91Ó°ÊÓ

When dealing with complex numbers, the modulus is a fundamental concept. It measures the absolute distance of a complex number from the origin in the complex plane. For a complex number expressed as \( z = x + yi \), the modulus \( |z| \) is calculated using the formula:

• \( |z| = \sqrt{x^2 + y^2} \)

This formula is essentially derived from the Pythagorean theorem, where \( x \) and \( y \) represent the real and imaginary components of \( z \).
The modulus offers a geometric representation and helps understand the size or magnitude of the complex number. For instance, in the exercise, we compute \( |z| \) for different complex numbers:

• For \( z = 1 \), \( |1| = 1 \)
• For \( z = 4i \), \( |4i| = 4 \)
• For \( z = 1+i \), \( |1+i| = \sqrt{2} \)

Understanding the modulus is crucial for various applications in physics, engineering, and mathematics, especially when dealing with wave functions, signal processing, and other domains requiring complex number manipulation.

The argument of a complex number provides the angle direction from the positive real axis to the line representing the complex number in the complex plane. For a complex number \( z = x + yi \), the argument, usually denoted as \( \operatorname{Arg}(z) \), can be calculated with:

• \( \operatorname{Arg}(z) = \tan^{-1}\left(\frac{y}{x}\right) \)

This value is typically given in radians and can be adjusted depending on the quadrant in which the point \( (x, y) \) is located.
In the original exercise, we find:

• \( \operatorname{Arg}(1) = 0 \) (on the real axis)
• \( \operatorname{Arg}(4i) = \frac{\pi}{2} \) (positive imaginary axis)
• \( \operatorname{Arg}(1+i) = \frac{\pi}{4} \) (first quadrant)

Calculating the argument is pivotal in converting complex numbers into polar form and is widely used in the analysis of AC circuits and signal transformations.

Logarithms play an essential role in transforming multiplicative relationships into additive ones. When applied to complex numbers, the logarithm, particularly of the modulus \( |z| \), helps in obtaining the real part of the complex function. For a function like \( f(z) = \log_{e}|z| + i \operatorname{Arg}(z) \), \( \log_{e}|z| \) feeds into the real component.

• The natural logarithm \( \log_{e} \) or \( \ln \) is commonly used.

In the solution, the logarithm part helped evaluate:

• For \( z = 1 \), \( \log_{e}(1) = 0 \)
• For \( z = 4i \), \( \log_{e}(4) \)
• For \( z = 1+i \), \( \log_{e}(\sqrt{2}) \), which simplifies to \( \frac{1}{2}\log_{e}(2) \)

Understanding logarithms in the context of complex functions is crucial for fields like signal processing and control systems, where phase and amplitude undergo transformations.

Evaluating complex functions is about understanding the structure and components of the complex number involved. In our exercise, the function \( f(z) = \log_{e}|z| + i \operatorname{Arg}(z) \) combines both the modulus and argument to form its outcome. Here's the step-by-step thought process:

• Identify the modulus \( |z| \) and argument \( \operatorname{Arg}(z) \) of the given complex number.
• Calculate the logarithm of the modulus.
• Multiply the argument by \( i \) to obtain the imaginary part.
• Combine both parts to conclude the evaluation.

For instance:

• At \( z = 1 \), the result is 0 as \( \log_{e}(1) = 0 \) and \( \operatorname{Arg}(1) = 0 \).
• At \( z = 4i \), results in \( \log_{e}(4) + i \frac{\pi}{2} \) as \( \operatorname{Arg}(4i) = \frac{\pi}{2} \).
• At \( z = 1+i \), results in \( \frac{1}{2}\log_{e}(2) + i \frac{\pi}{4} \) since \( |1+i| = \sqrt{2} \) and \( \operatorname{Arg}(1+i) = \frac{\pi}{4} \).

These evaluations are vital for understanding phases and magnitudes in complex planes, especially in engineering and physics applications.