Real Analysis
Prerak Srivastava
Agrim Rawat
Monsoon Semester @ IIITH '22
Complex Analysis
References
Complex Numbers Crash Course3B1B Video on Complex Analysis
Introduction
Problem
$$ \text{Find the solutions of the given equations} \\x^2 + 1 = 0 \\ x^2 + 4x + 5 = 0 \\ $$
Solution
• We define $i$ to be the "imaginary unit"
$$ i = \sqrt{-1} $$
Solution
$$ x^2 + 1 = 0 \\ x^2 = -1 \\ x = \pm i \\
\\ ------- \\ x^2 + 4x + 5 = 0
\\ D = b^2 - 4ac
\\ D = 4^2 - 4(1)(5)
\\ D = -4
\\ x = \frac{-b \pm \sqrt{D}}{2a}
\\ x = -2 \pm i $$
General Notation
• Cartesian Form - $a+ib$
• Polar Form - $r e^{i\theta}$
• $r = \sqrt{a^2 + b^2}$
• $\theta = \tan^{-1}(\frac{b}{a})$
Conventions and Notations
• The math is similar to that of vectors
• Real Part - $Re(z) = r\cos(\theta) = a$
• Imaginary Part - $Im(z) = r\sin(\theta) = b$
• Modulus - $|z| = r = \sqrt{a^2 + b^2}$ - always non-negative
• Argument - $arg(z) = \theta = \tan^{-1}(\frac{b}{a}) $
• Principle Argument - $arg(z) \in (-\pi, \pi] $ - pick smaller angle
• Anticlockwise angle is considered positive
Property of $i$
• $i^2 = -1$
• $i^3 = -i$
• $i^4 = 1$
• $i^5 = i$
$$
i^n =
\begin{cases}
1 & ,\ n = 4x \\
i & ,\ n = 4x + 1 \\
-1 & ,\ n = 4x + 2 \\
-i & ,\ n = 4x + 3 \\
\end{cases} \\ \\ \text{Here} \ x \in \mathbb{Z} \\
$$
Euler's Formula
Given
$$ |z| = r \\ Re(z) = r\cos(\theta)\\ Im(z) = r\sin(\theta)\\ z = r(\cos(\theta) + i\sin(\theta)) \\ $$
Taylor Series Expansion
$$ \text{Using taylor series expansion for sine and cosine, we get} \\ sin(x) = \sum_{j=1}^{\infty}\frac{(-1)^j}{(2j+1)!}x^{2j+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\ cos(x) = \sum_{j=0}^{\infty}\frac{(-1)^j}{(2j)!}x^{2j} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \\ $$
Proof
$$ cos(x) + isin(x) = 1 + ix - \frac{x^2}{2} - i\frac{x^3}{3} + \frac{x^4}{4} + i\frac{x^5}{5} - \frac{x^6}{6} - i\frac{x^7}{7} + \cdots \\ $$
Observation and Conclusion
$$ e^x = \sum_{j=0}^{\infty}\frac{x^j}{j!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots \\ e^{ix} = \sum_{j=0}^{\infty}\frac{(ix)^j}{j!} = 1 + ix - \frac{x^2}{2!} - i\frac{x^3}{3!} + \frac{x^4}{4!} + i\frac{x^5}{5!} - \frac{x^6}{6!} - i\frac{x^7}{7!} + \cdots \\ \therefore e^{ix} = cos(x) + isin(x) \\ $$
Properties of Modulus and Amplitude
Modulus Function
$$ |z_1z_2| = |z_1||z_2| \\ |\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|} \\ z - \bar{z} = |z|^2 = z\bar{z} \\ |z^n| = |z|^n $$
Amplitude Function
Amplitude function works similar to the logarithmic function $y = log(x)$ $$ arg(z_1z_2) = arg(z_1) + arg(z_2) \\ arg(\frac{z_1}{z_2}) = arg(z_1) - arg(z_2) \\ arg(z^n) = n \cdot arg(z) \\ arg(x) = 0 \ \forall \ x \in R^{+} \\ arg(x) = \pi \ \forall \ x \in R^{-} $$
Some More Properties of Complex Numbers
Addition and Subtraction
$$ z_1 + z_2 = (a_1 + b_1i) + (a_2 + b_2i) = (a_1 + a_2) + (b_1 + b_2)i \\ z_1 - z_2 = (a_1 + b_1i) - (a_2 + b_2i) = (a_1 - a_2) + (b_1 - b_2)i \\ $$
Multiplication and Division
$$ z_1z_2 = (a_1 + b_1i)(a_2 + b_2i) = (a_1a_2 - b_1b_2) + (a_1b_2 + a_2b_1)i \\ z_1z_2 = (r_1e^{i\theta_1})(r_2e^{i\theta_2}) = r_1r_2e^{i(\theta_1 + \theta_2)} \\ \frac{z_1}{z_2} = \frac{r_1e^{i\theta_1}}{r_2e^{i\theta_2}} = r_1e^{i(\theta_1 - \theta_2)} \\ \frac{a_1 + ib_1}{a_2 + ib_2} = \frac{a_1 + ib_1}{a_2 + ib_2} \cdot \frac{a_2 - ib_2}{a_2 - ib_2} = \frac{a_1a_2 + b_1b_2}{a_2^2 + b_2^2} + \frac{a_2b_1 - a_1b_2}{a_2^2 + b_2^2}i \\ $$
De-Moivre's Theorem
$$ z = r e^{i\theta} \\ z^n = r^n e^{i\theta n} \\ z^n = r^n(cos(n\theta ) + isin(n\theta)) \\ \text{At } r = 1 \\ (cos(\theta) + isin(\theta))^n = cos(n\theta) + isin(n\theta) \\ $$
Triangular Inequality
$$ |z_1 + z_2| = \sqrt{|z_1|^2 + |z_2|^2 + 2|z_1||z_2|\cos(\phi)} \\ \phi = arg(z_1 - z_2) \\ \therefore |z_1 + z_2| \leq |z_1| + |z_2| \\ \text{Equality is reached at } \phi = 0 \\ \text{The above formula can be generalized to} \\ \sum_{i=1}^n |z_i| \leq \sum_{i=1}^n |z_i| \\ $$
$n^{th}$ root
$$ \text{For 2 complex numbers} \ w \ \text{and} \ z \\ \text{Let } w^n = z \\ w = r_1e^{i\theta_1} \ \text{and} \ z = r_2e^{i\theta_2} \\ z^n = r_2^n e^{i\theta_2n} \\ \therefore r_1^n = r_2 \ \text{and} \ \theta_1n = \theta_2 + 2k\pi \\ 0 \leq k \leq n - 1 \\ \therefore w = r_2^{\frac{1}{n}}e^{i\frac{\theta_2 + 2k\pi}{n}} \\ $$
Examples
Find the modulus and amplitude of the following complex numbers
1. $0$
2. $i^i$
3. $z = a^i, a \in R$
4. $\log(-2)$
Solution
1.1
It is just a matter of convention, it can be considered to be 0 or undefined.