Week 4 – Complex Numbers
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Integration in the Complex Plane (Zill & Wright Chapter
Integration in the Complex Plane (Zill & Wright Chapter 18) 1016-420-02: Complex Variables∗ Winter 2012-2013 Contents 1 Contour Integrals 2 1.1 Definition and Properties . 2 1.2 Evaluation . 3 1.2.1 Example: R z¯ dz ............................. 3 C1 1.2.2 Example: R z¯ dz ............................. 4 C2 R 2 1.2.3 Example: C z dz ............................. 4 1.3 The ML Limit . 5 1.4 Circulation and Flux . 5 2 The Cauchy-Goursat Theorem 7 2.1 Integral Around a Closed Loop . 7 2.2 Independence of Path for Analytic Functions . 8 2.3 Deformation of Closed Contours . 9 2.4 The Antiderivative . 10 3 Cauchy's Integral Formulas 12 3.1 Cauchy's Integral Formula . 12 3.1.1 Example #1 . 13 3.1.2 Example #2 . 13 3.2 Cauchy's Integral Formula for Derivatives . 14 3.3 Consequences of Cauchy's Integral Formulas . 16 3.3.1 Cauchy's Inequality . 16 3.3.2 Liouville's Theorem . 16 ∗Copyright 2013, John T. Whelan, and all that 1 Tuesday 18 December 2012 1 Contour Integrals 1.1 Definition and Properties Recall the definition of the definite integral Z xF X f(x) dx = lim f(xk) ∆xk (1.1) ∆xk!0 xI k We'd like to define a similar concept, integrating a function f(z) from some point zI to another point zF . The problem is that, since zI and zF are points in the complex plane, there are different ways to get between them, and adding up the value of the function along one path will not give the same result as doing it along another path, even if they have the same endpoints. -
The Lorenz Curve
Charting Income Inequality The Lorenz Curve Resources for policy making Module 000 Charting Income Inequality The Lorenz Curve Resources for policy making Charting Income Inequality The Lorenz Curve by Lorenzo Giovanni Bellù, Agricultural Policy Support Service, Policy Assistance Division, FAO, Rome, Italy Paolo Liberati, University of Urbino, "Carlo Bo", Institute of Economics, Urbino, Italy for the Food and Agriculture Organization of the United Nations, FAO About EASYPol The EASYPol home page is available at: www.fao.org/easypol EASYPol is a multilingual repository of freely downloadable resources for policy making in agriculture, rural development and food security. The resources are the results of research and field work by policy experts at FAO. The site is maintained by FAO’s Policy Assistance Support Service, Policy and Programme Development Support Division, FAO. This modules is part of the resource package Analysis and monitoring of socio-economic impacts of policies. The designations employed and the presentation of the material in this information product do not imply the expression of any opinion whatsoever on the part of the Food and Agriculture Organization of the United Nations concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries. © FAO November 2005: All rights reserved. Reproduction and dissemination of material contained on FAO's Web site for educational or other non-commercial purposes are authorized without any prior written permission from the copyright holders provided the source is fully acknowledged. Reproduction of material for resale or other commercial purposes is prohibited without the written permission of the copyright holders. -
Niobrara County School District #1 Curriculum Guide
NIOBRARA COUNTY SCHOOL DISTRICT #1 CURRICULUM GUIDE th SUBJECT: Math Algebra II TIMELINE: 4 quarter Domain: Student Friendly Level of Resource Academic Standard: Learning Objective Thinking Correlation/Exemplar Vocabulary [Assessment] [Mathematical Practices] Domain: Perform operations on matrices and use matrices in applications. Standards: N-VM.6.Use matrices to I can represent and Application Pearson Alg. II Textbook: Matrix represent and manipulated data, e.g., manipulate data to represent --Lesson 12-2 p. 772 (Matrices) to represent payoffs or incidence data. M --Concept Byte 12-2 p. relationships in a network. 780 Data --Lesson 12-5 p. 801 [Assessment]: --Concept Byte 12-2 p. 780 [Mathematical Practices]: Niobrara County School District #1, Fall 2012 Page 1 NIOBRARA COUNTY SCHOOL DISTRICT #1 CURRICULUM GUIDE th SUBJECT: Math Algebra II TIMELINE: 4 quarter Domain: Student Friendly Level of Resource Academic Standard: Learning Objective Thinking Correlation/Exemplar Vocabulary [Assessment] [Mathematical Practices] Domain: Perform operations on matrices and use matrices in applications. Standards: N-VM.7. Multiply matrices I can multiply matrices by Application Pearson Alg. II Textbook: Scalars by scalars to produce new matrices, scalars to produce new --Lesson 12-2 p. 772 e.g., as when all of the payoffs in a matrices. M --Lesson 12-5 p. 801 game are doubled. [Assessment]: [Mathematical Practices]: Domain: Perform operations on matrices and use matrices in applications. Standards:N-VM.8.Add, subtract and I can perform operations on Knowledge Pearson Alg. II Common Matrix multiply matrices of appropriate matrices and reiterate the Core Textbook: Dimensions dimensions. limitations on matrix --Lesson 12-1p. 764 [Assessment]: dimensions. -
