How to do Theoretical Physics research

lecture notes on group theory in physics, theoretical particle physics lecture notes, what is theoretical physics and applied mathematics,what is theoretical nuclear physics pdf free download
Dr.ShaneMatts Profile Pic
Dr.ShaneMatts,United States,Teacher
Published Date:23-07-2017
Your Website URL(Optional)
Comment
Theoretical Physics Reference Release0.5 ˇ Ondrej ˇ Certík March 09, 2017CONTENTS 1 Introduction 1 1.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Contributors 3 3 Mathematics 5 3.1 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.4 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.5 Fourier Transform of a Periodic Function (e.g. in a Crystal) . . . . . . . . . . . . . . . . . . . . . . 34 3.6 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.7 Fast Fourier Transform (FFT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.8 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.9 Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.10 Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.11 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.12 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.13 Argument function, atan2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.14 Multiple Argument Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.15 Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.16 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.17 Variations and Functional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.18 Dirac Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.19 Homogeneous Functions (Euler’s Theorem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.20 Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.21 Binomial Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.22 Double Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.23 Triangle Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.24 Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.25 Incomplete Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.26 Factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.27 Double Factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.28 Fermi-Dirac Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.29 Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.30 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.31 Gaunt Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.32 Wigner 3j Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.33 Multipole Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.34 Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 i3.35 Feynman Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.36 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.37 Wigner D Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.38 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.39 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3.40 Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.41 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 4 ClassicalMechanics,SpecialandGeneralRelativity 185 4.1 Gravitation and Electromagnetism as a Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 185 4.2 Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 4.3 Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 5 ClassicalElectromagnetism 225 5.