Presentation on theme: "Chapter 2 Complex Numbers"— Presentation transcript:
1Chapter 2 Complex Numbers February 5 Complex numbers2.1 Introduction2.2 Real and imaginary parts of a complex number2.3 The complex plane2.4 Terminology and notationSolution to a quadratic equation:Example p46.A complex number has a real part and an imaginary part.
2Representation of a complex number on the complex plane: A complex number x + iy can be specified or represented by the following equivalent methods on the complex plane:The original rectangular form x + iy.A point with the coordinates (x, y).A vector that starts from the origin and ends at the point (x, y).The polar form reiq that satisfiesExample p48.Modulus (magnitude, absolute value) of a complex number:Angle (argument, phase) of a complex number:Example p50.
3Complex conjugateThe complex conjugate pair z=x + iy and z*=x − iy are symmetric with respect to the x-axis on the complex plane.Problems 4.3,9,18.
4Read: Chapter 2: 1-4Homework:2.4.1,3,5,15,18.Due: February 14
5February 7, 10 Complex algebra 2.1 Complex algebra A. Simplifying to x + iy formExamples p51.1-4; Problems 5.3,7.B. Complex conjugate of a complex expressionThe complex conjugate of any expression is just to change all i’s into –i.Example p53.1
6C. Finding the absolute value of z Example p53.2; Problems 5.26,28.D. Complex equationExample p54; Problems 5.36,43.E. GraphsComplex equations or inequalities have geometrical meanings.Example p55.1-4; Problems 5.52,53,59.
7F. Physical applications The position of a moving particle is represented by a vector. This vector also represents a complex number. Addition and subtraction of complex numbers is analogous to the addition and subtraction of vectors. Therefore the position, speed and acceleration of a particle can be well represented by complex numbers.However, because the multiplication of complex numbers is not in analogy with the multiplication of the vectors, physical principles involving multiplication of vectors can not be represented by complex algebra. Example: W=F·s.Example p56.
8Read: Chapter 2: 5Homework:2.5.2,7,23,26,33,36,47,59,68.Due: February 21
9February 14 Complex infinite series 2.6 Complex infinite series Convergence of a complex infinite series:Theorem: An absolutely convergent complex series converges.
10Theorem: Ration test for a complex series an: Example p57.1-2; Problems 6.2,6,7.
112.7 Complex power series; disk of convergence Disk of convergence: An area on which the series is convergent.Radius of convergence: The radius of the disk of convergence.Example p58.7.2a-c; Problems 7.5,8,15.Disk of convergence for the quotient of two power series:Example p59.
12Read: Chapter 2: 6-7Homework:2.6.3,5,6,13;2.7.8,11,15.Due: February 21
13February 17 Elementary functions 2.8 Elementary functions of complex numbersElementary functions: powers and roots; trigonometric and inverse trigonometric functions; logarithmic and exponential functions; and the combination of them.Functions of a complex variable can be defined using their corresponding infinite series.2.9 Euler’s formulaExamples p.62.
14Power of a complex number: Roots of a complex number: Multiplication and division of complex numbers using Euler’s formula:Example p.63; Problems 9.13,22,38.2.10 Powers and roots of complex numbersPower of a complex number:Example p.64.1.Roots of a complex number:Fundamental theorem of algebra:
15Notes about roots of a complex number: There are altogether n values forThe first root isAll roots are evenly distributed on the circle with a radius of Each root has a incremental phase change ofExample p ; Problem
16Read: Chapter 2: 8-10Homework:2.9.7,13,23,38;2.10.2,18,21,27.Due: February 28
17February 19 Exponential and trigonometric functions 2.11 The exponential and trigonometric functionsNotes on trigonometric functions:sinz and cosz are generally complex numbers. They can be more than 1 even if they are real.The trigonometric identities (such as ) and the derivative rules (such as ) still hold.Examples p ; Problems 11.6,9.
19Read: Chapter 2: 11-12Homework:2.11.6,8,10,11;2.12.1,3,11,32,36.Due: February 28
20February 21 Complex roots and powers 2.13 Logarithms Notes on logarithms:We use Lnr to represent the ordinary real logarithm of r.Because a complex number can have an infinite number of phases, it will have an infinite number of logarithms, differing by multiples of 2pi.The logarithm with the imaginary part 0q <2p is called the principle value.Examples p ; Problems 14.3,6,7.
212.14 Complex roots and powers Notes on roots and powers:There are many amplitudes as well as many phases for ab.For the amplitude in most cases only the principle value (0q a<2p and n=0) is needed.by=0, bx=m or 1/m gives us the real powers and real roots of a complex number.Examples p ; Problems 14.8,12,14.
222.15 Inverse trigonometric and hyperbolic functions Example p.75. 1; Problems 15.3.
23Read: Chapter 2: 13-15Homework:2.14.3,4,8,14,17;2.15.1,3,15.Due: February 28