Nguồn ngoài

Complex Methods for the Sciences

An introductory course in mathematical methods for science and engineering students. Topics covered include complex analysis, contour integration, Fourier analysis, and Green's function methods. Mathematical methods are illustrated with examples drawn from physics, with a focus on oscillations and waves. These notes are also available in the form of Jupyter notebooks on Github (GPLv3).

Mathematical Physics and Pedagogy Physics

Tác giả: Y. D. Chong
Nhà xuất bản: Nanyang Technological University
Nguồn: LibreTexts
Giấy phép: CC BY-SA
Ngôn ngữ: EN
Danh mục: Khoa học cơ bản

Đọc trực tuyến Tải PDF (nguồn mở)

Mục lục· 78 mục

01: Mathematical Functions
1.01: Real Functions
1.02: 1.2 The Exponential Function
1.03: 1.3 The Logarithm Function
1.04: Non-Natural Powers
1.05: 1.5 Trigonometric Functions
1.06: Hyperbolic Functions
1.07: Continuity
1.08: Exercises
02: Derivatives
2.01: Properties of Derivatives
2.02: Taylor Series
2.03: Ordinary Differential Equations
2.04: Partial Derivatives
2.05: Exercises
03: Integrals
3.01: Basic Properties of Definite Integrals
3.02: Integrals as Antiderivatives
3.03: Integration by Parts
3.04: Change of Variables
3.05: The Gaussian Integral
3.06: Differentiating Under the Integral Sign
3.07: Exercises
04: Complex Numbers
4.01: Complex Algebra
4.02: Conjugates and Magnitudes
4.03: Euler's Formula
4.04: The Complex Plane
4.05: Complex Functions
4.06: Trajectories in the Complex Plane
4.07: Why Complex Numbers
4.08: Exercises
05: Complex Oscillations
5.01: The Damped Harmonic Oscillator
5.02: Complex Solution
5.03: General Solution for the Damped Harmonic Oscillator
5.04: Stating the Solution in Terms of Initial Conditions
5.05: Exercises
06: Complex Waves
6.01: The Wave Equation
6.02: Real Solutions to the Wave Equation
6.03: Complex Solutions to the Wave Equation
6.04: Waves in 3D Space
6.05: Harmonic Waves
6.06: Exercises
07: Complex Derivatives
7.01: Complex Continuity and Differentiability
7.02: Analytic Functions
7.03: The Cauchy-Riemann Equations
7.04: Exercises
08: Branch Points and Branch Cuts
8.01: Non-Integer Powers as Multi-Valued Operations
8.02: Branches
8.03: Aside- The Meaning of Infinity for Complex Numbers
8.04: Branch Cuts for General Multi-Valued Operations
8.05: Exercises
09: Contour Integration
9.01: Contour Integrals
9.02: Cauchy's Integral Theorem
9.03: Poles
9.04: Using Contour Integration to Solve Definite Integrals
9.05: Exercises
10: Fourier Series and Fourier Transforms
10.01: Fourier Series
10.02: Fourier Transforms
10.03: Fourier Transforms for Time-Domain Functions
10.04: Basic Properties of the Fourier Transform
10.05: Fourier Transforms of Differential Equations
10.06: Common Fourier Transforms
10.07: The Delta Function
10.08: Multi-Dimensional Fourier Transforms
10.09: Exercises
11: Green's Functions
11.01: The Driven Harmonic Oscillator
11.02: Space-Time Green's Functions
11.03: Causality and the Time-Domain Green's Function
11.04: Looking Ahead
11.05: Exercises