91Ó°ÊÓ

Complex numbers are an extension of the real numbers and consist of two parts: a real part and an imaginary part. They are often expressed in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, with \( i \) being the imaginary unit. The imaginary unit \( i \) is defined such that \( i^2 = -1 \).

These numbers can be plotted on a two-dimensional plane called the complex plane. This plane has a horizontal axis representing the real part (real axis) and a vertical axis for the imaginary part (imaginary axis). In our exercise, the complex number is \( 1+i \). Here, both the real and imaginary parts are 1. Understanding complex numbers is essential because they allow us to solve equations that real numbers cannot, especially those involving the square roots of negative numbers.

While complex numbers like \( 1 + i \) can be expressed in Cartesian form, they can also be expressed in polar form. The polar form of a complex number emphasizes its magnitude and direction rather than its horizontal and vertical components.

The polar form is represented as \( r e^{i\theta} \), where \( r \) is the magnitude or modulus of the complex number, and \( \theta \) is the angle or argument. To calculate these for \( 1 + i \):

• Magnitude (\( r \)) is found using the formula \( r = \sqrt{a^2 + b^2} \). For our example, \( r = \sqrt{1^2 + 1^2} = \sqrt{2} \).
• Argument (\( \theta \)) is the angle made with the positive real axis, determined by \( \tan^{-1}(b/a) \). Here, \( \theta = \tan^{-1}(1/1) = \frac{\pi}{4} \).

Thus, \( 1 + i \) is expressed in polar form as \( \sqrt{2} e^{i\frac{\pi}{4}} \). This representation is particularly useful when applying operations like multiplication and division or finding roots and logarithms of complex numbers.

The transformation of a complex number from its standard form to polar form involves finding both the magnitude and argument, key components in the process of calculating complex logarithms.

In the context of logarithms, the magnitude \( r \) transforms to \( \ln r \), while the argument \( \theta \) becomes part of the imaginary component in the expression. Using these transformations, we apply the complex logarithm formula for a number in polar form, \( re^{i\theta} \), which gives us:

• Real part: \( \ln r \)
• Imaginary part: \( i(\theta + 2k\pi) \)

Thus, for \( 1 + i \), we substitute: \( \ln(\sqrt{2}) + i(\frac{\pi}{4} + 2k\pi) \). Simplifying \( \ln(\sqrt{2}) \), we use the identity \( \ln(a^b) = b\ln a \), producing \( \frac{1}{2}\ln(2) \). The final solution is \( \frac{1}{2}\ln(2) + i(\frac{\pi}{4} + 2k\pi) \), where \( k \) is an integer indicating the multiple values of the logarithm due to its periodic nature.