Introduction to Complex Numbers and Their Analysis

Complex numbers play a crucial role in mathematics and engineering, representing numbers that consist of a real part and an imaginary part. The argument and logarithm of complex numbers are essential concepts that help in understanding and manipulating these numbers. This article will delve into the process of finding the argument and logarithm of complex numbers, specifically focusing on the expression $frac{3i}{2-i} frac{3-i}{2i}$.

Step-by-Step Guide to Finding the Argument of a Complex Number

The argument (or phase angle) of a complex number $z a bi$ is the angle θ that the vector representing the complex number makes with the positive real axis. It can be calculated using the formula:

$text{arg}(z) arctan(frac{b}{a})$

Step 1: Simplify the Complex Expression

Let's start by simplifying the given complex expression:

$frac{3i}{2-i} frac{3-i}{2i}$

First, we rationalize the denominators:

$frac{3i}{2-i} frac{3i(2 i)}{(2-i)(2 i)} frac{6i 3i^2}{4 1} frac{6i - 3}{5} -frac{3}{5} frac{6}{5}i$

$frac{3-i}{2i} frac{(3-i)(-i)}{2i(-i)} frac{-3i i^2}{-2} frac{-i - 1}{-2} frac{1}{2} - frac{1}{2}i$

Adding these simplified expressions:

$(-frac{3}{5} frac{6}{5}i) (frac{1}{2} - frac{1}{2}i) -frac{3}{5} frac{1}{2} frac{6}{5}i - frac{1}{2}i frac{2}{10} frac{5}{10} frac{6}{5}i - frac{1}{2}i frac{7}{10} frac{6}{5}i - frac{1}{2}i frac{7}{10} frac{12}{10}i - frac{5}{10}i frac{7}{10} frac{7}{10}i frac{7}{10}(1 i)$

Step 2: Find the Argument

The simplified complex number is $frac{7}{10}(1 i)$. To find its argument:

$text{arg}left(frac{7}{10}(1 i)right) arctanleft(frac{1}{1}right) arctan(1) frac{pi}{4}$

The argument of the given complex expression is $frac{pi}{4}$ radians.

Understanding the Logarithm of a Complex Number

The logarithm of a complex number $z a bi$ can be expressed as:

$log(z) ln|z| itext{arg}(z)$

Step 1: Calculate the Magnitude

The magnitude (or modulus) of the complex number is:

$|z| sqrt{a^2 b^2}$

For $frac{7}{10}(1 i)$, the magnitude is:

$|frac{7}{10}(1 i)| frac{7}{10}sqrt{1^2 1^2} frac{7}{10}sqrt{2} frac{7sqrt{2}}{10}$

Step 2: Calculate the Logarithm

The natural logarithm of the magnitude, combined with the argument:

$log(frac{7}{10}(1 i)) lnleft(frac{7sqrt{2}}{10}right) ifrac{pi}{4}$

This expression represents the logarithm of the given complex number.

Conclusion

Understanding the argument and logarithm of complex numbers is crucial for a wide range of mathematical and engineering applications. This article has provided a detailed guide on how to find these values, specifically for the complex expression $frac{3i}{2-i} frac{3-i}{2i}$. By breaking down the steps and using the formulas, complex numbers become more manageable, enabling efficient problem-solving and analysis.

Related Keywords

argument complex number logarithm complex number complex number analysis