2.5 Exponential Functions and Trigonometric Functions
|
|
- Amos Park
- 6 years ago
- Views:
Transcription
1 5 CHAPTER. COMPLEX-VALUED FUNCTIONS.5 Exponential Functions and Trigonometric Functions Exponential Function and Its Properties By the theory of power series, we can define e z := which is called the exponential function. Here 0! = and n! =... (n ) n. e z has the following properties:. The radius of convergence R = +. In fact, R = lim n a n a n+ = lim n n = +. Therefore, e z is an analytic function defined on C.. When z = x R, the restriction e x is the usual real exponential function. This is because the Taylor series of e x is z n n! n! xn. 3. e z+w = e z e w, z, w C. In fact, e x+u = e x e u forall x, u R. Then by the uniqueness theorem.4.6, e z+w = e z e w holds for any z, w C. 4. As a function from C to C, e z is not a one-to-one function. Consider solutions of the equation e z =. It is equivalent to z = πki for any integer k. 6π 4π π e z 0 π 4π 6π 5. We have ( e z ) = e z by the power series definition.
2 .5. EXPONENTIAL FUNCTIONS AND TRIGONOMETRIC FUNCTIONS 53 n 6. e x = lim n ( + n) x, x R. 7. Euler s formula: e ix = cos x + i sin x, x R. (.0) From this, we have e z = r(cos θ + i sin θ) for any z C. Trigonometric Functions. From Euler s formula, taking conjugate: e ix = cos x i sin x so that cos x = eix + e ix, sin x = eix e ix. i which can be used to define complex-valued function cos z and sin z. Hence e iz = cons z + isin z, and cos z = cos z := eiz + e iz, sin z := eiz e iz. i (n)! zn, sin z = (n + )! zn+. Here the formulas are similar to the real case: cos x = (n+)! xn+. (n)! xn and sin x =. We shall see the difference between the real case and the complex case. In the real case, cos x and sin x ; in the complex case, cosz and sin z are no longer true. In fact, take z = iy where y R, we have 3. We can verify: sin z = ei(iy) e i(iy) = e y e y i cons z + sin z =, as y. cos(z + w) = cos z cos w sin z sin w, sin(z + w) = sin z cos w + cos z sin w. 4. Consider sin z = 0 e iz e iz = 0 s iz = z = πk where k Z z = πk where k Z. Similarly, cos z = 0 z = π + πk where k Z.
3 54 CHAPTER. COMPLEX-VALUED FUNCTIONS 5. We can also define tan z := sin z cos z, cos z cot z := sin z. Logarithm Function. Review Calculus: the inverse function of e x is log x. In this course we denote log x to be the natural log ln x. By the inverse function property, we have e log x = x, x (0, ); log e x = x, x R. (.) Denote y = e x. By the derivative formula of inverse function ((f ) = f, we have (log x) = (e y ) = e y = x. (.) Then by the fundamental theorem of calculus f(x) = x f (t)dt, we obtain a log x = x which is regarded as definition of log x in some textbook. dt (.3) t. We notice that e z is not one-to-one function on C (see above) so that we cannot define the inverse function log z of e z. However, e z could be one-to-one function on some domains and log z could be defined on some domain. Recall the logarithm function has power series expansion: log( + x) = n + xn+, x (, ). Here the radius of convergence of this power series is. Then we have log( + z) := n + zn+, z (). and, by replacing + z with z, log z = n + (z )n+, z (; ). If f : X Y is an one-to-one and onto map, it has inverse map f : Y X satisfying f (f(x)) = x, x X and f(f (y)) = y, y Y.
4 .5. EXPONENTIAL FUNCTIONS AND TRIGONOMETRIC FUNCTIONS 55 Since e log x = x, x (0, ), by applying the uniqueness theorem.4.6, it implies e log z = z, z (; ). 3. On the other hand, write z = re iθ = z e i arg z, by the addition property of logarithm, we obtain log z = log z + log e iarg z = log z + iarg z, z (; ). 4. The addition theorem of exponential functions implies log(z z ) = log(z ) + log(z ), arg(z z ) = arg(z ) + arg(z ). 5. At this moment, the analytic function log z is defined on the disk (, ). We ll show later that the domain of definition for log z can be extended. Historic Remark Logarithm function is closely related to its inverse function: exponential function by x = b y y = log b x. In 64, John Napier (550-67) published a book entitled Mirifici Logarithmorum Canonis Descriptio in which logarithm was introduced. The table of logarithms made by Napier contributed to the advance of science, and especially of astronomy, by making some difficult calculations possible (it turns multiplication into addition). 3 Early resistance to the use of logarithms was muted by Kepler s enthusiastic support and his publication of a clear and impeccable explanation of how they worked. By the way, Napier did not use a base as we now understand it, but his logarithms were, up to a scaling factor, effectively to base e. In 668, Nicholas Mercator (60-687), a Germany mathematician, first mentioned the natural logarithm ln x in his work Logarithmo-technia 4 In this treatise Mercator described the Mercator series, also independently discovered by Gregory Saint-Vincent: log( + x) = x x + 3 x3 4 x Joost Bürgi independently discovered logarithms; however, he did not publish his discovery until four years after Napier. 4 en.wikipedia.org: natural logarithm. The mathematics teacher John Speidell had already in 69 compiled a table on the natural logarithm.
