Problem 22 - Complex Analysis

Question

For \(z, w \in \mathbb{C}^{*}\) show $$ \begin{aligned} \log (z w) &=\log (z)+\log (w)+2 \pi \mathrm{i} k(z, w) \quad \text { where } \\\ k(z, w) &=\left\\{\begin{aligned} 0, & \text { if }-\pi<\operatorname{Arg} z+\operatorname{Arg} w \leq \pi \\ +1, & \text { if }-2 \pi<\operatorname{Arg} z+\operatorname{Arg} w \leq-\pi \\\ -1, & \text { if } \quad \pi<\operatorname{Arg} z+\operatorname{Arg} w \leq 2 \pi \end{aligned}\right. \end{aligned} $$

Step-by-Step Solution

Answer

The complex logarithm follows the principal value range with adjustments by \( 2\pi \text{i} k(z, w) \) to ensure \( \log(zw) = \log(z) + \log(w) + 2\pi i k(z, w) \).

Step 1: Understanding Argument Addition in Complex Multiplication

The argument of a product of two complex numbers \( z \) and \( w \) is the sum of their individual arguments. That is, \( \operatorname{Arg}(zw) = \operatorname{Arg}(z) + \operatorname{Arg}(w) \). However, due to the periodicity of the complex logarithm, we must consider the principal value of the argument, which lies in the range \((-\pi, \pi] \).

Step 2: Analyzing the Principal Value

Given \( \operatorname{Arg}(zw) = \operatorname{Arg}(z) + \operatorname{Arg}(w) \), we need to place this sum within the principal range \((-\pi, \pi] \). This is where the function \( k(z, w) \) comes into play, adjusting the argument by adding \( 2\pi \) or subtracting \( 2\pi \) if necessary, thus forming the correct principal value for \( \log(zw) \).

Step 3: Defining the Function k(z, w)

The function \( k(z, w) \) is designed to correct the total argument so it fits within the principal range. It steps the argument up or down by \( 2 \pi \) depending on the sum of \( \operatorname{Arg}(z) \) and \( \operatorname{Arg}(w) \). Hence, it evaluates as follows:- \( k(z, w) = 0 \), if \( -\pi < \operatorname{Arg}(z) + \operatorname{Arg}(w) \leq \pi \).- \( k(z, w) = +1 \), if \( -2\pi < \operatorname{Arg}(z) + \operatorname{Arg}(w) \leq -\pi \).- \( k(z, w) = -1 \), if \( \pi < \operatorname{Arg}(z) + \operatorname{Arg}(w) \leq 2\pi \).

Step 4: Combining All Parts of the Formula

Combine the terms following the adjusted argument rule. Thus, the natural logarithm in complex numbers satisfies:\[\log (z w) = \log(z) + \log(w) + 2\pi i k(z, w)\]This indicates the correct adjustment applied to ensure the principal value range for \( \operatorname{Arg}(zw) \).

Step 5: Verifying the Correctness of k(z, w)

Substitute example arguments and verify that \( k(z, w) \) adjusts the arguments correctly to remain within \((-\pi,\pi]\). For example, check if \( z=e^{i3\pi/4}, w=e^{i3\pi/4} \), their arguments sum to \(3\pi/2\), leading \( k(z, w) = -1 \), correcting the final argument to \(-\pi/2\).

Key Concepts

Argument of a Complex Number

In the realm of complex numbers, the argument is a key aspect that describes the angle a complex number makes with the positive direction of the real axis. For a complex number in the form of \( z = a + bi \), the argument \( \operatorname{Arg}(z) \) is typically given by \( \theta = \tan^{-1}(b/a) \). This angle is measured in radians and captures the directional nature of the complex number in the complex plane.
The argument is crucial when dealing with complex multiplication. Specifically, the argument of the product of two complex numbers \( z \) and \( w \) is simply the sum of their arguments, expressed as \( \operatorname{Arg}(zw) = \operatorname{Arg}(z) + \operatorname{Arg}(w) \). However, this sum must be adjusted to remain within a principal range, which is usually taken to be \((-\pi, \pi] \).
Understanding how to work with arguments not just helps in multiplication, but also in complex logarithms, where arguments dictate the "angle" part of polar representations.

Principal Value

Principal value is an important concept when working with periodic functions like the logarithm of a complex number. It is the selected value that falls within a specified range, making calculations more manageable and consistent.
For the argument of a complex number, the principal value is constrained to \((-\pi, \pi] \). When calculating the logarithm of a product of complex numbers, it’s necessary to bring the resulting argument back into this range if it doesn’t naturally fit after addition.
In the provided formula, the function \( k(z, w) \) accounts for this correction. It adjusts the argument by adding or subtracting \( 2\pi \) when the summation falls outside the principal range. This ensures that the logarithm equation remains valid while maintaining the simplicity that the principal value provides.
Using this understanding, we can efficiently keep our calculations within the specified, manageable range of \((-\pi, \pi] \), facilitating the multiplicative properties aligned with complex logarithms.

Complex Multiplication

Complex multiplication involves the multiplication of two complex numbers, combining aspects of both magnitude and direction (or argument) of each number. It can be visualized as scaling by the magnitudes of the numbers and rotating by their arguments.
For example, if you have two complex numbers \( z \) and \( w \), represented in polar form as \( z = r e^{i\theta} \) and \( w = s e^{i\phi} \), their product is \( zw = rs e^{i(\theta + \phi)} \). This multiplication results in the arguments being added: \( \operatorname{Arg}(zw) = \theta + \phi \).
However, after such operations, you may find the resulting arguments outside the intended range. Thus, it’s essential to use concepts like the principal value and the periodic nature of the complex plane to adjust this back within acceptable limits, making sure the angles remain within \((-\pi, \pi] \). This careful management is also integral when working with complex logarithms.

Periodic Nature of Logarithm

The periodic nature of logarithms, particularly in the context of complex numbers, is defined by the inherent periodicity of the argument in the complex plane. This periodicity arises because rotating a complex number by \( 2\pi \) (equivalent to a complete circle around the origin) results in a value that appears the same due to its angle but has effectively completed one full cycle.
This property of periodicity can affect calculations like those involving logarithms of complex numbers. For instance, the principal logarithm function, \( \log(z) \), is multi-valued, differing by multiples of \( 2\pi i \). When calculating \( \log(zw) \), the periodic return to the range \((-\pi, \pi] \) necessitates adjustments by terms like \( 2\pi i k(z, w) \) in our formula.
This adjustment ensures that the calculated argument maintains consistency with the defined principal value range. Overall, understanding the periodic nature of logarithms helps navigate the complex landscape of angle adjustments and ensures correctness in calculations.