not holomorphic. Furthermore, the magnitude of ∂f/∂z̄ can be seen as a measure of how far a function deviates from being holomorphic.

In essence, the Cauchy-Riemann equations and the condition ∂f/∂z̄ = 0 provide a rigorous framework for determining the holomorphicity of a complex function. They bridge the gap between the limit definition of the derivative and the practical computation of partial derivatives, allowing us to analyze complex functions more effectively. Understanding these equations is crucial for navigating the complexities of complex analysis and its applications.

Let's consider the function:

f(z) = z̄²/z for z ≠ 0 and f(0) = 0

This function provides a compelling example of a non-holomorphic function that nevertheless satisfies the Cauchy-Riemann equations at a specific point. This seemingly paradoxical behavior highlights the subtle distinctions between satisfying the Cauchy-Riemann equations and being complex differentiable. To analyze this function, we'll first express it in terms of its real and imaginary parts.

Let z = x + iy, where x and y are real variables. Then z̄ = x - iy, and we can rewrite the function as:

f(z) = (x - iy)² / (x + iy) = [(x² - y²) - 2ixy] / (x + iy)

To separate the real and imaginary parts, we multiply the numerator and denominator by the complex conjugate of the denominator:

f(z) = {[(x² - y²) - 2ixy] * (x - iy)} / (x² + y²) = [(x³ - 3xy²) + i(-3x²y + y³)] / (x² + y²)

Thus, we have u(x, y) = (x³ - 3xy²) / (x² + y²) and v(x, y) = (-3x²y + y³) / (x² + y²). Now, we can compute the partial derivatives of u and v with respect to x and y to check the Cauchy-Riemann equations.

This function is particularly interesting because it satisfies the Cauchy-Riemann equations at the origin (0, 0) but is not complex differentiable there. This illustrates that satisfying the Cauchy-Riemann equations is a necessary but not sufficient condition for complex differentiability. The failure of complex differentiability arises from the path dependence of the limit definition of the derivative at the origin.

To verify that the Cauchy-Riemann equations are satisfied at the origin, we need to compute the partial derivatives of u(x, y) and v(x, y) at (0, 0). Since the expressions for u and v are defined piecewise, we need to use the limit definition of the partial derivatives:

∂u/∂x|₀ = lim (h→0) [u(h, 0) - u(0, 0)] / h = lim (h→0) (h³/h²) / h = 1

∂v/∂y|₀ = lim (k→0) [v(0, k) - v(0, 0)] / k = lim (k→0) (k³/k²) / k = 1

∂u/∂y|₀ = lim (k→0) [u(0, k) - u(0, 0)] / k = lim (k→0) 0 / k = 0

∂v/∂x|₀ = lim (h→0) [v(h, 0) - v(0, 0)] / h = lim (h→0) 0 / h = 0

Thus, we have ∂u/∂x|₀ = 1, ∂v/∂y|₀ = 1, ∂u/∂y|₀ = 0, and ∂v/∂x|₀ = 0. Clearly, the Cauchy-Riemann equations ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x are satisfied at the origin.

This result might lead one to believe that the function is complex differentiable at the origin. However, as we will see, this is not the case. The satisfaction of the Cauchy-Riemann equations is a necessary condition for complex differentiability, but it is not sufficient. The function fails to be complex differentiable due to the path dependence of the limit definition of the derivative.

