1 Complex Numbers Sums and Products Basic Algebraic Properties Further Properties Vectors and Moduli Complex Conjugates Exponential Form Products and Powers in Exponential Form Arguments of Products and Quotients Roots of Complex Numbers Examples Regions in the Complex Plane 2 Analytic Functions Functions of a Complex Variable Mappings Mappings by the Exponential Function Limits Theorems on Limits Limits Involving the Point at Infinity Continuity Derivatives Differentiation Formulas Cauchy-Riemann Equations Sufficient Conditions for Differentiability Polar Coordinates Analytic Functions Examples Harmonic Functions Uniquely Determined Analytic Functions Reflection Principle 3 Elementary Functions The Exponential Function The Logarithmic Function Branches and Derivatives of Logarithms Some Identities Involving Logarithms Complex Exponents Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions 4 Integrals Derivatives of Functions w(t) Definite Integrals of Functions w(t) Contours Contour Integrals Some Examples Examples with Branch Cuts Upper Bounds for Moduli of Contour Integrals Antiderivatives Proof of the Theorem Cauchy-Goursat Theorem Proof of the Theorem Simply Connected Domains Multiply Connected Domains Cauchy Integral Formula An Extension of the Cauchy Integral Formula Some Consequences of the Extension Liouville's Theorem and the Fundamental Theorem of Algebra Maximum Modulus Principle 5 Series Convergence of Sequences Convergence of Series Taylor Series Proof of Taylor's Theorem Examples Laurent Series Proof of Laurent's Theorem Examples Absolute and Uniform Convergence of Power Series Continuity of Sums of Power Series Integration and Differentiation of Power Series Uniqueness of Series Representations Multiplication and Division of Power Series 6 Residues and Poles Isolated Singular Points Residues Cauchy's Residue Theorem Residue at Infinity The Three Types of Isolated Singular Points Residues at Poles Examples Zeros of Analytic Functions Zeros and Poles Behavior of Functions Near Isolated Singular Points 7 Applications of Residues Evaluation of Improper Integrals Example Improper Integrals from Fourier Analysis Jordan's Lemma Indented Paths An Indentation Around a Branch Point Integration Along a Branch Cut Definite Integrals Involving Sines and Cosines Argument Principle Rouche's Theorem Inverse Laplace Transforms Examples 8 Mapping by Elementary Functions Linear Transformations The Transformation w = 1/z Mappings by 1/z Linear Fractional Transformations An Implicit Form Mappings of the Upper Half Plane The Transformation w = sin z Mappings by z2 and Branches of z1/2 Square Roots of Polynomials Riemann Surfaces Surfaces for Related Functions 9 Conformal Mapping Preservation of Angles Scale Factors Local Inverses Harmonic Conjugates Transformations of Harmonic Functions Transformations of Boundary Conditions 10 Applications of Conformal Mapping Steady Temperatures Steady Temperatures in a Half Plane A Related Problem Temperatures in a Quadrant Electrostatic Potential Potential in a Cylindrical Space Two-Dimensional Fluid Flow The Stream Function Flows Around a Corner and Around a Cylinder 11 The Schwarz-Christoffel Transformation Mapping the Real Axis onto a Polygon Schwarz-Christoffel Transformation Triangles and Rectangles Degenerate Polygons Fluid Flow in a Channel Through a Slit Flow in a Channel with an Offset Electrostatic Potential about an Edge of a Conducting Plate 12 Integral Formulas of the Poisson Type Poisson Integral Formula Dirichlet Problem for a Disk Related Boundary Value Problems Schwarz Integral Formula Dirichlet Problem for a Half Plane Neumann Problems Appendixes Bibliography Table of Transformations of Regions Index

Complex variables and applications

Book Review: Elements of the Random Walk: An Introduction for Advanced Students and Researchers

In the 1990s, physicists started looking beyond their disciplinary boundaries by using their methods to study various problems usually thrown up by financial economics. This dissertation deals with this extension of physics outside its disciplinary borders. It seeks to determine what sort of discipline econophysics is in relation to physics and to economics, how its emergence was made possible, and what sort of knowledge it produces. Using a variety of evidence including bibliometric analysis Chapter 1 explores the field’s disciplinary identity as a branch of physics even though its intellectual heart is better seen as the re-emergence of a 1960s research programme initiated in economics. Chapter 2 is historical: it identifies the key role played by the Santa Fe Institute and its pioneering complexity research in the shaping of methodological horizons of econophysics. These are in turn investigated in Chapter 3, which argues that there are in fact three methodological strands: statistical econophysics, bottom-up agent-based econophysics, and top-down agent-based econophysics. Viewed from a Lakatosian perspective they all share a conceptual hard-core but articulate the protective belt in distinctly different ways. The last and final chapter is devoted to the way econophysicists produce and justify their knowledge. It shows that econophysics operates by proposing empirically adequate analogies between physical and other systems in exactly the ways emphasised by Pierre Duhem. The contrast between such use of analogy in econophysics and modeling practices implemented by financial economics explains why econophysics remains so controversial to economists.

