Find the real and imaginary parts of each of the following...

Find the real and imaginary parts of each of the following complex numbers: $$ \begin{gathered} \frac{\mathrm{i}-1}{\mathrm{i}+1} ; \quad \frac{3+4 \mathrm{i}}{1-2 \mathrm{i}} ; \mathrm{i}^{n}, n \in \mathbb{Z} ; \quad\left(\frac{1+\mathrm{i}}{\sqrt{2}}\right)^{n}, n \in \mathbb{Z} ; \\ \left(\frac{1+\mathrm{i} \sqrt{3}}{2}\right)^{n}, n \in \mathbb{Z} ; \quad \sum_{\nu=0}^{7}\left(\frac{1-\mathrm{i}}{\sqrt{2}}\right)^{\nu} ; \frac{(1+\mathrm{i})^{4}}{(1-\mathrm{i})^{3}}+\frac{(1-\mathrm{i})^{4}}{(1+\mathrm{i})^{3}} \end{gathered} $$

Find the real and imaginary parts of each of the following complex numbers:
$$
\begin{gathered}
\frac{\mathrm{i}-1}{\mathrm{i}+1} ; \quad \frac{3+4 \mathrm{i}}{1-2 \mathrm{i}} ; \mathrm{i}^{n}, n \in \mathbb{Z} ; \quad\left(\frac{1+\mathrm{i}}{\sqrt{2}}\right)^{n}, n \in \mathbb{Z} ; \\
\left(\frac{1+\mathrm{i} \sqrt{3}}{2}\right)^{n}, n \in \mathbb{Z} ; \quad \sum_{\nu=0}^{7}\left(\frac{1-\mathrm{i}}{\sqrt{2}}\right)^{\nu} ; \frac{(1+\mathrm{i})^{4}}{(1-\mathrm{i})^{3}}+\frac{(1-\mathrm{i})^{4}}{(1+\mathrm{i})^{3}}
\end{gathered}
$$

Key Concepts

Complex Number Arithmetic

This concept deals with operations on complex numbers, which are typically written in the form a + bi, where a represents the real part and b represents the imaginary part. It includes techniques for addition, subtraction, multiplication, and division, and it is fundamental for isolating the real and imaginary components of expressions.

Complex Conjugation and Rationalization

The complex conjugate of a complex number a + bi is a - bi. This tool is essential for rationalizing denominators in complex fractions: by multiplying the numerator and denominator by the conjugate of the denominator, one can eliminate the imaginary unit from the denominator, making the expression easier to analyze and simplifying the process of extracting real and imaginary parts.

Polar Form Representation

Expressing complex numbers in polar form involves rewriting them as r(cos? + i sin?), where r is the modulus (or absolute value) and ? is the argument (or angle). This representation is particularly useful for simplifying the multiplication, division, and exponentiation of complex numbers, as these operations translate into operations on r and ?.

Euler’s Formula

Euler’s formula, stated as e^(i?) = cos? + i sin?, provides a powerful connection between the exponential function and trigonometric functions. It allows complex numbers to be expressed in exponential form, which is especially beneficial when dealing with powers and roots, as it leads to compact and efficient computations.

De Moivre’s Theorem

De Moivre’s theorem is a direct consequence of Euler’s formula and states that (cos? + i sin?)^n = cos(n?) + i sin(n?) for an integer n. This theorem is indispensable for computing powers of complex numbers, particularly when they are represented in polar or exponential form, and it aids in identifying the real and imaginary parts of these powers.

Geometric Series of Complex Numbers

When summing powers of complex numbers, one often encounters a geometric series, where each term is obtained by multiplying the previous term by a fixed complex ratio. Using the geometric series formula allows for the summation of such sequences in closed form, facilitating the determination of the resulting complex expression’s real and imaginary components.

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