Cauchy-Riemann Equations And The Complex Logarithm Derivative

In the realm of complex analysis, the complex logarithm stands as a fundamental function, extending the familiar concept of logarithms from real numbers to complex numbers. This exploration delves into the fascinating properties of the complex logarithm, specifically focusing on its real and imaginary parts and their adherence to the Cauchy-Riemann equations. These equations serve as a cornerstone in determining the differentiability of complex functions, and their satisfaction by the complex logarithm underscores its significance in complex analysis. We will embark on a journey to demonstrate that the real and imaginary components of the function w = log z indeed fulfill the Cauchy-Riemann conditions when z is non-zero. Furthermore, we shall derive its derivative, providing a comprehensive understanding of this essential complex function.

The complex logarithm, denoted as log z, is defined as the inverse of the complex exponential function. Given a complex number z, the complex logarithm yields another complex number w such that e^w = z. However, unlike its real counterpart, the complex logarithm is a multi-valued function due to the periodic nature of the complex exponential. This multi-valuedness arises from the fact that adding integer multiples of 2πi to w does not alter the exponential value, as e^(w + 2πik) = e^w for any integer k. Consequently, we often consider the principal branch of the complex logarithm, denoted as Log z, which restricts the imaginary part of the logarithm to the interval (-π, π], thereby ensuring a single-valued function. This principal branch plays a crucial role in various applications and theoretical considerations within complex analysis.

The Cauchy-Riemann equations, at their core, provide a necessary condition for the differentiability of a complex function. These equations establish a relationship between the partial derivatives of the real and imaginary parts of the function. Specifically, if we express a complex function f(z) = u(x, y) + iv(x, y), where u and v are real-valued functions of the real variables x and y, then the Cauchy-Riemann equations state that ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x. These equations stem from the requirement that the derivative of f(z) must be independent of the direction in which the limit is taken in the complex plane. In other words, the rate of change of the function must be the same whether we approach a point along the real axis or the imaginary axis. This directional independence is a hallmark of complex differentiability and is elegantly captured by the Cauchy-Riemann equations.

The derivative of a complex function, similar to its real counterpart, quantifies the rate of change of the function with respect to its input. However, in the complex domain, the notion of a derivative is more nuanced due to the two-dimensional nature of the complex plane. The derivative of a complex function f(z) at a point z₀, denoted as f'(z₀), is defined as the limit of the difference quotient (f(z) - f(z₀))/(z - z₀) as z approaches z₀. For this limit to exist, it must be the same regardless of the path taken by z as it approaches z₀. This path independence is a stringent requirement and is intimately connected to the Cauchy-Riemann equations. If a complex function satisfies the Cauchy-Riemann equations and its partial derivatives are continuous, then the function is guaranteed to be differentiable. This connection between the Cauchy-Riemann equations and differentiability underscores their fundamental importance in complex analysis.

To delve into the intricate details of the complex logarithm and its adherence to the Cauchy-Riemann equations, we must first establish a clear understanding of its mathematical formulation. The complex logarithm, denoted as w = log z, is defined as the inverse of the complex exponential function. This means that if w = log z, then e^w = z. To express the complex logarithm in terms of its real and imaginary parts, we represent both z and w in their respective forms. Let z = x + iy be a non-zero complex number, where x and y are real numbers representing the real and imaginary parts of z, respectively. Similarly, let w = u + iv, where u and v are real numbers representing the real and imaginary parts of w, respectively. The goal is to express u and v in terms of x and y, thereby providing a concrete representation of the complex logarithm.

Substituting these expressions into the equation e^w = z, we get e^(u + iv) = x + iy. Using Euler's formula, which states that e^(iθ) = cos θ + i sin θ, we can rewrite the left-hand side as e^u * e^(iv) = e^u (cos v + i sin v). Thus, we have e^u (cos v + i sin v) = x + iy. Now, we can equate the real and imaginary parts of both sides of the equation. Equating the real parts gives us e^u cos v = x, and equating the imaginary parts gives us e^u sin v = y. These two equations form a system that we can solve for u and v in terms of x and y. Solving this system will provide us with the real and imaginary parts of the complex logarithm.