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 5.2 Semiconductor Device Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 6 ThermodynamicsandStatisticalPhysics 253 6.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 6.2 Statistical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 7 FluidDynamics 263 7.1 Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 7.2 MHD Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 7.3 Compressible Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 8 QuantumFieldTheoryandQuantumMechanics 297 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 8.2 Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 8.3 Quantum Electrodynamics (QED) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 8.4 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 8.5 Systematic Perturbation Theory in QM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 8.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 8.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 8.8 Radial Schrödinger and Dirac Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 8.9 Density Functional Theory (DFT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 8.10 Hartree-Fock (HF) Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 8.11 Projector Augmented-Wave Method (PAW) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 Bibliography 445 Index 447 iiCHAPTER ONE INTRODUCTION 1.1 Preface I have a very bad memory. I am able to memorize quite a lot of things short term, but I am not able to remember most formulas from quantum mechanics over the long term (e.g. like over the summer). I don’t remember formulas for perturbation theory (neither time dependent or time independent), I don’t remember Feynman rules in quantum field 2 theory, I don’t even remember the Dirac equation exactly (where the𝑖 should be, if there is𝑚 or𝑚 , ...). The thing about quantum field theory is not that some particular steps are difficult, but that there are so many of them and one has to master all of them at once, in order to really “get it”. I never got QFT, because once I mastered one part sufficiently, I forgot some other part and it took so much time to master that other part that I forgot the first part again. However, I was determined that I would get it. In order to do so, I realized I need to keep notes of things I understood, written in my own way. Then, when I relearn some parts that I forgot, it just takes me a few minutes to go over my reference notes to get into it quickly. My own style of understanding is that the notes should be complete (no need to consult external books), yet very short and getting directly to the point, and also with every single calculation carried out explicitly. See also the preface to the QFT part. If you want to study physics, learn math the physics way (as opposed to the usual mathematics way of a definition, theorem, proof, ...). When I was beginning my undergrad physics studies (and even on a high school), I also had this common misconception, that I need to study math and understand every proof and then I’ll be somehow prepared for physics. I was very wrong. I used to study calculus by myself and then trying to learn the proofs, and Lebesgue integral and I was learning that from the mathematics