1 the Complex Plane
Math 135A, Winter 2012 Complex numbers 1 The complex numbers C are important in just about every branch of mathematics. These notes present some basic facts about them. 1 The Complex Plane A complex number z is given by a pair of real numbers x and y and is written in the form z = x+iy, where i satisfies i2 = −1. The complex numbers may be represented as points in the plane, with the real number 1 represented by the point (1; 0), and the complex number i represented by the point (0; 1). The x-axis is called the \real axis," and the y-axis is called the \imaginary axis." For example, the complex numbers 1, i, 3 + 4i and 3 − 4i are illustrated in Fig 1a. 3 + 4i 6 + 4i 2 + 3i imag i 4 + i 1 real 3 − 4i Fig 1a Fig 1b Complex numbers are added in a natural way: If z1 = x1 + iy1 and z2 = x2 + iy2, then z1 + z2 = (x1 + x2) + i(y1 + y2) (1) It's just vector addition. Fig 1b illustrates the addition (4 + i) + (2 + 3i) = (6 + 4i). Multiplication is given by z1z2 = (x1x2 − y1y2) + i(x1y2 + x2y1) Note that the product behaves exactly like the product of any two algebraic expressions, keeping in mind that i2 = −1. Thus, (2 + i)(−2 + 4i) = 2(−2) + 8i − 2i + 4i2 = −8 + 6i We call x the real part of z and y the imaginary part, and we write x = Re z, y = Im z.(Remember: Im z is a real number.) The term \imaginary" is a historical holdover; it took mathematicians some time to accept the fact that i (for \imaginary," naturally) was a perfectly good mathematical object. -
Complex Eigenvalues
Quantitative Understanding in Biology Module III: Linear Difference Equations Lecture II: Complex Eigenvalues Introduction In the previous section, we considered the generalized two- variable system of linear difference equations, and showed that the eigenvalues of these systems are given by the equation… ± 2 4 = 1,2 2 √ − …where b = (a11 + a22), c=(a22 ∙ a11 – a12 ∙ a21), and ai,j are the elements of the 2 x 2 matrix that define the linear system. Inspecting this relation shows that it is possible for the eigenvalues of such a system to be complex numbers. Because our plants and seeds model was our first look at these systems, it was carefully constructed to avoid this complication [exercise: can you show that http://www.xkcd.org/179 this model cannot produce complex eigenvalues]. However, many systems of biological interest do have complex eigenvalues, so it is important that we understand how to deal with and interpret them. We’ll begin with a review of the basic algebra of complex numbers, and then consider their meaning as eigenvalues of dynamical systems. A Review of Complex Numbers You may recall that complex numbers can be represented with the notation a+bi, where a is the real part of the complex number, and b is the imaginary part. The symbol i denotes 1 (recall i2 = -1, i3 = -i and i4 = +1). Hence, complex numbers can be thought of as points on a complex plane, which has real √− and imaginary axes. In some disciplines, such as electrical engineering, you may see 1 represented by the symbol j. √− This geometric model of a complex number suggests an alternate, but equivalent, representation. -
ISO Basic Latin Alphabet
ISO basic Latin alphabet The ISO basic Latin alphabet is a Latin-script alphabet and consists of two sets of 26 letters, codified in[1] various national and international standards and used widely in international communication. The two sets contain the following 26 letters each:[1][2] ISO basic Latin alphabet Uppercase Latin A B C D E F G H I J K L M N O P Q R S T U V W X Y Z alphabet Lowercase Latin a b c d e f g h i j k l m n o p q r s t u v w x y z alphabet Contents History Terminology Name for Unicode block that contains all letters Names for the two subsets Names for the letters Timeline for encoding standards Timeline for widely used computer codes supporting the alphabet Representation Usage Alphabets containing the same set of letters Column numbering See also References History By the 1960s it became apparent to thecomputer and telecommunications industries in the First World that a non-proprietary method of encoding characters was needed. The International Organization for Standardization (ISO) encapsulated the Latin script in their (ISO/IEC 646) 7-bit character-encoding standard. To achieve widespread acceptance, this encapsulation was based on popular usage. The standard was based on the already published American Standard Code for Information Interchange, better known as ASCII, which included in the character set the 26 × 2 letters of the English alphabet. Later standards issued by the ISO, for example ISO/IEC 8859 (8-bit character encoding) and ISO/IEC 10646 (Unicode Latin), have continued to define the 26 × 2 letters of the English alphabet as the basic Latin script with extensions to handle other letters in other languages.[1] Terminology Name for Unicode block that contains all letters The Unicode block that contains the alphabet is called "C0 Controls and Basic Latin". -