To solve for u, we can square both equations and add them together. This eliminates the trigonometric functions and yields an equation involving only e^u. Squaring the first equation gives us (e^u cos v)² = x², and squaring the second equation gives us (e^u sin v)² = y². Adding these two equations, we get e^(2u) cos² v + e^(2u) sin² v = x² + y². Factoring out e^(2u), we have e^(2u) (cos² v + sin² v) = x² + y². Since cos² v + sin² v = 1, this simplifies to e^(2u) = x² + y². Taking the natural logarithm of both sides, we get 2u = ln(x² + y²), and therefore, u = (1/2) ln(x² + y²). This expression gives us the real part of the complex logarithm in terms of x and y. Notably, x² + y² represents the square of the magnitude of the complex number z, so we can also write u = ln|z|, where |z| denotes the magnitude of z.

To solve for v, we can divide the second equation (e^u sin v = y) by the first equation (e^u cos v = x). This eliminates e^u and gives us an equation involving only the trigonometric functions. Dividing the two equations, we get (e^u sin v) / (e^u cos v) = y / x, which simplifies to tan v = y / x. The solution to this equation is v = arctan(y / x) + 2πk, where k is an integer. However, we must be careful when using the arctangent function, as it only provides values in the interval (-π/2, π/2). To obtain the correct value of v, we need to consider the quadrant in which the complex number z = x + iy lies. The angle v represents the argument of z, denoted as arg(z). The argument is the angle between the positive real axis and the line connecting the origin to the point z in the complex plane. Therefore, the imaginary part of the complex logarithm, v, is given by v = arg(z). The principal value of the argument, denoted as Arg(z), lies in the interval (-π, π]. Thus, the principal branch of the complex logarithm is given by Log z = ln|z| + i Arg(z).

Now that we have expressed the complex logarithm w = log z in terms of its real and imaginary parts, u and v, respectively, we can proceed to verify whether it satisfies the Cauchy-Riemann equations. As we derived in the previous section, the real part u is given by u(x, y) = (1/2) ln(x² + y²), and the imaginary part v is given by v(x, y) = arg(z) = arctan(y/x), considering the appropriate quadrant for the argument. The Cauchy-Riemann equations state that for a complex function to be differentiable, its real and imaginary parts must satisfy the following two conditions: ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x. To verify these equations, we need to compute the partial derivatives of u and v with respect to x and y.

Let's first compute the partial derivatives of the real part u(x, y) = (1/2) ln(x² + y²). The partial derivative of u with respect to x is given by ∂u/∂x = (1/2) * (1 / (x² + y²)) * (2x) = x / (x² + y²). Similarly, the partial derivative of u with respect to y is given by ∂u/∂y = (1/2) * (1 / (x² + y²)) * (2y) = y / (x² + y²). These are the partial derivatives of the real part of the complex logarithm.

Next, we compute the partial derivatives of the imaginary part v(x, y) = arctan(y/x). The partial derivative of v with respect to x is given by ∂v/∂x = (1 / (1 + (y/x)²)) * (-y/x²) = -y / (x² + y²). The partial derivative of v with respect to y is given by ∂v/∂y = (1 / (1 + (y/x)²)) * (1/x) = x / (x² + y²). These are the partial derivatives of the imaginary part of the complex logarithm.

Now, we can compare the partial derivatives to verify the Cauchy-Riemann equations. The first Cauchy-Riemann equation states that ∂u/∂x = ∂v/∂y. From our calculations, we have ∂u/∂x = x / (x² + y²) and ∂v/∂y = x / (x² + y²). Clearly, these two expressions are equal, so the first Cauchy-Riemann equation is satisfied. The second Cauchy-Riemann equation states that ∂u/∂y = -∂v/∂x. From our calculations, we have ∂u/∂y = y / (x² + y²) and ∂v/∂x = -y / (x² + y²). Therefore, -∂v/∂x = y / (x² + y²), which is equal to ∂u/∂y. Thus, the second Cauchy-Riemann equation is also satisfied. Since both Cauchy-Riemann equations are satisfied by the real and imaginary parts of the complex logarithm, we can conclude that the complex logarithm is differentiable when z is not zero.

It is important to note that the Cauchy-Riemann equations provide a necessary but not sufficient condition for differentiability. In addition to satisfying the Cauchy-Riemann equations, the partial derivatives must also be continuous for the function to be differentiable. In the case of the complex logarithm, the partial derivatives we calculated are continuous everywhere except at the origin (0, 0), where the denominator x² + y² becomes zero. This is consistent with the fact that the complex logarithm is not defined at z = 0. Therefore, the complex logarithm satisfies the conditions for differentiability for all non-zero complex numbers.