books. At the university, I always did all my math exams first (as far as I remember, I always got A from those), hoping that would be a good start for the physics exams, but I always found out that it was mostly useless. Now I know that the only way to study physics is to go and do physics directly and learn the math on the way as needed. The math section of this book reviews all the math, that is necessary for studying theoretical physics (graduate level). There are actually quite a lot of good math books written by physicists as well as many excellent physics books, covering everything that I cover here. But I really like to have all the theoretical physics and the corresponding math explained in one book, and to keep it as short as possible. Also everyone has a bit different style and amount of rigor and I have not found a book that would perfectly suite my own style, thus I wrote one. 1.2 Introduction The Theoretical Physics Reference is an attempt to derive all theoretical physics equations (that are ever needed for applications) from the general and special relativity and the standard model of particle physics. The goals are: • All calculations are very explicit, with no intermediate steps left out. 1Theoretical Physics Reference, Release 0.5 • Start from the most general (and correct) physical theories (general relativity or standard model) and derive the specialized equations from them (e.g. the Schrödinger equation). • Math is developed in the math section (not in the physics section). • Theory should be presented as short and as explicitly as possible. Then there should be arbitrary number of examples, to show how the theory is used. • There should be just one notation used throughout the book. • It should serve as a reference to any physics equation (exact derivation where it comes from) and the reader should be able to understand how things work from this book, and be ready to understand specialized literature. This is a work in progress and some chapters don’t conform to the above goals yet. Usually first some derivation is written, as we understood it, then the mathematical tools are extracted and put into the math section, and the rest is fit where it belongs. Sometimes we don’t understand some parts yet, then those are currently left there as they are. There are many excellent books about theoretical physics, that one can consult about particular details. The goal of this book (when completed) is to show where things come from and serve as a reference to any particular field, so that one doesn’t get lost when reading specialized literature. Here is an incomplete list of some of the best books in theoretical physics (we only picked those that we actually read): 1. Landau, L. D.; Lifshitz, E. M: Course of Theoretical Physics 2. Richard Feynman: The Feynman Lectures on Physics 3. Walter Greiner: “Classical Theoretical Physics” series of texts 4. Herbert Goldstein: Classical Mechanics 5. J.D. Jackson: Classical Electrodynamics 6. Charles W. Misner, Kip S. Thorne, John Wheeler: Gravitation 7. Bernard Schutz: A First Course in General Relativity 8. Carrol S.: The Lecture Notes on General Relativity 9. J.J. Sakurai: Advanced Quantum Mechanics 10. Brown L. S.: Quantum Field Theory 11. Mark Srednicki: Quantum Field Theory 12. Claude Itzykson, Jean-Bernard