When Physics Became Undisciplined An Essay on Econophysics

We discuss USA stock market data from 1789 until 2020, focusing our attention on the S&P 500 index (1957–2020). We find that the data can be split into two periods, (1789–1948) and (1948–2020), displaying roughly 2% and 7% growth rates, respectively. The index variations from each trend appear similar, suggesting some degree of stationarity in market fluctuations. We then correlate market behavior to macroeconomic data, such as world (and USA) population growth and gross domestic product (GDP), on different time horizons. The analysis signals that the S&P 500 might be overvalued, possibly undergoing a series of bubbles, since the 1990s. To understand this behavior, we introduce a model for bubbles, showing that they can be caused by a lack of correlations between stock prices and a virtual market index, the latter calculated self-consistently from the stock prices. We argue that variations, $ \Delta\gamma $, in the "bubble parameter" (or decoupling factor $ \gamma $), are anticorrelated to variations of the Federal Funds Rate (FFR), which may trigger a bubble phenomenon ($ \gamma\to1 $) when persistent rate cuts become too pronounced. The FFR are confronted with the consumer price index (CPI) in the period (1955–2020) as an attempt to complete the picture. Our analyses suggest that the strong departure of the S&P 500 from historical fundamental trends within (1990–2020) may reflect the development of financial anomalies, in part related to monetary policies, which should be carefully addressed in the near future.

Estimating market index valuation from macroeconomic trends

Following our previous investigation of the USA Standard and Poor index anti-bubble that started in August 2000, we analyze thirty eight world stock market indices and identify 21 anti-bubble. An ``anti-bubble'' is defined as a self-fulfilling decreasing price created by positive price-to-price feedbacks feeding overall pessimism and negative market sentiment further strengthened by inter-personal interactions. We mathematically characterize anti-bubbles by a power law decrease of the price (or of the logarithm of the price) as a function of time and by decelerating/expanding log-periodic oscillations. The majority of European and Western stock market indices as well as other stock indices exhibit practically the same log-periodic power law anti-bubble structure as found for the USA S&P500 index. These anti-bubbles are found to start approximately at the same time, August 2000, in all these markets. This shows a remarkable degree of synchronization worldwide. The descent of the worldwide stock markets since 2000 is thus an international event, suggesting the strengthening of globalization.

Evidence of a Worldwide Stock Market Log-Periodic Anti-Bubble Since Mid-2000

Financial market models are able to help the investors foresee the risk of a financial market crash and reduce the probability of its occurrence. Modelling in financial markets is categorized into microscopic models and macroscopic models. The microscopic models study the mechanisms behind the market and their behaviour. These models assist in an understanding of the causes of financial market crashes. Macroscopic models find the disciplines and rules from the historical macroscopic data for the prediction of market trends, directions and crashes. They aim to provide forecasts of when the crashes should occur. Except for large fluctuations, the stock market price or index movements can be characterized by a random walk. The stock price trajectory possesses the properties of self-similarity and scale invariance, and hence mimics fractals. Furthermore, the stock price movement is a manifestation of the actions and interactions of stock traders. The hierarchical model with fractal structures, representing the interaction structure of the stock traders, is applicable to the study of the microscopic mechanisms of stock price movements. In the language of statistical mechanics, stock market crashes are viewed as critical phenomena where a crash occurs only at the critical point with a phase transition. The renormalization group is a mathematical apparatus that allows the decomposition of a macroscopic problem viewed at different scales. The renormalization group analysis of the hierarchical model finds that the time to reach the critical point of a system is a function of the interaction degree of stock traders in a power law. Moreover, the hierarchical model with the renormalization group formalism shows that the behavior of the fraction of all stock traders putting buy orders over time follows a power law coupled with log-periodic oscillations. Based on the renormalization group analysis results of the hierarchical model, a log-periodic power law model is derived by constructing an renormalization group formalism from the risk-driven model. In the log-periodic power law model, the expected time to crash is always finite. Therefore, market crashes are inevitable, but the existence of bubbles and crash risks is foreseen by the model.

https://summit.sfu.ca/system/files/iritems1/12362/etd7319_TWu.pdf

Tiankuang Tim Wu

Papers

Critical phenomena with renormalization group analysis of a hierarchical model of financial crashes