The Integral Resulting in a Logarithm of a Complex Number Dr. E. Jacobs in first Year Calculus You Learned About the Formula: Z B 1 Dx = Ln |B| − Ln |A| a X
The Integral Resulting in a Logarithm of a Complex Number Dr. E. Jacobs In first year calculus you learned about the formula: Z b 1 dx = ln jbj ¡ ln jaj a x It is natural to want to apply this formula to contour integrals. If z1 and z2 are complex numbers and C is a path connecting z1 to z2, we would expect that: Z 1 dz = log z2 ¡ log z1 C z However, you are about to see that there is an interesting complication when we take the formula for the integral resulting in logarithm and try to extend it to the complex case. First of all, recall that if f(z) is an analytic function on a domain D and C is a closed loop in D then : I f(z) dz = 0 This is referred to as the Cauchy-Integral Theorem. So, for example, if C His a circle of radius r around the origin we may conclude immediately that 2 2 C z dz = 0 because z is an analytic function for all values of z. H If f(z) is not analytic then C f(z) dz is not necessarily zero. Let’s do an 1 explicit calculation for f(z) = z . If we are integrating around a circle C centered at the origin, then any point z on this path may be written as: z = reiθ H 1 iθ Let’s use this to integrate C z dz. If z = re then dz = ireiθ dθ H 1 Now, substitute this into the integral C z dz. -
B1. the Generating Functional Z[J]
B1. The generating functional Z[J] The generating functional Z[J] is a key object in quantum field theory - as we shall see it reduces in ordinary quantum mechanics to a limiting form of the 1-particle propagator G(x; x0jJ) when the particle is under thje influence of an external field J(t). However, before beginning, it is useful to look at another closely related object, viz., the generating functional Z¯(J) in ordinary probability theory. Recall that for a simple random variable φ, we can assign a probability distribution P (φ), so that the expectation value of any variable A(φ) that depends on φ is given by Z hAi = dφP (φ)A(φ); (1) where we have assumed the normalization condition Z dφP (φ) = 1: (2) Now let us consider the generating functional Z¯(J) defined by Z Z¯(J) = dφP (φ)eφ: (3) From this definition, it immediately follows that the "n-th moment" of the probability dis- tribution P (φ) is Z @nZ¯(J) g = hφni = dφP (φ)φn = j ; (4) n @J n J=0 and that we can expand Z¯(J) as 1 X J n Z Z¯(J) = dφP (φ)φn n! n=0 1 = g J n; (5) n! n ¯ so that Z(J) acts as a "generator" for the infinite sequence of moments gn of the probability distribution P (φ). For future reference it is also useful to recall how one may also expand the logarithm of Z¯(J); we write, Z¯(J) = exp[W¯ (J)] Z W¯ (J) = ln Z¯(J) = ln dφP (φ)eφ: (6) 1 Now suppose we make a power series expansion of W¯ (J), i.e., 1 X 1 W¯ (J) = C J n; (7) n! n n=0 where the Cn, known as "cumulants", are just @nW¯ (J) C = j : (8) n @J n J=0 The relationship between the cumulants Cn and moments gn is easily found. -
Complex Sinusoids Copyright C Barry Van Veen 2014
Complex Sinusoids Copyright c Barry Van Veen 2014 Feel free to pass this ebook around the web... but please do not modify any of its contents. Thanks! AllSignalProcessing.com Key Concepts 1) The real part of a complex sinusoid is a cosine wave and the imag- inary part is a sine wave. 2) A complex sinusoid x(t) = AejΩt+φ can be visualized in the complex plane as a vector of length A that is rotating at a rate of Ω radians per second and has angle φ relative to the real axis at time t = 0. a) The projection of the complex sinusoid onto the real axis is the cosine A cos(Ωt + φ). b) The projection of the complex sinusoid onto the imaginary axis is the cosine A sin(Ωt + φ). AllSignalProcessing.com 3) The output of a linear time invariant system in response to a com- plex sinusoid input is a complex sinusoid of the of the same fre- quency. The system only changes the amplitude and phase of the input complex sinusoid. a) Arbitrary input signals are represented as weighted sums of complex sinusoids of different frequencies. b) Then, the output is a weighted sum of complex sinusoids where the weights have been multiplied by the complex gain of the system. 4) Complex sinusoids also simplify algebraic manipulations. AllSignalProcessing.com 5) Real sinusoids can be expressed as a sum of a positive and negative frequency complex sinusoid. 6) The concept of negative frequency is not physically meaningful for real data and is an artifact of using complex sinusoids. -
Percent R, X and Z Based on Transformer KVA