5 56 CHAPTER. COMPLEX-VALUED FUNCTIONS which was treated as the area of the hyperbola x dt = x+ dt (see (.3)). It was also 0 +t t in this treatise that the first known use of the term natural log for the natural log appears, in the Latin form log naturalis; his use of this term is somewhat surprising, since it predates the development of calculus, in which the most natural properties of this logarithm appear. In 697, Jacob Bernoulli studied lim n (+ n )n, which had been implicit in earlier work on natural logarithms. In 748, Euler defined the two functions: 5 ( e x = lim + x n, log x = lim n n) n( n x ). (.4) n and also proved his famous formula: e ix = cos x + i sin x (.5) Euler proved that the infinite series (.5) of both sides being equal. 50 years later, the view of complex numbers as points in the complex plane arose. Until Euler the trigonometrical quantities sine, cosine, tangent, etc., were regarded as lines connected with the circle rather than functions. Even the derivation of the series expansion for the sine in dependence of the arc by Newton and Leibniz did not change this view. It was Euler who introduced the functional point of view. 6 By the way, in 988, readers of the Mathematical Intelligencer voted it the Most Beautiful Mathematical Formula Ever. In total, Euler was responsible for three of the top five formulas in that poll. 7 It was a difficulty problem how to define log a when a is a negative number. Between 7 and 73, Bernoulli held the view that log a = log( a) because d( x) = dx, while x x Leibniz believed that log( a) must be imaginary. Between 77 and 73, the question was taken up by Bernoulli and Euler without at a solution. It was only in 749/75, Euler developed ideas far enough to study logarithm so that it leads to a satisfying solution. Euler s approach is to deal with logarithm of general complex numbers. Euler s argument is as follows: by the definition (.4), log z = lim n n( n z ). For each positive integer n, n( n z ) has n different roots (Euler realized it!) so that the limit indeed has infinitely many different values. When z =, he got log = ±πki, k =,, 3, Morris Kline, Mathematical Thought from Ancient to Modern Times, volume, New York Oxford, Oxford University Press, 97, p Hans Niels Jahnke (editor), A History of Analysis, AMS, 003, p Here is the list of the top five:. Euler s formula;. Euler s formula for a polyhedron: V + F = E The number of primes is infinite. 4. There are 5 regular polyhedrons = π 6 (Euler). cf., David Wells, Are these the most beautiful? Mathematical Intelligencer (3)(990), Hans Niels Jahnke (editor), A History of Analysis, AMS, 003, p.7-8.
12 Logarithmic Function
12 Logarithmic Function The definition of logarithmic function From Calculus, the logarithmic function, log x : (0, ) R, is defined to be the inverse function of e x : R (0, ) gievn by 42 e log x = x,
More informationCHAPTER 3 ELEMENTARY FUNCTIONS 28. THE EXPONENTIAL FUNCTION. Definition: The exponential function: The exponential function e z by writing
CHAPTER 3 ELEMENTARY FUNCTIONS We consider here various elementary functions studied in calculus and define corresponding functions of a complex variable. To be specific, we define analytic functions of
More informationMA 201 Complex Analysis Lecture 6: Elementary functions
MA 201 Complex Analysis : The Exponential Function Recall: Euler s Formula: For y R, e iy = cos y + i sin y and for any x, y R, e x+y = e x e y. Definition: If z = x + iy, then e z or exp(z) is defined
More informationChapter 3 Elementary Functions
Chapter 3 Elementary Functions In this chapter, we will consier elementary functions of a complex variable. We will introuce complex exponential, trigonometric, hyperbolic, an logarithmic functions. 23.