Having established that the complex logarithm satisfies the Cauchy-Riemann equations and is therefore differentiable for non-zero z, we can now proceed to determine its derivative. The derivative of a complex function f(z) = u(x, y) + iv(x, y) can be expressed in terms of its partial derivatives as f'(z) = ∂u/∂x + i ∂v/∂x or equivalently as f'(z) = ∂v/∂y - i ∂u/∂y. These expressions are a direct consequence of the Cauchy-Riemann equations and provide a convenient way to compute the derivative once the partial derivatives are known. In our case, f(z) = log z, u(x, y) = (1/2) ln(x² + y²), and v(x, y) = arg(z).

We have already computed the partial derivatives of u and v in the previous section. We found that ∂u/∂x = x / (x² + y²), ∂v/∂x = -y / (x² + y²), ∂u/∂y = y / (x² + y²), and ∂v/∂y = x / (x² + y²). Using the first expression for the derivative, f'(z) = ∂u/∂x + i ∂v/∂x, we can substitute the values of the partial derivatives to obtain f'(z) = (x / (x² + y²)) + i (-y / (x² + y²)) = (x - iy) / (x² + y²). This is an expression for the derivative of the complex logarithm in terms of the real and imaginary parts x and y of z.

To express the derivative in a more compact form, we can recognize that x - iy is the complex conjugate of z = x + iy, denoted as z̄. Also, x² + y² is the square of the magnitude of z, which we can write as |z|² = z z̄. Therefore, we can rewrite the derivative as f'(z) = z̄ / |z|² = z̄ / (z z̄). Canceling the common factor z̄, we obtain the derivative of the complex logarithm as f'(z) = 1/z. This elegant result shows that the derivative of the complex logarithm is simply the reciprocal of the complex number z, which is analogous to the derivative of the natural logarithm in the real domain.

It is worth noting that the derivative f'(z) = 1/z is defined for all non-zero complex numbers, which is consistent with the differentiability of the complex logarithm. The derivative is undefined at z = 0, which corresponds to the singularity of the complex logarithm at the origin. This singularity arises from the multi-valued nature of the complex logarithm and the fact that the argument of 0 is undefined. The derivative 1/z is a fundamental result in complex analysis and is used extensively in various applications, such as contour integration and solving complex differential equations.

Using the alternative expression for the derivative, f'(z) = ∂v/∂y - i ∂u/∂y, we can also verify our result. Substituting the values of the partial derivatives, we get f'(z) = (x / (x² + y²)) - i (y / (x² + y²)) = (x - iy) / (x² + y²), which is the same expression we obtained earlier. This confirms the consistency of our calculations and the validity of the Cauchy-Riemann equations in determining the derivative of the complex logarithm.

In this exploration, we have embarked on a comprehensive journey to understand the properties of the complex logarithm and its adherence to the Cauchy-Riemann equations. We began by defining the complex logarithm as the inverse of the complex exponential function and expressing it in terms of its real and imaginary parts. We then computed the partial derivatives of these real and imaginary parts and demonstrated that they satisfy the Cauchy-Riemann equations for all non-zero complex numbers. This verification confirmed the differentiability of the complex logarithm in the complex plane, excluding the origin.

Furthermore, we derived the derivative of the complex logarithm using the Cauchy-Riemann equations and found that it is given by f'(z) = 1/z. This result mirrors the familiar derivative of the natural logarithm in the real domain and highlights the close relationship between real and complex analysis. The derivative 1/z is a fundamental result in complex analysis and plays a crucial role in various applications.

This analysis underscores the significance of the Cauchy-Riemann equations in determining the differentiability of complex functions. These equations provide a powerful tool for verifying whether a complex function possesses a derivative and for computing that derivative when it exists. The complex logarithm, as a fundamental function in complex analysis, serves as an excellent example of how the Cauchy-Riemann equations can be applied to understand the behavior of complex functions.

The study of the complex logarithm and its properties extends beyond theoretical considerations. It has practical applications in various fields, including electrical engineering, fluid dynamics, and quantum mechanics. The complex logarithm is used in signal processing, circuit analysis, and the study of wave phenomena. Its ability to represent multi-valued functions and its connection to the complex exponential make it an indispensable tool in these areas. Understanding the complex logarithm and its derivative is essential for anyone working with complex functions and their applications.

In conclusion, we have successfully shown that the real and imaginary parts of the complex logarithm satisfy the Cauchy-Riemann equations when z is not zero. We have also derived its derivative, further solidifying our understanding of this important complex function. This exploration highlights the power of complex analysis and its ability to extend familiar concepts from the real domain to the complex domain, providing a rich and versatile framework for mathematical analysis and its applications.