Complex Analysis Notes
1. Definition & Field Structure
The set of complex numbers is defined as:
This forms a field under the following operations[cite: 1, 2]:
- Addition: \[ (a + ib) + (c + id) = (a + c) + i(b + d) \]
- Multiplication: \[ (a + ib) \cdot (c + id) = (ac – bd) + i(ad + bc) \]
2. Multiplicative Inverse
[cite_start]For a non-zero complex number \( z = a + ib \), the inverse is given by[cite: 4]:
Let \( (a+ib)^{-1} = c+id \). Since the product must range the identity \( I \): \[ \begin{aligned} (a+ib)(c+id) &= 1 \\ (a-ib)(a+ib)(c+id) &= 1 \cdot (a-ib) \\ (a^2 + b^2)(c+id) &= a – ib \\ c+id &= \frac{a-ib}{a^2+b^2} \end{aligned} \]
Examples:
-
[cite_start]
- Ex 1: \( (1+i)^{-1} = \frac{1-i}{1^2+1^2} = \frac{1-i}{2} \) [cite: 5] [cite_start]
- Ex 2: \( (2+3i)^{-1} = \frac{2-3i}{2^2+3^2} = \frac{2-3i}{13} \) [cite: 5]
3. The Argand Plane
Every point \( z = x + iy \in C \) can be represented as a coordinate \( (x,y) \) in \( \mathbb{R}^2 \). [cite_start]This Cartesian Plane is called the Complex Plane or Argand Plane [cite: 6-9].
[cite_start]Example: If \( z = 1 + 2i \), then the coordinates are \( (1, 2) \)[cite: 10, 11].
4. Polar Form
[cite_start]For \( z = x + iy \), the polar representation is [cite: 13-18]:
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)
- \( r = \sqrt{x^2 + y^2} \) (Modulus)
- \( \theta = \) Angle from positive x-axis
Note 2: \( \theta, \theta+2\pi, \dots \) represent the same position. [cite_start]We focus on the Principal Argument [cite: 21-25].
5. Principal Argument (\( \text{Arg } z \))
Defined for \( z = x + iy \) where \( z \neq 0 \). [cite_start]The range is[cite: 26, 41]:
Algorithm to Find Argument[cite: 27, 28]:
- Find reference angle \( \alpha = \tan^{-1} |\frac{y}{x}| \) such that \( 0 \le \alpha < \frac{\pi}{2} \).
- Determine \( \text{Arg } z \) based on the quadrant:
| Condition ($x, y$) | Quadrant / Axis | [cite_start]Arg \( z \) Formula [cite: 29-37] |
|---|---|---|
| \( x > 0, y > 0 \) | 1st Quadrant | \( \alpha \) |
| \( x < 0, y > 0 \) | 2nd Quadrant | \( \pi – \alpha \) |
| \( x < 0, y < 0 \) | 3rd Quadrant | \( -(\pi – \alpha) \) |
| \( x > 0, y < 0 \) | 4th Quadrant | \( -\alpha \) |
| \( x > 0, y = 0 \) | + Real Axis | \( 0 \) |
| \( x = 0, y > 0 \) | + Imaginary Axis | \( \frac{\pi}{2} \) |
| \( x < 0, y = 0 \) | – Real Axis | \( \pi \) |
| \( x = 0, y < 0 \) | – Imaginary Axis | \( -\frac{\pi}{2} \) |
6. Practice Questions on Principal Argument
Determine the principal argument for the following complex numbers:
1. Identify \( x = 1, y = \sqrt{3} \).
2. Calculate \( \alpha = \tan^{-1}|\frac{\sqrt{3}}{1}| = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \).
3. Since \( x > 0, y > 0 \) (1st Quadrant), formula is \( \alpha \).
Ans: \( \frac{\pi}{3} \)
1. Identify \( x = -1, y = 1 \).
2. Calculate \( \alpha = \tan^{-1}|\frac{1}{-1}| = \tan^{-1}(1) = \frac{\pi}{4} \).
3. Since \( x < 0, y > 0 \) (2nd Quadrant), formula is \( \pi – \alpha \).
Ans: \( \pi – \frac{\pi}{4} = \frac{3\pi}{4} \)