More informationChapter 1. Complex Numbers. 1.1 Complex Numbers. Did it come from the equation x = 0 (1.1)
Chapter 1 Complex Numbers 1.1 Complex Numbers Origin of Complex Numbers Did it come from the equation Where did the notion of complex numbers came from? x 2 + 1 = 0 (1.1) as i is defined today? No. Very
More informationLecture 9. = 1+z + 2! + z3. 1 = 0, it follows that the radius of convergence of (1) is.
The Exponential Function Lecture 9 The exponential function 1 plays a central role in analysis, more so in the case of complex analysis and is going to be our first example using the power series method.
More informationCHAPTER 4. Elementary Functions. Dr. Pulak Sahoo
CHAPTER 4 Elementary Functions BY Dr. Pulak Sahoo Assistant Professor Department of Mathematics University Of Kalyani West Bengal, India E-mail : sahoopulak1@gmail.com 1 Module-4: Multivalued Functions-II
More informationSolutions to Tutorial for Week 3
The University of Sydney School of Mathematics and Statistics Solutions to Tutorial for Week 3 MATH9/93: Calculus of One Variable (Advanced) Semester, 08 Web Page: sydney.edu.au/science/maths/u/ug/jm/math9/
More informationThe modulus, or absolute value, of a complex number z a bi is its distance from the origin. From Figure 3 we see that if z a bi, then.
COMPLEX NUMBERS _+i _-i FIGURE Complex numbers as points in the Arg plane i _i +i -i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with
More information3 Elementary Functions
3 Elementary Functions 3.1 The Exponential Function For z = x + iy we have where Euler s formula gives The note: e z = e x e iy iy = cos y + i sin y When y = 0 we have e x the usual exponential. When z
More informationIII.2. Analytic Functions
III.2. Analytic Functions 1 III.2. Analytic Functions Recall. When you hear analytic function, think power series representation! Definition. If G is an open set in C and f : G C, then f is differentiable
More informationCOMPLEX ANALYSIS TOPIC V: HISTORICAL REMARKS
COMPLEX ANALYSIS TOPIC V: HISTORICAL REMARKS PAUL L. BAILEY Historical Background Reference: http://math.fullerton.edu/mathews/n2003/complexnumberorigin.html Rafael Bombelli (Italian 1526-1572) Recall
More informationComplex Analysis Homework 1: Solutions
Complex Analysis Fall 007 Homework 1: Solutions 1.1.. a) + i)4 + i) 8 ) + 1 + )i 5 + 14i b) 8 + 6i) 64 6) + 48 + 48)i 8 + 96i c) 1 + ) 1 + i 1 + 1 i) 1 + i)1 i) 1 + i ) 5 ) i 5 4 9 ) + 4 4 15 i ) 15 4
More informationComplex Variables. Instructions Solve any eight of the following ten problems. Explain your reasoning in complete sentences to maximize credit.
Instructions Solve any eight of the following ten problems. Explain your reasoning in complete sentences to maximize credit. 1. The TI-89 calculator says, reasonably enough, that x 1) 1/3 1 ) 3 = 8. lim
More informationM361 Theory of functions of a complex variable
M361 Theory of functions of a complex variable T. Perutz U.T. Austin, Fall 2012 Lecture 4: Exponentials and logarithms We have already been making free use of the sine and cosine functions, cos: R R, sin:
More informationExercises for Part 1
MATH200 Complex Analysis. Exercises for Part Exercises for Part The following exercises are provided for you to revise complex numbers. Exercise. Write the following expressions in the form x + iy, x,y
More informationComplex Variables. Cathal Ormond
Complex Variables Cathal Ormond Contents 1 Introduction 3 1.1 Definition: Polar Form.............................. 3 1.2 Definition: Length................................ 3 1.3 Definitions.....................................
More informationMTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1.