Zuber: Quantum Field Theory 13. Zee A.: Quantum Field Theory in a Nutshell 14. Steven Weinberg: The Quantum Theory of Fields 15. L.H. Ryder: Quantum Field Theory 16. Jirí ˇ Horejší: ˇ Fundamentals of Electroweak Theory 17. Michele Maggiore: A Modern Introduction to Quantum Field Theory 18. M.E. Peskin & D.V. Schroeder: An Introduction to Quantum Field Theory 19. J.W. Negele, H. Orland: Quantum Many-Particle Systems 20. X-G. Wen: Quantum Field Theory of Many-Body Systems 21. Dirac, P.A.M.: General Theory of Relativity 2 Chapter 1. IntroductionCHAPTER TWO CONTRIBUTORS A list of people who contributed patches or text for the book in the order of the date of contribution: ˇ • Ondˇ rej Certík • Karel Výborný • Jean Jordaan • Lei Ma 3Theoretical Physics Reference, Release 0.5 4 Chapter 2. ContributorsCHAPTER THREE MATHEMATICS 3.1 Integration This chapter doesn’t assume any knowledge about differential geometry. The most versatile way to do integration over manifolds is explained in the differential geometry section. 3.1.1 General Case 𝑛 We want to integrate a function𝑓 over a𝑘-manifold in R , parametrized as: ⎛ ⎞ 𝜙 (𝑡 ,𝑡 ,...,𝑡 ) 1 1 2 𝑘 ⎜ ⎟ 𝜙 (𝑡 ,𝑡 ,...,𝑡 ) 2 1 2 𝑘 ⎜ ⎟ 𝑘 𝑛 𝜙 : R → R 𝜙 (𝑡 ,𝑡 ,...,𝑡 ) =⎜ ⎟ 1 2 𝑘 . . ⎝ ⎠ . 𝜙 (𝑡 ,𝑡 ,...,𝑡 ) 𝑛 1 2 𝑘 then the integral of𝑓(𝑥 ,𝑥 ,...,𝑥 ) over𝜙 is: 1 2 𝑛 ∫︁ ∫︁ √ 𝑓(𝑥 ,𝑥 ,...,𝑥 ) d𝑆 = 𝑓(𝜙 (𝑡 ,𝑡 ,...,𝑡 )) det G d𝑡 d𝑡 ··· d𝑡 1 2 𝑛 1 2 𝑘 1 2 𝑘 𝑛 𝑀 R where G is called a Gram matrix and J is a Jacobian: 𝜕𝜙 𝜕𝜙 𝑘 𝑘 𝑇 (G) = (J J) =𝐽 𝐽 = 𝑖𝑗 𝑖𝑗 𝑖𝑘 𝑘𝑗 𝜕𝑡 𝜕𝑡 𝑖 𝑗 ⎛ ⎞ 𝜕𝜙 𝜕𝜙 𝜕𝜙 ··· 𝜕𝑡 𝜕𝑡 𝜕𝑡 1 2 𝑘 ⎜ ⎟ . . . . . . . . ⎜ ⎟ . . . . 𝜕𝜙 𝑖 ⎜ ⎟ (J) = = ⎜ ⎟ 𝑖𝑗 . . . . 𝜕𝑡 ⎜ . . . . ⎟ 𝑗 . . . . ⎝ ⎠ . . . . . . . . . . . . The idea behind this comes from the fact that the volume of the𝑘-dimensional parallelepiped spanned by the vectors 𝜕𝜙 𝜕𝜙 ,..., 𝜕𝑡 𝜕𝑡 1 𝑘 is given by √ 𝑇 𝑉 = det J J where J is an𝑛× 𝑘 matrix having those vectors as its column vectors. 5Theoretical Physics Reference, Release 0.5 Example Let’s integrate a function𝑓(𝑥,𝑦,𝑧) over the surface of a sphere in 3D (e.g. 𝑘 = 2 and𝑛 = 3): ⎛ ⎞ 𝑟 sin𝜃 cos𝜑 ⎝ ⎠ 𝜙 (𝜃,𝜑 ) = 𝑟 sin𝜃 sin𝜑 𝑟 cos𝜃 ⎛ ⎞ − 𝑟 sin𝜃 sin𝜑 𝑟 cos𝜃 cos𝜑 ⎝ ⎠ 𝑟 sin𝜃 cos𝜑 𝑟 cos𝜃 sin𝜑 J = 0 − 𝑟 sin𝜃 ⎛ ⎞ (︂ )︂ (︂ )︂ − 𝑟 sin𝜃 sin𝜑 𝑟 cos𝜃 cos𝜑 2 2 − 𝑟 sin𝜃 sin𝜑 𝑟 sin𝜃 cos𝜑 0 𝑟 sin 𝜃 0 𝑇 ⎝ ⎠ G = J J = 𝑟 sin𝜃 cos𝜑 𝑟 cos𝜃 sin𝜑 = 2 𝑟 cos𝜃 cos𝜑 𝑟 cos𝜃 sin𝜑 − 𝑟 sin𝜃 0 𝑟 0 − 𝑟 sin𝜃 4 2 det G =𝑟 sin 𝜃 √ 2 det G =𝑟 sin𝜃 ∫︁ ∫︁ 2 𝑓(𝑥,𝑦,𝑧)d𝑆 = 𝑓(𝑟 sin𝜃 cos𝜑,𝑟 sin𝜃 sin𝜑,𝑟 cos𝜃 )𝑟 sin𝜃 d𝜃 d𝜑 = 𝑛 𝑀 R ∫︁ ∫︁ 𝜋 2𝜋 2 = d𝜃 d𝜑𝑓(𝑟 sin𝜃 cos𝜑,𝑟 sin𝜃 sin𝜑,𝑟 cos𝜃 )𝑟 sin𝜃 0 0 Let’s say we want to calculate the surface area of a sphere, so we set𝑓(𝑥,𝑦,𝑧) = 1 and get: ∫︁ ∫︁ ∫︁ ∫︁ 𝜋 2𝜋 𝜋 2 2 2 d𝑆 = d𝜃 d𝜑𝑟 sin𝜃 = 2𝜋𝑟 d𝜃 sin𝜃 = 4𝜋𝑟 𝑀 0 0 0 3.1.2 Special Cases k = n 𝑅 2 det G = det J J = (det J) d𝑆 = det J d𝑡 d𝑡 ··· d𝑡 1 2 𝑘 k = 1 (︃ )︃ (︂ )︂ (︂ )︂ ⃒ ⃒ 2 2 2 ⃒ ⃒ d𝜙 d𝜙 d𝜙 1 2 ⃒ ⃒ det G = det + +··· = ⃒ ⃒ d𝑡 d𝑡 d𝑡 ⃒ ⃒ ⃒ ⃒ d𝜙 ⃒ ⃒ d𝑆 = d𝑡 ⃒ ⃒ d𝑡 6 Chapter 3. MathematicsTheoretical Physics Reference, Release 0.5 k = n - 1 𝑅 det G = det J J = 2 2 2 = det(··· ) + det(··· ) +··· + det(... ) = ⃒ ⃒ ⎛ ⎞ 2 𝜕𝜙 𝜕𝜙 𝜕𝜙 ⃒ ⃒ ··· e 1 𝜕𝑡 𝜕𝑡 𝜕𝑡 ⃒ 1 2 𝑘 ⃒ ⃒ ⎜ ⎟⃒ . . . . . . . . ⃒ ⎜ ⎟⃒ . . . . e 2 ⃒ ⎜ ⎟⃒ 2 =⃒ det⎜ ⎟⃒ ≡ 𝜔 𝜙 . . . . . ⃒ ⎜ . . . . . ⎟⃒ . . . . . ⃒ ⃒ ⎝ ⎠ ⃒ ⃒ . . . . ⃒ . . . . ⃒ . . . . e 𝑛 d𝑆 =𝜔 d𝑡 d𝑡 ··· d𝑡 𝜙 1 2 𝑘 𝜔 is a generalization of a vector cross product. The