Problem 49 - Advanced Engineering Mathematics

Question

Since \(|z|=\sqrt{x^{2}+y^{2}}\) and \(\operatorname{Arg} z=\tan ^{-1} \frac{y}{x}\) for \(x>0\) we have \\[ \operatorname{Ln} z=\log _{e}|z|+i \operatorname{Arg} z=\log _{e}\left(x^{2}+y^{2}\right)^{1 / 2}+i \tan ^{-1} \frac{y}{x}=\frac{1}{2} \log _{e}\left(x^{2}+y^{2}\right)+i \tan ^{-1} \frac{y}{x} \\]

Step-by-Step Solution

Answer

The expression for \(\operatorname{Ln} z\) is \(\frac{1}{2} \log_e (x^2 + y^2) + i \tan^{-1}(\frac{y}{x})\).

Step 1: Understanding Magnitude

The magnitude \(|z|\) of a complex number \(z=x+iy\) is given by \(|z|=\sqrt{x^2 + y^2}\). This formula calculates the distance from \(z\) to the origin in the complex plane.

Step 2: Understanding Argument

The argument \(\operatorname{Arg} z\) of the complex number \(z=x+iy\) is computed as \(\tan^{-1}(\frac{y}{x})\) when \(x > 0\). This gives the angle formed with the positive real axis.

Step 3: Logarithm of Magnitude

Now, compute the natural logarithm of the magnitude: \(\log_e |z| = \log_e \sqrt{x^2 + y^2}\). Since \(\log_e (a^{1/2}) = \frac{1}{2}\log_e(a)\), we rewrite it as \(\frac{1}{2} \log_e (x^2 + y^2)\).

Step 4: Combine to Find Ln[z]

The complex logarithm \(\operatorname{Ln} z\) is given by combining the logarithm of the magnitude and the argument, expressed as \(\operatorname{Ln} z = \frac{1}{2} \log_e (x^2 + y^2) + i \tan^{-1}(\frac{y}{x})\). This represents the logarithm in its standard form for complex numbers.

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are typically written in the form \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part. The imaginary unit \( i \) is defined as \( \sqrt{-1} \).
Complex numbers are represented on a plane called the complex plane, where the horizontal axis is the real part, and the vertical axis is the imaginary part. This makes it easier to visualize the addition, subtraction, multiplication, and division of complex numbers.

• Addition: Add the real parts together and the imaginary parts together.
• Subtraction: Subtract the real parts and the imaginary parts.
• Multiplication: Multiply each part according to distributive property, keeping in mind \( i^2 = -1 \).
• Division: Involves multiplying the numerator and denominator by the conjugate of the denominator to simplify the expression.

Using complex numbers, many phenomena in physics and engineering can be represented, particularly involving wave behavior.

Magnitude of Complex Numbers

The magnitude, also known as the modulus, of a complex number \( z = x + iy \) is its absolute distance to the origin \( (0,0) \) on the complex plane. This is calculated using the Pythagorean theorem and is given by the formula: \[ |z| = \sqrt{x^2 + y^2} \] This formula essentially measures how 'long' the complex number vector is from the origin to the point \( z \). It is always a non-negative number and provides a sense of the size or 'absolute value' of the complex number.
Understanding the magnitude is crucial as it translates the complex number into a real number, which can be used in diverse applications, such as calculating power in electrical circuits or determining signal strength.
The graphical interpretation in the complex plane also helps in understanding how the magnitude relates to geometric distance, making it easier to connect algebraic results with graphical ones.

Argument of Complex Numbers

The argument of a complex number refers to the angle that its line forms with the positive direction of the real axis on the complex plane. For a complex number \( z = x + iy \) with \( x > 0 \), the argument is calculated using the arctangent function: \[ \operatorname{Arg} z = \tan^{-1} \left( \frac{y}{x} \right) \] This angle is usually measured in radians and can take values between \(-\pi\) and \(\pi\). The argument helps identify the direction in which the vector representing the complex number is pointing, thus conveying critical information about the orientation of \( z \).
It is important to note that when interpreting the argument, other values can also represent the same direction due to periodicity, but typically the principal value between \(-\pi\) and \(\pi\) is chosen.
Knowing the argument is essential for aspects like rotation in complex multiplication and calculating certain Fourier transforms, making it a fundamental concept in both pure and applied mathematics.