MTH4101 CALCULUS II REVISION NOTES 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) 1.1 Introduction Types of numbers (natural, integers, rationals, reals) The need to solve quadratic equations:
More informationCourse 2BA1: Hilary Term 2006 Section 7: Trigonometric and Exponential Functions
Course 2BA1: Hilary Term 2006 Section 7: Trigonometric and Exponential Functions David R. Wilkins Copyright c David R. Wilkins 2005 Contents 7 Trigonometric and Exponential Functions 1 7.1 Basic Trigonometric
More informationMath 421 Midterm 2 review questions
Math 42 Midterm 2 review questions Paul Hacking November 7, 205 () Let U be an open set and f : U a continuous function. Let be a smooth curve contained in U, with endpoints α and β, oriented from α to
More informationExercises involving elementary functions
017:11:0:16:4:09 c M. K. Warby MA3614 Complex variable methods and applications 1 Exercises involving elementary functions 1. This question was in the class test in 016/7 and was worth 5 marks. a) Let
More informationComplex numbers, the exponential function, and factorization over C
Complex numbers, the exponential function, and factorization over C 1 Complex Numbers Recall that for every non-zero real number x, its square x 2 = x x is always positive. Consequently, R does not contain
More informationSyllabus: for Complex variables
EE-2020, Spring 2009 p. 1/42 Syllabus: for omplex variables 1. Midterm, (4/27). 2. Introduction to Numerical PDE (4/30): [Ref.num]. 3. omplex variables: [Textbook]h.13-h.18. omplex numbers and functions,
More informationMath 411, Complex Analysis Definitions, Formulas and Theorems Winter y = sinα
Math 411, Complex Analysis Definitions, Formulas and Theorems Winter 014 Trigonometric Functions of Special Angles α, degrees α, radians sin α cos α tan α 0 0 0 1 0 30 π 6 45 π 4 1 3 1 3 1 y = sinα π 90,
More informationMATH 135: COMPLEX NUMBERS
MATH 135: COMPLEX NUMBERS (WINTER, 010) The complex numbers C are important in just about every branch of mathematics. These notes 1 present some basic facts about them. 1. The Complex Plane A complex
More informationax 2 +bx+c = 0, M = R 4 X
Complex Numbers Why shall we study complex analysis? We list several examples to illustrate why shall we study complex analysis. (Algebra) If we only limit ourselves to real numbers, the quadratic equation
More informationFourth Week: Lectures 10-12
Fourth Week: Lectures 10-12 Lecture 10 The fact that a power series p of positive radius of convergence defines a function inside its disc of convergence via substitution is something that we cannot ignore
More informationSection 7.2. The Calculus of Complex Functions
Section 7.2 The Calculus of Complex Functions In this section we will iscuss limits, continuity, ifferentiation, Taylor series in the context of functions which take on complex values. Moreover, we will
More informationBernoulli Numbers and their Applications
Bernoulli Numbers and their Applications James B Silva Abstract The Bernoulli numbers are a set of numbers that were discovered by Jacob Bernoulli (654-75). This set of numbers holds a deep relationship
More information2.2 The derivative as a Function
2.2 The derivative as a Function Recall: The derivative of a function f at a fixed number a: f a f a+h f(a) = lim h 0 h Definition (Derivative of f) For any number x, the derivative of f is f x f x+h f(x)
More informationA Library of Functions
LibraryofFunctions.nb 1 A Library of Functions Any study of calculus must start with the study of functions. Functions are fundamental to mathematics. In its everyday use the word function conveys to us
More informationChapter 3 Differentiation Rules
Chapter 3 Differentiation Rules Derivative constant function if c is any real number, then Example: The Power Rule: If n is a positive integer, then Example: Extended Power Rule: If r is any real number,
More informationComplex Variable Outline. Richard Koch
Complex Variable Outline Richard Koch March 6, 206 Contents List of Figures 5 Preface 8 2 Preliminaries and Examples 0 2. Review of Complex Numbers.......................... 0 2.2 Holomorphic Functions..............................
More informationCOMPLEX NUMBERS AND SERIES
COMPLEX NUMBERS AND SERIES MIKE BOYLE Contents 1. Complex Numbers 1 2. The Complex Plane 2 3. Addition and Multiplication of Complex Numbers 2 4. Why Complex Numbers Were Invented 3 5. The Fundamental
More informationConsidering our result for the sum and product of analytic functions, this means that for (a 0, a 1,..., a N ) C N+1, the polynomial.