det(··· ) symbol means a determinant of a matrix with one row 𝜙 removed (first term in the sum has first row removed, second term has second row removed, etc.). k = 2, n = 3 ⃒ ⃒ 2 ⃒ ⃒ 𝜕𝜙 𝜕𝜙 ⃒ ⃒ det G = × ⃒ ⃒ 𝜕𝑡 𝜕𝑡 1 2 ⃒ ⃒ ⃒ ⃒ 𝜕𝜙 𝜕𝜙 ⃒ ⃒ d𝑆 = × d𝑡 d𝑡 1 2 ⃒ ⃒ 𝜕𝑡 𝜕𝑡 1 2 y = f(x, z) (︂ )︂ (︂ )︂ 2 2 𝜕𝑓 𝜕𝑓 det G = 1 + + 𝜕𝑥 𝜕𝑧 √︃ (︂ )︂ (︂ )︂ 2 2 𝜕𝑓 𝜕𝑓 d𝑆 = 1 + + d𝑥 d𝑧 𝜕𝑥 𝜕𝑧 in general for𝑥 =𝑓(𝑥 ,𝑥 ,...,𝑥 ) we get: 𝑗 1 2 𝑛 (︂ )︂ (︂ )︂ 2 2 𝜕𝑓 𝜕𝑓 det G = 1 + + +··· 𝜕𝑥 𝜕𝑥 1 2 √︃ (︂ )︂ (︂ )︂ 2 2 𝜕𝑓 𝜕𝑓 d𝑆 = 1 + + +··· d𝑥 d𝑥 ··· d𝑥 1 2 𝑛 𝜕𝑥 𝜕𝑥 1 2 The “𝑥 ” term is missing in the sums above. 𝑗 Implicit Surface For a surface given implicitly by 𝐹 (𝑥 ,𝑥 ,...,𝑥 ) = 0 1 2 𝑛 we get: ⃒ ⃒ ⃒ ⃒ 𝜕𝐹 ⃒ ⃒ d𝑆 =∇𝐹 d𝑥 ··· d𝑥 1 𝑛− 1 ⃒ ⃒ 𝜕𝑥 𝑛 3.1. Integration 7Theoretical Physics Reference, Release 0.5 Orthogonal Coordinates If the coordinate vectors are orthogonal to each other: 𝜕𝜙 𝜕𝜙 · = 0 for𝑖̸=𝑗 𝜕𝑡 𝜕𝑡 𝑖 𝑖 we get: ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ 𝜕𝜙 𝜕𝜙 𝜕𝜙 ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ d𝑆 = ··· d𝑡 ··· d𝑡 1 𝑘 ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ 𝜕𝑡 𝜕𝑡 𝜕𝑡 1 2 𝑘 3.1.3 Motivation Let the𝑘-dimensional parallelepiped P be spanned by the vectors 𝜕𝜙 𝜕𝜙 ,..., 𝑡 𝑡 1 𝑘 and let J is𝑛× 𝑘 matrix having these vectors as its column vectors. Then the area of P is √ 𝑇 𝑉 = det J J so the definition of the integral over a manifold is just approximating the surface by infinitesimal parallelepipeds and integrating over them. 3.1.4 Example Let’s calculate the total distance traveled by a body in 1D, whose position is given by𝑠(𝑡): ⃒ ⃒ ∫︁ ∫︁ 𝑡 2 ⃒ ⃒ d𝑠 ⃒ ⃒ 𝑙 = d𝑠 = d𝑡 = ⃒ ⃒ d𝑡 𝛾 𝑡 1 ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ ∫︁ ′ ∫︁ ′′ ∫︁ 𝑡 𝑡 𝑡 2 ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ d𝑠 d𝑠 d𝑠 ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ = d𝑡 + d𝑡 +··· + d𝑡 = ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ d𝑡 ′ d𝑡 ′′′′··· d𝑡 𝑡 𝑡 𝑡 1 ′ ′′ ′ ′′′′··· =𝑠(𝑡 )− 𝑠(𝑡 ) +𝑠(𝑡 )− 𝑠(𝑡 ) +··· +𝑠(𝑡 )− 𝑠(𝑡 ) 1 2 ⃒ ⃒ ′ ′′ d𝑠 ⃒ ⃒ where 𝑡 , 𝑡 , ... are all the points at which = 0, so each of the integrals in the above sum has either positive or d𝑡 negative integrand. 3.2 Complex Numbers We start by defining arg(𝑧) by its principal value, then everything else follows from this definition. We could have also used any other branch, but then most results in this chapter would need to be updated with the new convention. Then we define exponential, logarithm, power and so on using simple natural formulas. From these definitions, everything else follows using a very simple algebra manipulation, all the “messy” features are hidden in the definition and properties of the real atan 2 function. In the derivation of each formula, only formulas introduced before (above) are used. Every formula in this chapter holds for all complex numbers, unless explicitly specified otherwise. 8 Chapter 3. MathematicsTheoretical Physics Reference, Release 0.5 3.2.1 Real and Imaginary Part A complex number𝑧 can be written using its real and imaginary parts: 𝑧 = Re𝑧 +𝑖 Im𝑧 The absolute value𝑧 is defined as: √︀ 2 2 𝑧 = Re 𝑧 + Im 𝑧 3.2.2 Argument Function Principal value of arg(𝑧) is defined as arg𝑧 = atan 2(Im𝑧, Re𝑧) Thus we have− 𝜋 arg𝑧≤ 𝜋 . All operations with arg𝑧 are then derived using the properties of the real atan 2 function. 