Understanding the Argument of a Complex Number Squared

In the context of complex numbers, the concept of the argument is fundamental. The argument of a complex number (z), denoted as (arg(z)), is the angle formed between the positive real axis and the line representing the complex number on the complex plane. Given a complex number (z) expressed in polar form as (z re^{itheta}), where (r) is the modulus and (theta) is the argument, we can explore how the argument of (z^2) is calculated.

Calculating the Argument of (z^2)

Starting with the expression for (z) in polar form:

[z re^{itheta}]

When we square (z), we get:

[z^2 (re^{itheta})^2 r^2 e^{2itheta}]

This means that the argument of (z^2) is:

[arg (z^2) 2theta]

This relationship is a direct consequence of the properties of exponents and Euler's formula. It tells us that the argument of the square of a complex number is twice the argument of the original complex number.

Additional Insights and Considerations

The above explanation is based on the assumption that the original angle (theta) is in the first quadrant, where both the real and imaginary parts of the complex number are positive. However, in more general cases, the argument might need adjustments to maintain it within the principal range. Different conventions for the principal argument can lead to slight variations in the argument of the squared complex number.

For example, when working with two complex numbers (Z_1) and (Z_2), the argument of their product is the sum of their individual arguments. If (arg(Z_1) x) and (arg(Z_2) y), then:

[arg(Z_1 cdot Z_2) x y]

Applying this to the square of a complex number (z):

[arg(Z^2) arg(Z) arg(Z) 2theta]

Conclusion

In summary, the argument of (z^2) is (2theta), where (theta) is the argument of the original complex number (z). This result is derived directly from the polar form of complex numbers and the properties of exponential functions. However, it's essential to consider the principal range conventions when working with complex arguments in more general settings.

Understanding these concepts is crucial for advanced applications in mathematics, physics, and engineering that involve complex numbers.