Lecture 3 Usual complex functions MATH-GA 245.00 Complex Variables Polynomials. Construction f : z z is analytic on all of C since its real and imaginary parts satisfy the Cauchy-Riemann relations and
More information2.4 Lecture 7: Exponential and trigonometric
154 CHAPTER. CHAPTER II.0 1 - - 1 1 -.0 - -.0 - - - - - - - - - - 1 - - - - - -.0 - Figure.9: generalized elliptical domains; figures are shown for ǫ = 1, ǫ = 0.8, eps = 0.6, ǫ = 0.4, and ǫ = 0 the case
More informationTAYLOR AND MACLAURIN SERIES
TAYLOR AND MACLAURIN SERIES. Introduction Last time, we were able to represent a certain restricted class of functions as power series. This leads us to the question: can we represent more general functions
More information1. Complex Numbers. John Douglas Moore. July 1, 2011
1. Complex Numbers John Douglas Moore July 1, 2011 These notes are intended to supplement the text, Fundamentals of complex analysis, by Saff and Snider [5]. Other often-used references for the theory
More informationMath 185 Fall 2015, Sample Final Exam Solutions
Math 185 Fall 2015, Sample Final Exam Solutions Nikhil Srivastava December 12, 2015 1. True or false: (a) If f is analytic in the annulus A = {z : 1 < z < 2} then there exist functions g and h such that
More informationExponential and Trigonometric functions
Exponential and Trigonometric functions Our toolkit of concrete holomorphic functions is woefully small. We will now remedy this by introducing the classical exponential and trigonometric functions using
More informationMA3111S COMPLEX ANALYSIS I
MA3111S COMPLEX ANALYSIS I 1. The Algebra of Complex Numbers A complex number is an expression of the form a + ib, where a and b are real numbers. a is called the real part of a + ib and b the imaginary
More information1 Review of complex numbers
1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely
More informationSome commonly encountered sets and their notations
NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS (This notes are based on the book Introductory Mathematics by Ng Wee Seng ) LECTURE SETS & FUNCTIONS Some commonly encountered sets and their
More information3 + 4i 2 + 3i. 3 4i Fig 1b
The introduction of complex numbers in the 16th century was a natural step in a sequence of extensions of the positive integers, starting with the introduction of negative numbers (to solve equations of
More informationCandidates are expected to have available a calculator. Only division by (x + a) or (x a) will be required.
Revision Checklist Unit C2: Core Mathematics 2 Unit description Algebra and functions; coordinate geometry in the (x, y) plane; sequences and series; trigonometry; exponentials and logarithms; differentiation;
More informationCourse 214 Section 2: Infinite Series Second Semester 2008
Course 214 Section 2: Infinite Series Second Semester 2008 David R. Wilkins Copyright c David R. Wilkins 1989 2008 Contents 2 Infinite Series 25 2.1 The Comparison Test and Ratio Test.............. 26
More information1 + lim. n n+1. f(x) = x + 1, x 1. and we check that f is increasing, instead. Using the quotient rule, we easily find that. 1 (x + 1) 1 x (x + 1) 2 =
Chapter 5 Sequences and series 5. Sequences Definition 5. (Sequence). A sequence is a function which is defined on the set N of natural numbers. Since such a function is uniquely determined by its values
More information18.04 Practice problems exam 1, Spring 2018 Solutions
8.4 Practice problems exam, Spring 8 Solutions Problem. omplex arithmetic (a) Find the real and imaginary part of z + z. (b) Solve z 4 i =. (c) Find all possible values of i. (d) Express cos(4x) in terms
More informationComplex Numbers. Integers, Rationals, and Reals. The natural numbers: The integers:
Complex Numbers Integers, Rationals, and Reals The natural numbers: N {... 3, 2,, 0,, 2, 3...} The integers: Z {... 3, 2,, 0,, 2, 3...} Note that any two integers added, subtracted, or multiplied together
More information18.03 LECTURE NOTES, SPRING 2014
18.03 LECTURE NOTES, SPRING 2014 BJORN POONEN 7. Complex numbers Complex numbers are expressions of the form x + yi, where x and y are real numbers, and i is a new symbol. Multiplication of complex numbers
More informationsin cos 1 1 tan sec 1 cot csc Pre-Calculus Mathematics Trigonometric Identities and Equations
Pre-Calculus Mathematics 12 6.1 Trigonometric Identities and Equations Goal: 1. Identify the Fundamental Trigonometric Identities 2. Simplify a Trigonometric Expression 3. Determine the restrictions on