3.2.3 Exponential Exponential is defined using: 𝑧 Re𝑧+𝑖 Im𝑧 Re𝑧 𝑒 =𝑒 =𝑒 (cos Im𝑧 +𝑖 sin Im𝑧) It follows: 𝑎+𝑏 Re(𝑎+𝑏) 𝑒 =𝑒 (cos Im(𝑎 +𝑏) +𝑖 sin Im(𝑎 +𝑏)) = Re𝑎 Re𝑏 =𝑒 𝑒 (cos(Im𝑎) cos(Im𝑏)− sin(Im𝑎) sin(Im𝑏) +𝑖 sin(Im𝑎) cos(Im𝑏) +𝑖 cos(Im𝑎) sin(Im𝑏)) = Re𝑎 Re𝑏 =𝑒 (cos Im𝑎 +𝑖 sin Im𝑎)𝑒 (cos Im𝑏 +𝑖 sin Im𝑏) = 𝑎 𝑏 =𝑒 𝑒 Any complex number can be written in a polar form as follows: (︂ )︂ Re𝑧 Im𝑧 𝑧 = Re𝑧 +𝑖 Im𝑧 =𝑧 +𝑖 = 𝑧 𝑧 (︃ )︃ Re𝑧 Im𝑧 =𝑧 √︀ +𝑖√︀ = 2 2 2 2 Re 𝑧 + Im 𝑧 Re 𝑧 + Im 𝑧 =𝑧 (cos atan 2(Im𝑧, Re𝑧) +𝑖 sin atan 2(Im𝑧, Re𝑧)) = =𝑧 (cos arg𝑧 +𝑖 sin arg𝑧) = 𝑖 arg𝑧 =𝑧𝑒 The following formula holds: 𝑧 Re𝑧 𝑖 Im𝑧 𝑖 Im𝑧 arg𝑒 = arg𝑒 𝑒 = arg𝑒 = = arg(cos Im𝑧 +𝑖 sin Im𝑧) = = atan 2(sin Im𝑧, cos Im𝑧) = ⌊︂ ⌋︂ 𝜋 − Im𝑧 = Im𝑧 + 2𝜋 2𝜋 3.2. Complex Numbers 9Theoretical Physics Reference, Release 0.5 and also: 𝑖 arg𝑎 𝑖 arg𝑏 arg𝑎𝑏 = arg(𝑎𝑒 𝑏𝑒 ) = 𝑖(arg𝑎+arg𝑏) = arg(𝑎𝑏𝑒 ) = 𝑖(arg𝑎+arg𝑏) = arg(𝑒 ) = = arg(cos(arg𝑎 + arg𝑏) +𝑖 sin(arg𝑎 + arg𝑏)) = = atan 2(sin(arg𝑎 + arg𝑏), cos(arg𝑎 + arg𝑏)) = ⌊︂ ⌋︂ 𝜋 − arg𝑎− arg𝑏 = arg𝑎 + arg𝑏 + 2𝜋 2𝜋 and ⌊︂ ⌋︂ 1 𝜋 + arg𝑧 arg =− arg𝑧 + 2𝜋 𝑧 2𝜋 and ⃒ ⃒ (︂ )︂ ⃒ ⃒ 𝑎 1 1 𝑖 arg𝑎 𝑖 arg ⃒ ⃒ 𝑏 arg = arg 𝑎𝑒 𝑒 = ⃒ ⃒ 𝑏 𝑏 ⃒ ⃒ (︂ )︂ ⃒ ⃒ 𝜋 +arg𝑏 1 𝑖(arg𝑎− arg𝑏)+2𝜋𝑖 ⌊ ⌋ ⃒ ⃒ 2𝜋 = arg 𝑎 𝑒 = ⃒ ⃒ 𝑏 𝑖(arg𝑎− arg𝑏) = arg(𝑒 ) = = arg(cos(arg𝑎− arg𝑏) +𝑖 sin(arg𝑎− arg𝑏)) = = atan 2(sin(arg𝑎− arg𝑏), cos(arg𝑎− arg𝑏)) = ⌊︂ ⌋︂ 𝜋 − arg𝑎 + arg𝑏 = arg𝑎− arg𝑏 + 2𝜋 2𝜋 3.2.4 Logarithm The logarithm is defined as: log𝑧 = log𝑧 +𝑖 arg𝑧 (3.1) The motivation is from the following formula: 𝑖 arg𝑧 log𝑧 𝑖 arg𝑧 log𝑧+𝑖 arg𝑧 𝑧 =𝑧𝑒 =𝑒 𝑒 =𝑒 which using our definition becomes: log𝑧+𝑖 arg𝑧 log𝑧 𝑧 =𝑒 =𝑒 (3.2) so a logarithm is an inverse function to an exponential. The formula (3.2) would be satisfied even if we add a factor of 2𝜋𝑖𝑛 (where𝑛 is an integer) to the right hand side of (3.1). However, the convention is to define logarithm using the equation (3.1) exactly. We can now derive a few important formulas: 𝑧 Re𝑧 𝑖 Im𝑧 Re𝑧 log𝑒 = log𝑒 𝑒 = log𝑒 = Re𝑧 (︂ ⌊︂ ⌋︂)︂ ⌊︂ ⌋︂ 𝜋 − Im𝑧 𝜋 − Im𝑧 𝑧 𝑧 𝑧 log𝑒 = log𝑒 +𝑖 arg𝑒 = Re𝑧 +𝑖 Im𝑧 + 2𝜋 =𝑧 + 2𝜋𝑖 2𝜋 2𝜋 10 Chapter 3. MathematicsTheoretical Physics Reference, Release 0.5 and log𝑎𝑏 = log𝑎𝑏 +𝑖 arg𝑎𝑏 = (3.3) ⌊︂ ⌋︂ 𝜋 − arg𝑎− arg𝑏 = log𝑎 + log𝑏 +𝑖 arg𝑎 +𝑖 arg𝑏 + 2𝜋𝑖 = 2𝜋 ⌊︂ ⌋︂ 𝜋 − arg𝑎− arg𝑏 = log𝑎 + log𝑏 + 2𝜋𝑖 2𝜋 and ⃒ ⃒ 𝑎 𝑎 𝑎 ⃒ ⃒ log = log +𝑖 arg = (3.4) ⃒ ⃒ 𝑏 𝑏 𝑏 ⌊︂ ⌋︂ 𝜋 − arg𝑎 + arg𝑏 = log𝑎− log𝑏 +𝑖 arg𝑎− 𝑖 arg𝑏 + 2𝜋𝑖 = 2𝜋 ⌊︂ ⌋︂ 𝜋 − arg𝑎 + arg𝑏 = log𝑎− log𝑏 + 2𝜋𝑖 2𝜋 3.2.5 Power A power of two complex numbers is defined as: 𝑎 𝑎 log𝑧 𝑧 =𝑒 𝑎 From above we can also write the power𝑧 in two different ways: (︀ )︀ 𝑎 𝑎 𝑎 log𝑧 log𝑧 𝑧 = 𝑒 =𝑒 𝑎 But these two cannot be used as a definition of a power, because both require the knowledge of𝑥 , which we are trying log𝑧 to define, where𝑥 =𝑧 or𝑥 =𝑒 . It follows: ⌊︂ ⌋︂ 𝜋 − Im𝑎 log𝑥 𝑎 𝑎 log𝑥 log𝑥 = log𝑒 =𝑎 log𝑥 + 2𝜋𝑖 (3.5) 2𝜋 and 𝑎 𝜋 − Im𝑎 log𝑥 𝑎 𝑏 𝑏 log𝑥 𝑏 𝑎 log𝑥+2𝜋𝑖 ( ⌊ ⌋) 2𝜋 (3.6) (𝑥 ) =𝑒 =𝑒 = 𝜋 − Im𝑎 log𝑥 𝑎𝑏 log𝑥 2𝜋𝑖𝑏⌊ ⌋ 2𝜋 =𝑒 𝑒 = 𝜋 − Im𝑎 log𝑥 𝑎𝑏 2𝜋𝑖𝑏 ⌊ ⌋ 2𝜋 =𝑥 𝑒 As a special case for𝑥 =𝑒 one gets: 𝜋 − Im𝑎 𝑎 𝑏 𝑎𝑏 2𝜋𝑖𝑏⌊ ⌋ 2𝜋 (3.7) (𝑒 ) =𝑒 𝑒 Similarly: 𝜋 − arg𝑥− arg𝑦 𝑎 𝑎 log𝑥𝑦 𝑎 log𝑥+𝑎 log𝑦+2𝜋𝑖𝑎 ⌊ ⌋ 2𝜋 (𝑥𝑦) =𝑒 =𝑒 = 𝜋 − arg𝑥− arg𝑦 𝑎 𝑎 2𝜋𝑖𝑎 ⌊ ⌋ 2𝜋 =𝑥 𝑦 𝑒 3.2.6 Examples For integer𝑛 we get from (3.6): 𝜋 − Im𝑎 log𝑥 𝑎 𝑛 𝑎𝑛 2𝜋𝑖𝑛⌊ ⌋ 𝑎𝑛 2𝜋 (𝑥 ) =𝑥 𝑒 =𝑥 3.2. Complex Numbers 11Theoretical Physics Reference, Release 0.5 Using (3.6): √ 𝜋 − Im 2 log𝑥 𝜋 − 2 arg𝑥 𝜋 − 2 arg𝑥 