To demonstrate that f(z) is not complex differentiable at the origin, we need to show that the limit

lim (z→0) [f(z) - f(0)] / (z - 0) = lim (z→0) f(z) / z

does not exist or depends on the path of approach. Let's consider two different paths:

Path 1: Along the real axis (y = 0)

Along the real axis, z = x, and z̄ = x. The limit becomes:

lim (x→0) f(x) / x = lim (x→0) (x²/x) / x = lim (x→0) 1 = 1

Path 2: Along the line y = x

Along the line y = x, z = x + ix and z̄ = x - ix. The limit becomes:

lim (x→0) f(x + ix) / (x + ix) = lim (x→0) [(x - ix)² / (x + ix)] / (x + ix) = lim (x→0) (x - ix)² / (x + ix)²

• = lim (x→0) (x²(1 - i)²) / (x²(1 + i)²) = lim (x→0) (-2i) / (-4) = i/2*

Since the limits along these two paths are different (1 and i/2), the limit

lim (z→0) f(z) / z

does not exist. Therefore, the function f(z) = z̄²/z is not complex differentiable at the origin, even though it satisfies the Cauchy-Riemann equations there.

This example vividly illustrates the crucial distinction between satisfying the Cauchy-Riemann equations and being complex differentiable. The path independence of the limit definition of the derivative is a stringent requirement that this function fails to meet. This failure underscores the importance of considering the limit definition of the derivative when analyzing the complex differentiability of a function.

The core difference between ∂f/∂z and the limit definition of a derivative for a non-holomorphic function lies in what they capture and how they are computed. The partial derivative ∂f/∂z is a formal expression derived from the partial derivatives with respect to x and y. It provides a snapshot of how the function changes along the complex plane's real and imaginary axes. However, it does not, by itself, guarantee complex differentiability.

The limit definition of the derivative, on the other hand, embodies the essence of complex differentiability. It demands that the rate of change of the function must be the same regardless of the direction from which we approach a point in the complex plane. This path independence is a critical requirement that ∂f/∂z alone cannot ensure.

For holomorphic functions, there is a beautiful harmony between these two concepts. If a function is holomorphic, the limit definition of the derivative exists, and it can be computed using ∂f/∂z. However, for non-holomorphic functions, this harmony breaks down. The partial derivative ∂f/∂z may exist, and the function may even satisfy the Cauchy-Riemann equations at a point, but the limit definition of the derivative may fail to exist due to path dependence.

Consider the example function f(z) = z̄²/z. At the origin, we showed that the Cauchy-Riemann equations are satisfied, which implies that ∂f/∂z can be computed. However, by examining different paths of approach to the origin, we demonstrated that the limit definition of the derivative does not exist. This discrepancy highlights that while ∂f/∂z provides valuable information about the function's behavior, it is not the ultimate arbiter of complex differentiability.

In essence, ∂f/∂z is a local measure of change along specific directions, while the limit definition is a global criterion that considers all possible directions. For non-holomorphic functions, this distinction is crucial for a complete understanding of their behavior. The limit definition reveals the function's non-differentiable nature, while ∂f/∂z provides insights into its partial derivatives, which can still be useful in other contexts.

Understanding the subtle differences between ∂f/∂z and the limit definition of a derivative is essential for navigating the world of complex analysis, especially when dealing with non-holomorphic functions. While ∂f/∂z provides a valuable tool for analyzing the partial derivatives of a complex function, the limit definition embodies the fundamental concept of complex differentiability, demanding path independence. The Cauchy-Riemann equations offer a bridge between these two concepts, providing a necessary (but not sufficient) condition for holomorphicity.

The example function f(z) = z̄²/z vividly illustrates the divergence between satisfying the Cauchy-Riemann equations and being complex differentiable. This function satisfies the Cauchy-Riemann equations at the origin, suggesting that ∂f/∂z might provide a meaningful measure of change. However, the limit definition of the derivative fails due to path dependence, revealing the function's non-holomorphic nature at that point. This discrepancy underscores the importance of considering the limit definition when assessing complex differentiability.

In conclusion, while ∂f/∂z offers a convenient way to compute partial derivatives, the limit definition of the derivative remains the gold standard for determining complex differentiability. For non-holomorphic functions, this distinction is crucial for a comprehensive understanding of their behavior and limitations. By carefully considering both ∂f/∂z and the limit definition, we can gain deeper insights into the intricacies of complex functions and their applications in various fields.