More informationAP Calculus BC. Course Overview. Course Outline and Pacing Guide
AP Calculus BC Course Overview AP Calculus BC is designed to follow the topic outline in the AP Calculus Course Description provided by the College Board. The primary objective of this course is to provide
More informationCompletion Date: Monday February 11, 2008
MATH 4 (R) Winter 8 Intermediate Calculus I Solutions to Problem Set #4 Completion Date: Monday February, 8 Department of Mathematical and Statistical Sciences University of Alberta Question. [Sec..9,
More informationPhysics 307. Mathematical Physics. Luis Anchordoqui. Wednesday, August 31, 16
Physics 307 Mathematical Physics Luis Anchordoqui 1 Bibliography L. A. Anchordoqui and T. C. Paul, ``Mathematical Models of Physics Problems (Nova Publishers, 2013) G. F. D. Duff and D. Naylor, ``Differential
More informationu = 0; thus v = 0 (and v = 0). Consequently,
MAT40 - MANDATORY ASSIGNMENT #, FALL 00; FASIT REMINDER: The assignment must be handed in before 4:30 on Thursday October 8 at the Department of Mathematics, in the 7th floor of Niels Henrik Abels hus,
More informationINTEGRATION WORKSHOP 2003 COMPLEX ANALYSIS EXERCISES
INTEGRATION WORKSHOP 23 COMPLEX ANALYSIS EXERCISES DOUGLAS ULMER 1. Meromorphic functions on the Riemann sphere It s often useful to allow functions to take the value. This exercise outlines one way to
More information01 Harmonic Oscillations
Utah State University DigitalCommons@USU Foundations of Wave Phenomena Library Digital Monographs 8-2014 01 Harmonic Oscillations Charles G. Torre Department of Physics, Utah State University, Charles.Torre@usu.edu
More information4.1 Exponential and Logarithmic Functions
. Exponential and Logarithmic Functions Joseph Heavner Honors Complex Analysis Continued) Chapter July, 05 3.) Find the derivative of f ) e i e i. d d e i e i) d d ei ) d d e i ) e i d d i) e i d d i)
More information7.1. Calculus of inverse functions. Text Section 7.1 Exercise:
Contents 7. Inverse functions 1 7.1. Calculus of inverse functions 2 7.2. Derivatives of exponential function 4 7.3. Logarithmic function 6 7.4. Derivatives of logarithmic functions 7 7.5. Exponential
More informationMath Homework 1. The homework consists mostly of a selection of problems from the suggested books. 1 ± i ) 2 = 1, 2.
Math 70300 Homework 1 September 1, 006 The homework consists mostly of a selection of problems from the suggested books. 1. (a) Find the value of (1 + i) n + (1 i) n for every n N. We will use the polar
More informationSummary: Primer on Integral Calculus:
Physics 2460 Electricity and Magnetism I, Fall 2006, Primer on Integration: Part I 1 Summary: Primer on Integral Calculus: Part I 1. Integrating over a single variable: Area under a curve Properties of
More informationPower series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) a n z n. n=0
Lecture 22 Power series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) Recall a few facts about power series: a n z n This series in z is centered at z 0. Here z can
More information11.10a Taylor and Maclaurin Series
11.10a 1 11.10a Taylor and Maclaurin Series Let y = f(x) be a differentiable function at x = a. In first semester calculus we saw that (1) f(x) f(a)+f (a)(x a), for all x near a The right-hand side of
More informationIn this chapter we study several functions that are useful in calculus and other areas of mathematics.
Calculus 5 7 Special functions In this chapter we study several functions that are useful in calculus and other areas of mathematics. 7. Hyperbolic trigonometric functions The functions we study in this
More informationWith this expanded version of what we mean by a solution to an equation we can solve equations that previously had no solution.
M 74 An introduction to Complex Numbers. 1 Solving equations Throughout the calculus sequence we have limited our discussion to real valued solutions to equations. We know the equation x 1 = 0 has distinct
More information1. COMPLEX NUMBERS. z 1 + z 2 := (a 1 + a 2 ) + i(b 1 + b 2 ); Multiplication by;
1. COMPLEX NUMBERS Notations: N the set of the natural numbers, Z the set of the integers, R the set of real numbers, Q := the set of the rational numbers. Given a quadratic equation ax 2 + bx + c = 0,
More informationHere are brief notes about topics covered in class on complex numbers, focusing on what is not covered in the textbook.