1 1 1 2𝜋𝑖 𝜋𝑖 2 · 2 ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ 2 2 2 2 2𝜋 2𝜋 2𝜋 𝑥 = (𝑥 ) =𝑥 𝑒 =𝑥𝑒 = (− 1) 𝑥 Using (3.7): √ 𝜋 − Im𝑥 𝜋 − Im𝑥 1 𝑥 𝑥 𝜋𝑖 𝑥 ⌊ ⌋ ⌊ ⌋ 𝑥 2 2 2𝜋 2𝜋 2 𝑒 = (𝑒 ) =𝑒 𝑒 = (− 1) 𝑒 Using (3.3): ⌊︂ ⌋︂ 𝜋 − 𝜋 − 𝜋 0 = log 1 = log(− 1)(− 1) = log(− 1) + log(− 1) + 2𝜋𝑖 = 2𝜋 ⌊︀ ⌋︀ 1 =𝑖𝜋 +𝑖𝜋 + 2𝜋𝑖 − =𝑖𝜋 +𝑖𝜋 − 2𝜋𝑖 = 0 2 Code: from math import floor, pi from cmath import log log((-1) (-1)) 0j log(-1)+log(-1)+2 pi 1j floor((pi-pi-pi)/(2 pi)) 0j Another example: 2 𝜋 𝑖 𝑖 log𝑖 𝑖 arg𝑖 − 2 𝑖 =𝑒 =𝑒 =𝑒 Code: from math import exp, pi 1j 1j (0.20787957635076193+0j) exp(-pi/2) 0.20787957635076193 Another example, using (3.5): ⌊︂ ⌋︂ 1 √ 𝜋 − Im log𝑧 1 1 2 2 log( 𝑧) = log(𝑧 ) = log𝑧 + 2𝜋𝑖 = 2 2𝜋 ⌊︂ ⌋︂ 1 𝜋 − arg𝑧 1 2 1 = log𝑧 + 2𝜋𝑖 = log𝑧 2 2 2𝜋 and ⌊︂ ⌋︂ 𝜋 − Im 2 log𝑧 2 log(𝑧 ) = 2 log𝑧 + 2𝜋𝑖 = 2𝜋 ⌊︂ ⌋︂ 𝜋 − 2 arg𝑧 = 2 log𝑧 + 2𝜋𝑖 2𝜋 and (︂ )︂ ⌊︂ ⌋︂ 1 𝜋 − Im(− 1) log𝑧 − 1 log = log(𝑧 ) =− log𝑧 + 2𝜋𝑖 = 𝑧 2𝜋 ⌊︂ ⌋︂ 𝜋 + arg𝑧 =− log𝑧 + 2𝜋𝑖 2𝜋 Another example, following from (3.1) and (3.4): (︂ ⌊︂ ⌋︂)︂ 1 1 𝑧 𝜋 − arg𝑧 + arg𝑧 1 𝑧 arg𝑧 = (log𝑧− log𝑧) = log − 2𝜋𝑖 = log 𝑖 𝑖 𝑧 2𝜋 𝑖 𝑧 12 Chapter 3. MathematicsTheoretical Physics Reference, Release 0.5 3.2.7 Complex Conjugate The complex conjugate is defined by: 𝑧 = Re𝑧 +𝑖 Im𝑧 𝑧  = Re𝑧− 𝑖 Im𝑧 Now we can solve for Re𝑧 and Im𝑧: 1 Re𝑧 = (𝑧 +𝑧 ) 2 𝑖 Im𝑧 = (− 𝑧 +𝑧 ) 2 Any complex function𝑓 can be written using Re𝑧 and Im𝑧, i.e. 𝑓 =𝑓(Re𝑧, Im𝑧) or using𝑧 and𝑧 , i.e. 𝑓 =𝑓(𝑧,𝑧 ). Examples √︃ (︂ )︂ (︂ )︂ 2 2 √︀ √ 1 𝑖 2 2 𝑧 = Re 𝑧 + Im 𝑧 = (𝑧 +𝑧 ) + (− 𝑧 +𝑧 ) = 𝑧𝑧  2 2 √︁ √︀ 2 2 2 2 𝑧  = Re 𝑧  + Im 𝑧  = Re 𝑧 + (− Im𝑧) =𝑧 (︂ )︂ 𝑖 1 arg𝑧 = atan 2(Im𝑧, Re𝑧) = atan 2 (− 𝑧 +𝑧 ), (𝑧 +𝑧 ) = atan 2 (𝑖(− 𝑧 +𝑧 ),𝑧 +𝑧 ) 2 2 ⌊︂ ⌋︂ ⌊︂ ⌋︂ atan 2(Im𝑧, Re𝑧) +𝜋 arg𝑧 +𝜋 arg𝑧  = atan 2(− Im𝑧, Re𝑧) =− atan 2(Im𝑧, Re𝑧) + 2𝜋 =− arg𝑧 + 2𝜋 2𝜋 2𝜋 ⌊︂ ⌋︂ ⌊︂ ⌋︂ arg𝑧 +𝜋 arg𝑧 +𝜋 log𝑧 = log𝑧− 𝑖 arg𝑧 = log𝑧  +𝑖 arg𝑧 − 2𝜋𝑖 = log𝑧 − 2𝜋𝑖 2𝜋 2𝜋 1 1 1 1 1 √ arg𝑧+𝜋 log𝑧 (log𝑧+𝑖 arg𝑧) (log𝑧− 𝑖 arg𝑧) log𝑧 ¯+𝑖 arg𝑧 ¯− 2𝜋𝑖 ( ⌊ ⌋) 2𝜋 2 2 2 2 2 𝑧 =𝑧 =𝑒 =𝑒 =𝑒 =𝑒 = 1 √ arg𝑧+𝜋 arg𝑧+𝜋 − 𝜋𝑖 ⌊ ⌋ ⌊ ⌋ 2𝜋 2𝜋 2 = (𝑧 ) 𝑒 = (− 1) 𝑧  3.2.8 Complex Derivatives The complex derivative is defined by d𝑓 𝑓(𝑧 +ℎ)− 𝑓(𝑧) (3.8) = lim d𝑧 ℎ→0 ℎ 3.2. Complex Numbers 13Theoretical Physics Reference, Release 0.5 𝑖𝜃 Let’s calculate the complex derivative in the direction𝜃 , i.e. we useℎ =𝑡𝑒 with real𝑡 and we introduce𝑓 =𝑓(𝑥,𝑦) with𝑥 = Re𝑧,𝑦 = Im𝑧 to simplify the notation: 𝑖𝜃 d𝑓 𝑓(𝑧 +𝑡𝑒 )− 𝑓(𝑧) = lim = 𝑖𝜃 𝑡→0 d𝑧 𝑡𝑒 𝑓(𝑥 +𝑡 cos𝜃,𝑦 +𝑡 sin𝜃 )− 𝑓(𝑥,𝑦) − 𝑖𝜃 = lim 𝑒 = 𝑡→0 𝑡 d − 𝑖𝜃 = 𝑓(𝑥 +𝑡 cos𝜃,𝑦 +𝑡 sin𝜃 )𝑒 = d𝑡 (︂ )︂ 𝜕𝑓 𝜕𝑓 − 𝑖𝜃 = cos𝜃 + sin𝜃 𝑒 = 𝜕𝑥 𝜕𝑦 (︂ )︂ 𝑖𝜃 − 𝑖𝜃 𝑖𝜃 − 𝑖𝜃 𝜕𝑓 𝑒 +𝑒 𝜕𝑓 𝑒 − 𝑒 − 𝑖𝜃 = + 𝑒 = 𝜕𝑥 2 𝜕𝑦 2𝑖 − 2𝑖𝜃 − 2𝑖𝜃 𝜕𝑓 1 +𝑒 𝜕𝑓 1− 𝑒 = + = 𝜕𝑥 2 𝜕𝑦 2𝑖 (︂ )︂ (︂ )︂ 𝜕𝑓 𝜕𝑓 𝜕𝑓 𝜕𝑓 1 1 − 2𝑖𝜃 = − 𝑖 + +𝑖 𝑒 = 2 2 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜕𝑓 𝜕𝑓 − 2𝑖𝜃 = + 𝑒 𝜕𝑧 𝜕𝑧  In the last step we have expressed the derivatives with respect to𝑥,𝑦 in terms of derivatives with respect to𝑧,𝑧 , using the relations: 𝜕𝑓 𝜕𝑥𝜕𝑓 𝜕𝑦𝜕𝑓 = + = (3.9) 𝜕𝑧 𝜕𝑧 𝜕𝑥 𝜕𝑧 𝜕𝑦 1𝜕𝑓 𝑖 𝜕𝑓 = − = 2𝜕𝑥 2𝜕𝑦 (︂ )︂ 𝜕𝑓 𝜕𝑓 1 = − 𝑖 2 𝜕𝑥 𝜕𝑦 𝜕𝑓 𝜕𝑥𝜕𝑓 𝜕𝑦𝜕𝑓 = + = (3.10) 𝜕𝑧  𝜕𝑧 𝜕𝑥 𝜕𝑧 𝜕𝑦 1𝜕𝑓 𝑖 𝜕𝑓 = + = 2𝜕𝑥 2𝜕𝑦 (︂ )︂ 𝜕𝑓 𝜕𝑓 1 = +𝑖 2 𝜕𝑥 𝜕𝑦 Let’s repeat the important result: d𝑓(𝑧,𝑧 ) 𝜕𝑓(𝑧,𝑧 ) 𝜕𝑓(𝑧,𝑧 ) − 2𝑖𝜃 (3.11) = + 𝑒 d𝑧 𝜕𝑧 𝜕𝑧  The equation (3.11) states that the complex derivative along the direction𝜃 of any function can be calculated, but the 𝜕𝑓 