SHORT CIRCUIT FAULT CALCULATIONS Short circuit fault calculations as required to be performed on all electrical service entrances by National Electrical Code 110-9, 110-10. These calculations are made to assure that the service equipment will clear a fault in case of short circuit. To perform the fault calculations the following information must be obtained: 1. Available Power Company Short circuit KVA at transformer primary : Contact Power Company, may also be given in terms of R + jX. 2. Length of service drop from transformer to building, Type and size of conductor, ie., 250 MCM, aluminum. 3. Impedance of transformer, KVA size. A. %R = Percent Resistance B. %X = Percent Reactance C. %Z = Percent Impedance D. KVA = Kilovoltamp size of transformer. ( Obtain for each transformer if in Bank of 2 or 3) 4. If service entrance consists of several different sizes of conductors, each must be adjusted by (Ohms for 1 conductor) (Number of conductors) This must be done for R and X Three Phase Systems Wye Systems: 120/208V 3∅, 4 wire 277/480V 3∅ 4 wire Delta Systems: 120/240V 3∅, 4 wire 240V 3∅, 3 wire 480 V 3∅, 3 wire Single Phase Systems: Voltage 120/240V 1∅, 3 wire. Separate line to line and line to neutral calculations must be done for single phase systems. Voltage in equations (KV) is the secondary transformer voltage, line to line. Base KVA is 10,000 in all examples. Only those components actually in the system have to be included, each component must have an X and an R value. Neutral size is assumed to be the same size as the phase conductors. -
Inequality Measurement Development Issues No
Development Strategy and Policy Analysis Unit w Development Policy and Analysis Division Department of Economic and Social Affairs Inequality Measurement Development Issues No. 2 21 October 2015 Comprehending the impact of policy changes on the distribu- tion of income first requires a good portrayal of that distribution. Summary There are various ways to accomplish this, including graphical and mathematical approaches that range from simplistic to more There are many measures of inequality that, when intricate methods. All of these can be used to provide a complete combined, provide nuance and depth to our understanding picture of the concentration of income, to compare and rank of how income is distributed. Choosing which measure to different income distributions, and to examine the implications use requires understanding the strengths and weaknesses of alternative policy options. of each, and how they can complement each other to An inequality measure is often a function that ascribes a value provide a complete picture. to a specific distribution of income in a way that allows direct and objective comparisons across different distributions. To do this, inequality measures should have certain properties and behave in a certain way given certain events. For example, moving $1 from the ratio of the area between the two curves (Lorenz curve and a richer person to a poorer person should lead to a lower level of 45-degree line) to the area beneath the 45-degree line. In the inequality. No single measure can satisfy all properties though, so figure above, it is equal to A/(A+B). A higher Gini coefficient the choice of one measure over others involves trade-offs. -
The Deloitte Global 2021 Millennial and Gen Z Survey
A call for accountability and action T HE D ELO IT T E GLOB A L 2021 M IL LE N N IA L AND GEN Z SUR V E Y 1 Contents 01 06 11 INTRODUCTION CHAPTER 1 CHAPTER 2 Impact of the COVID-19 The effect on mental health pandemic on daily life 15 27 33 CHAPTER 3 CHAPTER 4 CONCLUSION How the past year influenced Driven to act millennials’ and Gen Zs’ world outlooks 2 Introduction Millennials and Generation Zs came of age at the same time that online platforms and social media gave them the ability and power to share their opinions, influence distant people and institutions, and question authority in new ways. These forces have shaped their worldviews, values, and behaviors. Digital natives’ ability to connect, convene, and create disruption via their keyboards and smartphones has had global impact. From #MeToo to Black Lives Matter, from convening marches on climate change to the Arab Spring, from demanding eco-friendly products to challenging stakeholder capitalism, these generations are compelling real change in society and business. The lockdowns resulting from the COVID-19 pandemic curtailed millennials’ and Gen Zs’ activities but not their drive or their desire to be heard. In fact, the 2021 Deloitte Global Millennial Survey suggests that the pandemic, extreme climate events, and a charged sociopolitical atmosphere may have reinforced people’s passions and given them oxygen. 01 Urging accountability Last year’s report1 reflected the results of two Of course, that’s a generality—no group of people is surveys—one taken just before the pandemic and a homogeneous.