Phys374, Spring 2008, Prof. Ted Jacobson Department of Physics, University of Maryland Complex numbers version 5/21/08 Here are brief notes about topics covered in class on complex numbers, focusing on
More informationExponential, Logarithmic &Trigonometric Derivatives
1 U n i t 9 12CV Date: Name: Exponential, Logarithmic &Trigonometric Derivatives Tentative TEST date Big idea/learning Goals The world s population experiences exponential growth the rate of growth becomes
More informationRe(z) = a, For example, 3 + 2i = = 13. The complex conjugate (or simply conjugate") of z = a + bi is the number z defined by
F COMPLEX NUMBERS In this appendi, we review the basic properties of comple numbers. A comple number is a number z of the form z = a + bi where a,b are real numbers and i represents a number whose square
More informationMTH 362: Advanced Engineering Mathematics
MTH 362: Advanced Engineering Mathematics Lecture 1 Jonathan A. Chávez Casillas 1 1 University of Rhode Island Department of Mathematics September 7, 2017 Course Name and number: MTH 362: Advanced Engineering
More informationBackground Complex Analysis (1A) Young Won Lim 9/2/14
Background Complex Analsis (1A) Copright (c) 2014 Young W. Lim. Permission is granted to cop, distribute and/or modif this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationFrom trigonometry to elliptic functions
From trigonometry to elliptic functions Zhiqin Lu The Math Club University of California, Irvine March 3, 200 Zhiqin Lu, The Math Club University of California, Irvine From trigonometry to elliptic functions
More informationPHYS 3900 Homework Set #02
PHYS 3900 Homework Set #02 Part = HWP 2.0, 2.02, 2.03. Due: Mon. Jan. 22, 208, 4:00pm Part 2 = HWP 2.04, 2.05, 2.06. Due: Fri. Jan. 26, 208, 4:00pm All textbook problems assigned, unless otherwise stated,
More informationProject 1: Riemann Sums
MS 00 Integral Calculus and Differential Equations 1 Project 1: Riemann Sums In this project you prove some summation identities and then apply them to calculate various integrals from first principles.
More informationComplex Function. Chapter Complex Number. Contents
Chapter 6 Complex Function Contents 6. Complex Number 3 6.2 Elementary Functions 6.3 Function of Complex Variables, Limit and Derivatives 3 6.4 Analytic Functions and Their Derivatives 8 6.5 Line Integral
More informationClassical transcendental curves
Classical transcendental curves Reinhard Schultz May, 2008 In his writings on coordinate geometry, Descartes emphasized that he was only willing to work with curves that could be defined by algebraic equations.
More information1 Discussion on multi-valued functions
Week 3 notes, Math 7651 1 Discussion on multi-valued functions Log function : Note that if z is written in its polar representation: z = r e iθ, where r = z and θ = arg z, then log z log r + i θ + 2inπ
More informationz = x + iy ; x, y R rectangular or Cartesian form z = re iθ ; r, θ R polar form. (1)
11 Complex numbers Read: Boas Ch. Represent an arb. complex number z C in one of two ways: z = x + iy ; x, y R rectangular or Cartesian form z = re iθ ; r, θ R polar form. (1) Here i is 1, engineers call
More informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES In section 11.9, we were able to find power series representations for a certain restricted class of functions. INFINITE SEQUENCES AND SERIES
More informationSection Taylor and Maclaurin Series
Section.0 Taylor and Maclaurin Series Ruipeng Shen Feb 5 Taylor and Maclaurin Series Main Goal: How to find a power series representation for a smooth function us assume that a smooth function has a power
More informationPURE MATHEMATICS AM 27
AM SYLLABUS (2020) PURE MATHEMATICS AM 27 SYLLABUS 1 Pure Mathematics AM 27 (Available in September ) Syllabus Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics
More information( ) a (graphical) transformation of y = f ( x )? x 0,2π. f ( 1 b) = a if and only if f ( a ) = b. f 1 1 f
Warm-Up: Solve sinx = 2 for x 0,2π 5 (a) graphically (approximate to three decimal places) y (b) algebraically BY HAND EXACTLY (do NOT approximate except to verify your solutions) x x 0,2π, xscl = π 6,y,,
More information18.04 Practice problems exam 2, Spring 2018 Solutions
8.04 Practice problems exam, Spring 08 Solutions Problem. Harmonic functions (a) Show u(x, y) = x 3 3xy + 3x 3y is harmonic and find a harmonic conjugate. It s easy to compute: u x = 3x 3y + 6x, u xx =
More informationMATH 106 HOMEWORK 4 SOLUTIONS. sin(2z) = 2 sin z cos z. (e zi + e zi ) 2. = 2 (ezi e zi )