result in general depends on 𝜃 . The derivatives for all possible angles 𝜃 lie on a circle, with the center and the 𝜕𝑧 ⃒ ⃒ ⃒ ⃒ 𝜕𝑓 𝜕𝑓 radius⃒ ⃒ . When the derivative has different values for different𝜃 , i.e. when ̸= 0, it means that the complex limit 𝜕𝑧 ¯ 𝜕𝑧 ¯ 𝜕𝑓 (3.8) does not exist. On the other hand, if the derivative does not depend on 𝜃 , i.e. when = 0, then the complex 𝜕𝑧 ¯ limit (3.8) exists, and the function has a complex derivative — such functions are called analytic. Analytic functions thus do not depend on𝑧  and we can write just𝑓 =𝑓(𝑧) for those. 𝜕𝑓 𝜕𝑓 The and are called Wirtinger derivatives. 𝜕𝑧 𝜕𝑧 ¯ We can see that the function is analytic (i.e. has a complex derivative) if and only if: (︂ )︂ 𝜕𝑓 𝜕𝑓 𝜕𝑓 1 = +𝑖 = 0 2 𝜕𝑧  𝜕𝑥 𝜕𝑦 14 Chapter 3. MathematicsTheoretical Physics Reference, Release 0.5 We can write𝑓 =𝑢 +𝑖𝑣: 𝜕𝑓 𝜕𝑓 +𝑖 = 0 𝜕𝑥 𝜕𝑦 𝜕(𝑢 +𝑖𝑣) 𝜕(𝑢 +𝑖𝑣) +𝑖 = 0 𝜕𝑥 𝜕𝑦 (︂ )︂ (︂ )︂ 𝜕𝑢 𝜕𝑣 𝜕𝑢 𝜕𝑣 − +𝑖 + = 0 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑥 both the real and imaginary parts must be equal to zero: 𝜕𝑢 𝜕𝑣 = 𝜕𝑥 𝜕𝑦 𝜕𝑢 𝜕𝑣 =− 𝜕𝑦 𝜕𝑥 These are called the Cauchy-Riemann equations. We can derive the chain rule: d𝑓(𝑔) 𝜕𝑓(𝑔) 𝜕𝑓(𝑔) − 2𝑖𝜃 (3.12) = + 𝑒 = d𝑧 𝜕𝑧 𝜕𝑧  (︂ )︂ (︂ )︂ 𝜕𝑓 𝜕𝑔 𝜕𝑓 𝜕𝑔  𝜕𝑓 𝜕𝑔 𝜕𝑓 𝜕𝑔  − 2𝑖𝜃 = + + + 𝑒 = 𝜕𝑔 𝜕𝑧 𝜕𝑔 𝜕𝑧 𝜕𝑔 𝜕𝑧  𝜕𝑔 𝜕𝑧  (︂ )︂ (︂ )︂ 𝜕𝑓 𝜕𝑔 𝜕𝑔 𝜕𝑓 𝜕𝑔  𝜕𝑔  − 2𝑖𝜃 − 2𝑖𝜃 = + 𝑒 + + 𝑒 = 𝜕𝑔 𝜕𝑧 𝜕𝑧  𝜕𝑔  𝜕𝑧 𝜕𝑧  𝜕𝑓 d𝑔 𝜕𝑓 d𝑔  = + 𝜕𝑔 d𝑧 𝜕𝑔  d𝑧 Another useful formula is the derivative of a conjugate function:    d𝑓 𝜕𝑓 𝜕𝑓 𝜕𝑓 𝜕𝑓 − 2𝑖𝜃 − 2𝑖𝜃 (3.13) = + 𝑒 = + 𝑒 = d𝑧 𝜕𝑧 𝜕𝑧  𝜕𝑧  𝜕𝑧 (︂ )︂ 𝜕𝑓 𝜕𝑓 d𝑓 − 2𝑖𝜃 − 2𝑖𝜃 − 2𝑖𝜃 = 𝑒 + 𝑒 = 𝑒 𝜕𝑧  𝜕𝑧 d𝑧 Using (3.13), the chain rule (3.12) can also be written as: d𝑓(𝑔) 𝜕𝑓 d𝑔 𝜕𝑓 d𝑔  𝜕𝑓 d𝑔 𝜕𝑓 d𝑔 − 2𝑖𝜃 (3.14) = + = + 𝑒 d𝑧 𝜕𝑔 d𝑧 𝜕𝑔  d𝑧 𝜕𝑔 d𝑧 𝜕𝑔  d𝑧 d𝑔 Which has the advantage that only the derivative is needed, the rest is just conjugation and multiplication. If𝑓 is d𝑧 𝜕𝑓 analytic, then = 0, the second term vanishes and the chain rule is analogous to real functions. 𝜕𝑔 ¯ 3.2. Complex Numbers 15Theoretical Physics Reference, Release 0.5 Examples d𝑧 𝜕𝑧 𝜕𝑧 − 2𝑖𝜃 = + 𝑒 = 1 d𝑧 𝜕𝑧 𝜕𝑧  d𝑧  𝜕𝑧  𝜕𝑧  − 2𝑖𝜃 − 2𝑖𝜃 = + 𝑒 =𝑒 d𝑧 𝜕𝑧 𝜕𝑧  1 1 1 d Re𝑧 d (𝑧 +𝑧 ) 𝜕 (𝑧 +𝑧 ) 𝜕 (𝑧 +𝑧 ) 2 2 2 − 2𝑖𝜃 1 1 − 2𝑖𝜃 = = + 𝑒 = + 𝑒 2 2 d𝑧 d𝑧 𝜕𝑧 𝜕𝑧  𝑖 𝑖 𝑖 d (− 𝑧 +𝑧 ) 𝜕 (− 𝑧 +𝑧 ) 𝜕 (− 𝑧 +𝑧 ) d Im𝑧 𝑖 𝑖 2 2 2 − 2𝑖𝜃 − 2𝑖𝜃 = = + 𝑒 =− + 𝑒 d𝑧 d𝑧 𝜕𝑧 𝜕𝑧  2 2 √ √ √ − 2𝑖𝜃 − 2𝑖𝜃 d𝑧 d 𝑧𝑧  𝜕 𝑧𝑧  𝜕 𝑧𝑧  𝑧  +𝑧𝑒 𝑧  +𝑧𝑒 − 2𝑖𝜃 = = + 𝑒 = √ = d𝑧 d𝑧 𝜕𝑧 𝜕𝑧  2𝑧 2 𝑧𝑧  ¯ d𝑓 d𝑓   d𝑓(𝑧) 𝜕𝑓 d𝑓 𝜕𝑓 d𝑓 𝑓 +𝑓 d𝑧 d𝑧 = + =  d𝑧 𝜕𝑓 d𝑧 𝜕𝑓 d𝑧 2𝑓 d arg𝑧 d atan 2 (𝑖(− 𝑧 +𝑧 ),𝑧 +𝑧 ) 𝜕 atan 2 (𝑖(− 𝑧 +𝑧 ),𝑧 +𝑧 ) 𝜕 atan 2 (𝑖(− 𝑧 +𝑧 ),𝑧 +𝑧 ) − 2𝑖𝜃 = = + 𝑒 = d𝑧 d𝑧 𝜕𝑧 𝜕𝑧  (𝑧 +𝑧 )(− 𝑖)− 𝑖(− 𝑧 +𝑧 ) (𝑧 +𝑧 )𝑖− 𝑖(− 𝑧 +𝑧 ) − 2𝑖𝜃 = + 𝑒 = 4𝑧𝑧  4𝑧𝑧  (︂ )︂ (︂ )︂ − 2𝑖𝜃 𝑖 1 1 𝑖 − 𝑧  +𝑧𝑒 − 2𝑖𝜃 = − + 𝑒 = 2 2 𝑧 𝑧  2 𝑧 − 2𝑖𝜃 − 2𝑖𝜃 d log𝑧 1 𝑧  +𝑧𝑒 𝑧  +𝑧𝑒 = = 2 d𝑧 𝑧 2𝑧 2𝑧 (︂ )︂ − 2𝑖𝜃 − 2𝑖𝜃 d log𝑧 d(log𝑧 +𝑖 arg𝑧) 𝑧  +𝑧𝑒 𝑖 − 𝑧  +𝑧𝑒 𝑧  𝑧  1 = = +𝑖 = = = 2 2 2 d𝑧 d𝑧 2𝑧 2 𝑧 𝑧 𝑧𝑧  𝑧 dlog𝑧 𝜕log𝑧 𝜕log𝑧 𝜕 log𝑧 𝜕 log𝑧 1 − 2𝑖𝜃 − 2𝑖𝜃 − 2𝑖𝜃 = + 𝑒 = + 𝑒 = 𝑒 d𝑧 𝜕𝑧 𝜕𝑧  𝜕𝑧  𝜕𝑧 𝑧  d log𝑧 dlog𝑧 1 1 − 2𝑖𝜃 − 2𝑖𝜃 d log𝑧 log𝑧 + log𝑧 log𝑧 + (log𝑧)𝑒 𝑧 log𝑧 +𝑧(log𝑧)𝑒 d𝑧 d𝑧 𝑧 𝑧 ¯ = = = d𝑧 2 log𝑧 2 log𝑧 2𝑧𝑧  log𝑧 Note that if𝑧 is real, i.e. 𝑧 =𝑧 , we recover the real derivative results by setting𝜃 = 0, i.e. taking the derivative along 16 Chapter 3. Mathematics

Advise: Why You Wasting Money in Costly SEO Tools, Use World's Best Free SEO Tool Ubersuggest.