1. Identify \( x = -1, y = -\sqrt{3} \).
2. Calculate \( \alpha = \tan^{-1}|\frac{-\sqrt{3}}{-1}| = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \).
3. Since \( x < 0, y < 0 \) (3rd Quadrant), formula is \( -(\pi - \alpha) \).
Ans: \( -(\pi – \frac{\pi}{3}) = -\frac{2\pi}{3} \)
1. Identify \( x = 1, y = -1 \).
2. Calculate \( \alpha = \tan^{-1}|\frac{-1}{1}| = \tan^{-1}(1) = \frac{\pi}{4} \).
3. Since \( x > 0, y < 0 \) (4th Quadrant), formula is \( -\alpha \).
Ans: \( -\frac{\pi}{4} \)
1. Identify \( x = 0, y = -5 \).
2. Since \( x = 0 \) and \( y < 0 \), this lies on the Negative Imaginary Axis.
3. Apply specific axis rule.
Ans: \( -\frac{\pi}{2} \)
Complex Exponents & Logarithms
Based on Pages 13–14 of the provided document.
1. Definitions (Page 13)
We define complex powers using the complex logarithm.
If \(z_1\) and \(z_2\) are complex numbers, then \(z_1^{z_2}\) is defined as: \[ z_1^{z_2} = e^{z_2 \cdot \log(z_1)} \]
The Complex Logarithm
Since the argument of a complex number is multi-valued (\(\theta + 2n\pi\)), the logarithm is also multi-valued:
where \( \ln|z| \) is the standard natural log of the modulus.
Principal Value (P.V.)
The Principal Value is obtained by taking the principal branch of the argument (where \(n=0\)):
2. Numerical Examples (Page 14)
The notes provide two specific examples calculating the Principal Value of complex exponents.
Example 1: Find P.V. of \((1+i)^i\)
Step 1: Convert Base to Polar FormFor \(z = 1+i\): \[ |z| = \sqrt{1^2 + 1^2} = \sqrt{2} \] \[ \text{Arg } z = \tan^{-1}(1/1) = \frac{\pi}{4} \]
Step 2: Calculate Principal Logarithm\[ \log(1+i) = \ln(\sqrt{2}) + i\left(\frac{\pi}{4}\right) \]
Step 3: Apply Exponent FormulaWe want \(e^{i \cdot \log(1+i)}\): \[ \text{Exponent} = i \left[ \ln\sqrt{2} + i\frac{\pi}{4} \right] \] \[ = i\ln\sqrt{2} + i^2\frac{\pi}{4} \] \[ = i\ln\sqrt{2} – \frac{\pi}{4} \]
Step 4: Final Result\[ (1+i)^i = e^{-\pi/4 + i\ln\sqrt{2}} \] \[ = e^{-\pi/4} \left[ \cos(\ln\sqrt{2}) + i\sin(\ln\sqrt{2}) \right] \]
Example 2: Find P.V. of \((1+i)^{1+i}\)
Step 1: SetupUsing the same log value from Example 1: \(\log(1+i) = \ln\sqrt{2} + i\frac{\pi}{4}\).
We calculate:
\[ (1+i)^{1+i} = e^{(1+i) \cdot \log(1+i)} \]
\[ (1+i) \left( \ln\sqrt{2} + i\frac{\pi}{4} \right) \] Expand the product: \[ = 1(\ln\sqrt{2}) + i\frac{\pi}{4} + i(\ln\sqrt{2}) + i^2\frac{\pi}{4} \] \[ = \ln\sqrt{2} + i\frac{\pi}{4} + i\ln\sqrt{2} – \frac{\pi}{4} \]
Step 3: Group Real and Imaginary Parts\[ \text{Real Part} = \ln\sqrt{2} – \frac{\pi}{4} \] \[ \text{Imaginary Part} = \frac{\pi}{4} + \ln\sqrt{2} \]
Step 4: Final Result\[ (1+i)^{1+i} = e^{\left(\ln\sqrt{2} – \frac{\pi}{4}\right)} \cdot e^{i\left(\frac{\pi}{4} + \ln\sqrt{2}\right)} \]
The Difference: ln vs log
Domain: Real Numbers Only
What it is: The standard Natural Logarithm (base \(e\)).
Input: It takes the Modulus (Length) of the number. Must be a positive real number.
Output: A Real Number.
Example:
\( \ln(\sqrt{2}) \approx 0.346 \)
Domain: Complex Numbers
What it is: The multi-valued Complex Logarithm.
Input: It takes the Entire Complex Number (The vector \(z\)).
Output: A Complex Number (Has Real + Imaginary parts).
Example:
\( \log(1+i) = 0.346 + i\frac{\pi}{4} \)
How they fit together:
In your specific example equation:
“The Complex Log is built by taking the Real ln of the length, and adding the imaginary direction.”