1) Introduction: Entering the Complex Plane 📚
Complex analysis extends calculus to complex numbers, focusing on contour integration along paths in the complex plane and the residue theorem for evaluating integrals. These tools are essential in physics (e.g., quantum mechanics), engineering, and number theory, offering elegant solutions to real integrals.
We’ll cover:
- Complex Functions: Analyticity and Cauchy-Riemann equations.
- Contour Integration: Integrating along closed curves.
- Residue Theorem: Computing integrals using residues.
Quick Recap: Real integrals are familiar; now we generalize to complex domains.
2) Complex Functions and Analyticity 🎓
A complex function \( f(z) = u + iv \) is analytic if it has a derivative.
Definition 31.1: Analytic Function
\( f(z) \) is analytic if \( f'(z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} = \frac{\partial v}{\partial y} - i \frac{\partial u}{\partial y} \), satisfying Cauchy-Riemann equations.
Example 1: Check Analyticity
Is \( f(z) = z^2 \) analytic?
- \( z = x + iy \), \( f = (x + iy)^2 = x^2 - y^2 + 2ixy \).
- \( u = x^2 - y^2 \), \( v = 2xy \).
- \( \frac{\partial u}{\partial x} = 2x \), \( \frac{\partial v}{\partial y} = 2x \); \( \frac{\partial u}{\partial y} = -2y \), \( \frac{\partial v}{\partial x} = 2y \).
- Yes, analytic.
Answer: Yes.
Example 2: Non-Analytic Function
Is \( f(z) = \bar{z} \) analytic?
- \( \bar{z} = x - iy \), \( u = x \), \( v = -y \).
- \( \frac{\partial u}{\partial x} = 1 \neq -1 = \frac{\partial v}{\partial y} \).
- No, not analytic.
Answer: No.
3) Contour Integration 📐
Contour integration evaluates integrals along paths in the complex plane.
Definition 31.2: Contour Integral
\( \int_C f(z) \, dz = \int_a^b f(z(t)) z'(t) \, dt \), where \( C \) is parameterized by \( z(t) \), \( t \in [a, b] \).
Example 3: Simple Contour
Evaluate \( \int_C z \, dz \) where \( C \) is \( z = e^{it} \), \( 0 \leq t \leq 2\pi \).
- \( z' = i e^{it} \), \( \int_0^{2\pi} e^{it} \cdot i e^{it} \, dt = i \int_0^{2\pi} e^{2it} \, dt \).
- \( = i \left[\frac{e^{2it}}{2i}\right]_0^{2\pi} = 0 \).
Answer: \( 0 \).
Example 4: Real Integral via Contour
Compute \( \int_{-\infty}^{\infty} \frac{1}{1 + x^2} \, dx \).
- Use \( f(z) = \frac{1}{1 + z^2} \), contour as semicircle in upper half-plane.
- Pole at \( z = i \), residue = \( \frac{1}{2i} \).
- \( 2\pi i \cdot \frac{1}{2i} = \pi \).
Answer: \( \pi \).
4) Residue Theorem 🔍
The residue theorem evaluates integrals using singularities.
Definition 31.3: Residue Theorem
\( \oint_C f(z) \, dz = 2\pi i \sum \text{Res}(f, a_k) \), where \( a_k \) are poles inside \( C \), and residue is \( \lim_{z \to a} (z - a) f(z) \) for simple poles.
Example 5: Single Pole
Evaluate \( \oint_{|z|=2} \frac{1}{z-1} \, dz \).
- Pole at \( z = 1 \), residue = 1.
- \( 2\pi i \cdot 1 = 2\pi i \).
Answer: \( 2\pi i \).
Example 6: Multiple Poles
Compute \( \int_{-\infty}^{\infty} \frac{1}{(z^2 + 1)^2} \, dz \).
- Poles at \( z = \pm i \), order 2.
- Residue at \( z = i \): \( \lim_{z \to i} \frac{d}{dz} (z - i)^2 \frac{1}{(z^2 + 1)^2} = \frac{-1}{4i} \).
- Total: \( 2\pi i \cdot \frac{-1}{4i} = \frac{\pi}{2} \).
Answer: \( \frac{\pi}{2} \).
5) Applications in Real Integrals 🔍
Complex analysis solves real integrals unavailable via real methods.
Definition 31.4: Real Integral Application
Use closed contours to capture residues for \( \int_{-\infty}^{\infty} f(x) \, dx \).
Example 7: Gaussian Integral
Evaluate \( \int_{-\infty}^{\infty} e^{-x^2} \, dx \).
- Use \( f(z) = e^{-z^2} \), contour as square, no poles in finite plane.
- Relate to \( \iint e^{-(x^2 + y^2)} \, dA = \pi \).
Answer: \( \sqrt{\pi} \).
Example 8: Trigonometric Integral
Compute \( \int_0^{\infty} \frac{\cos x}{1 + x^2} \, dx \).
- Use \( f(z) = \frac{e^{iz}}{1 + z^2} \), pole at \( z = i \).
- Residue, imaginary part gives result.
Answer: \( \frac{\pi}{2e} \).
6) Practice Questions 🎯
Fundamental Practice Questions 🌱
Instructions: Evaluate the contour integrals or check analyticity. 📚
Is \( f(z) = e^z \) analytic?
\( \int_{|z|=1} z^2 \, dz \)
Is \( f(z) = \ln|z| + i \arg(z) \) analytic?
\( \int_{|z|=2} \frac{1}{z} \, dz \)
\( \int_0^{\infty} \frac{1}{1 + x^2} \, dx \) (complex contour)
Residue of \( \frac{1}{z^2 - 1} \) at \( z = 1 \)
\( \int_{|z|=3} \frac{z}{z^2 + 4} \, dz \)
Is \( f(z) = \bar{z}^2 \) analytic?
\( \int_{-\infty}^{\infty} \frac{x}{x^2 + 9} \, dx \) (complex method)
\( \oint_{|z|=1} \frac{e^z}{z^2} \, dz \)
Residue of \( \frac{1}{(z-2)^3} \) at \( z = 2 \)
Challenging Practice Questions 🌟
Instructions: Solve these advanced complex analysis problems. 🧠
Evaluate \( \int_0^{\infty} \frac{\sin x}{x(x^2 + 1)} \, dx \) using contour integration.
Find the residues of \( \frac{z^2}{(z-1)^2(z+2)} \) and compute \( \oint_{|z|=3} f(z) \, dz \).
Determine \( \int_{-\infty}^{\infty} \frac{e^{iax}}{x^2 + 1} \, dx \) for real \( a \) using the residue theorem.
Compute \( \oint_{|z|=2} \frac{\cos z}{z^4 - 1} \, dz \) and identify all poles.
Evaluate \( \int_0^{\infty} \frac{x^{a-1}}{1 + x} \, dx \) for \( 0 < a < 1 \) using a keyhole contour.
7) Summary & Cheat Sheet 📋
7.1) Analyticity
Requires Cauchy-Riemann equations: \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \), \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \).
7.2) Contour Integration
\( \int_C f(z) \, dz = \int_a^b f(z(t)) z'(t) \, dt \).
7.3) Residue Theorem
\( \oint_C f(z) \, dz = 2\pi i \sum \text{Res}(f, a_k) \).
You’ve mastered complex analysis basics! Next, we’ll tackle advanced problem solving. 🎉