MATH 16 HOMEWORK 4 SOLUTIONS 1 Show directly from the definition that sin(z) = ezi e zi i sin(z) = sin z cos z = (ezi e zi ) i (e zi + e zi ) = sin z cos z Write the following complex numbers in standard
More informationCHAPTER 1 Prerequisites for Calculus 2. CHAPTER 2 Limits and Continuity 58
CHAPTER 1 Prerequisites for Calculus 2 1.1 Lines 3 Increments Slope of a Line Parallel and Perpendicular Lines Equations of Lines Applications 1.2 Functions and Graphs 12 Functions Domains and Ranges Viewing
More informationUNIVERSITY OF CAMBRIDGE Faculty of Mathematics MATHEMATICS WORKBOOK
UNIVERSITY OF CAMBRIDGE Faculty of Mathematics MATHEMATICS WORKBOOK August, 07 Introduction The Mathematical Tripos is designed to be accessible to students who are familiar with the the core A-level syllabus
More informationADDITIONAL MATHEMATICS
ADDITIONAL MATHEMATICS GCE NORMAL ACADEMIC LEVEL (016) (Syllabus 4044) CONTENTS Page INTRODUCTION AIMS ASSESSMENT OBJECTIVES SCHEME OF ASSESSMENT 3 USE OF CALCULATORS 3 SUBJECT CONTENT 4 MATHEMATICAL FORMULAE
More informationTopic 2 Notes Jeremy Orloff
Topic 2 Notes Jeremy Orloff 2 Analytic functions 2.1 Introduction The main goal of this topic is to define and give some of the important properties of complex analytic functions. A function f(z) is analytic
More informationAn Appreciation of Euler's Formula
Rose-Hulman Undergraduate Mathematics Journal Volume 18 Issue 1 Article 17 An Appreciation of Euler's Formula Caleb Larson North Dakota State University Follow this and additional works at: http://scholar.rose-hulman.edu/rhumj
More informationCourse Notes for Signals and Systems. Krishna R Narayanan
Course Notes for Signals and Systems Krishna R Narayanan May 7, 018 Contents 1 Math Review 5 1.1 Trigonometric Identities............................. 5 1. Complex Numbers................................
More informationMathematical Review for Signal and Systems
Mathematical Review for Signal and Systems 1 Trigonometric Identities It will be useful to memorize sin θ, cos θ, tan θ values for θ = 0, π/3, π/4, π/ and π ±θ, π θ for the above values of θ. The following
More informationMath 20B Supplement August 2017 version Linked to Calculus: Early Transcendentals, by Jon Rogawski, Edition 3, c 2015
Math 0B Supplement August 07 version Linked to Calculus: Early Transcendentals, by Jon Rogawski, Edition 3, c 05 Written by Ed Bender, Bill Helton and John Eggers Contributions by Magdelana Musat and Al
More informationAP Calculus AB. Course Overview. Course Outline and Pacing Guide
AP Calculus AB Course Overview AP Calculus AB is designed to follow the topic outline in the AP Calculus Course Description provided by the College Board. The primary objective of this course is to provide
More informationLogarithm as Inverse of Exponentiation. 1-E1 Precalculus
Logarithm as Inverse of Exponentiation 1-E1 1-E2 What should one know? Inverse function: Which criteria must be satisfied by a function having an inverse function? Why do we need to know an inverse function?
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES Before starting this section, you might need to review the trigonometric functions. DIFFERENTIATION RULES In particular, it is important to remember that,
More information6.5 Trigonometric Equations
6. Trigonometric Equations In this section, we discuss conditional trigonometric equations, that is, equations involving trigonometric functions that are satisfied only by some values of the variable (or
More informationChapter 3a Topics in differentiation. Problems in differentiation. Problems in differentiation. LC Abueg: mathematical economics
Chapter 3a Topics in differentiation Lectures in Mathematical Economics L Cagandahan Abueg De La Salle University School of Economics Problems in differentiation Problems in differentiation Problem 1.
More informationADDITIONAL MATHEMATICS
ADDITIONAL MATHEMATICS GCE Ordinary Level (Syllabus 4018) CONTENTS Page NOTES 1 GCE ORDINARY LEVEL ADDITIONAL MATHEMATICS 4018 2 MATHEMATICAL NOTATION 7 4018 ADDITIONAL MATHEMATICS O LEVEL (2009) NOTES
More informationSolutions to Exercises 1.1
Section 1.1 Complex Numbers 1 Solutions to Exercises 1.1 1. We have So a 0 and b 1. 5. We have So a 3 and b 4. 9. We have i 0+ 1i. i +i because i i +i 1 {}}{ 4+4i + i 3+4i. 1 + i 3 7 i 1 3 3 + i 14 1 1
More information2. Determine the domain of the function. Verify your result with a graph. f(x) = 25 x 2
29 April PreCalculus Final Review 1. Find the slope and y-intercept (if possible) of the equation of the line. Sketch the line: y = 3x + 13 2. Determine the domain of the function. Verify your result with
More information