complex variables and applications Chapter

1、Chapter 1 Complex number1: Sums and ProductsComplex number can be defined as ordered pairs of real numbers that are to be interpreted as points in the complex plane ,with rectangular coordinate and . It is customary to write a complex number as . Ie. ,where are known as the real and imaginary parts

2、of ,and we denote by , by . Two complex numbers and are equal whenever ,and Two complex numbers means and correspond to the same point in the complex plane ,or z plane. The sum and product are defined as follows:,.2: Basic algebraic propertiesCommutative laws: Associative laws: Distributive law : Th

3、ere is associated with each complex number an additive inverse ,. And additive inverses are used to defined subtraction :.For any nonzero complex number ,there is an unique complex number ,such that we call the multiplicative inverse of ,and as a consequence of an easy computation.3: Further propert

4、iesDivision by a nonzero complex number is defined as follows, and is called a multiplying numerator ,and is called a denominator .There are many identities as follows: ;Example.Computations such as the following are now justified: Finally,we note that the binomial formula involving real numbers rem

5、ains valid with complex numbers ,ie. 4: ModuliIt is natural to associated any nonzero complex number with a vector ,from the origin to the point that represents .The following diagram interprets the sum and the difference between and .The modulus of is defined as a nonnegative real number ,denoted b

6、y .Geometrically, the number is the distance between the origine and the point ,and is the distance between points and,ie. . And we call the set of points satisfying the circle with center and radius .In view of ,when ,There are some relations involving mouduli of complex numbers:1.2. 3. 4.And accor

7、ding to the diagram above:5. 5: Complex conjugatesThe complex conjugate of a complex number is defined as the complex number ,and denoted by ,and conjugates of complex numbers hold the following properties:1. ,()2.Proof: If ,then .In like manner,it is easy to show the others.Second week Wednesday 35

8、The new ladder classroom No. 603Teaching purpose and requirement: Grasp the plural exponential form, Learn to go on operation of product、 quotient and extraction in exponential form, know the concept of regions in the complex plane.Emphases: Roots of complex numbers; neighborhood of a given point Di

9、fficult point:the argument of a complex number. interior point of STeaching way: Office coachingExercises: P13 7,8,P21 2,11,P29 8,7 P32 8,10Content of courses :6: Exponential formLet and be the polar coordinates of the point that represents a nonzero complex number .Since can be written in polar for

10、m as ,In complex analysis,the real number is not allowed to be negative and is the length of the radius vector for ;that is .The real number has an infinite possible values, includes negative ones ,that differ by integral multiples of ,each value of is called an argument of ,and the set of all such

11、values is denoted by .The principal value of ,denoted by ,is the unique such that ,and .If ,the coordinate is undefined;and so it is always understood that whenever is discussed.We can write the expression as by means of Euler formula ,and so . ( is measured in radians)Example:The complex number ,wh

12、ich lies in the third quadrant,has principal argument .That is .so Here think about a question:Can we write The number has exponential form ,Note that the equation is a parametric representation of the circle ,centered at the origin with radius R.More generally,the circel ,whose center is and whose

13、radius is R,has the parametric representation yx0 7: Products and quotients in exponential formSimple trigonometry tells us that has the familiar additive property of the exponential function in calculus:Thus ,if then (1) Moreover Expression above yields an important identity involving arguments:,es

14、pecially,Example1:In order to find the principal argument Arg z when Observe that Since One value of arg z is ;and,because is between and ,we find that .Example 2:In order to put in rectangular form,one need only write 8: Roots of complex numbersTwo nonzero complex numbers are equal if and only if ，

15、where is some integer,.Now let consider the equation with respect to z:Where .If is a root of ,then we have ,and thus we can obtain that so where this radical denotes the unique positive nth root of ,and,(*)Consequently,the complex numbersare the nth root of ,and we write as .We are able to see imme

16、diately from this exponential form of the roots that they all lie on the circle about the orgin and are equally spaced every radians,starting with argument .Evdently,then,all of the distinct roots obtained.9: ExamplesExample 1: In order to determine the nth roots of unity,we write and find that . If

17、 we write ,all the distinct roots of unity one are Here we consider a question: if c is any particular nth root of a nonzero complex number ,can the set of nth roots be put in the form ?Example 2. Example 3. By some steps computation,we get the two roots of are ).Can you do it? 10: Regions in the co

18、mplex planeAn neighborhood of a given point is the set .It consists of all points lying inside but not on the circle centered at and with a specified positive radiu .A deleted neighborhood of is the set of .It consists of all points lying inside but not on the circle centered at with radius except f

19、or the point itself.A point is said to be an interior point of S whenever there exists some neighborhood of that is contained in the set S.A point is said to be an exterior point of S whenever there exists some neighborhood of containing no points of S.A point is said to be an boundary point of S whenever for any neighborhood of there are both points of S and those of .All boundary points of