资源描述
Functions of Complex Variable and Integral Transforms Gai YunyingDepartment of MathematicsHarbin Institutes of TechnologyPreface There are two parts in this course.The first part is Functions of complex variable(the complex analysis).In this part,the theory of analytic functions of complex variable will be introduced.The complex analysis that is the subject of this course was developed in the nineteenth century,mainly by Augustion Cauchy(1789-1857),later his theory was made more rigorous and extended by such mathematicians as Peter Dirichlet(1805-1859),Karl Weierstrass(1815-1897),and Georg Friedrich Riemann(1826-1866).Complex analysis has become an indispensable and standard tool of the working mathematician,physicist,and engineer.Neglect of it can prove to be a severe handicap in most areas of research and application involving mathematical ideas and techniques.The first part includes Chapter 1-6.The second part is Integral Transforms:the Fourier Transform and the Laplace Transform.The second part includes Chapter 7-8.1Chapter 1 Complex Numbers and Functions of Complex Variable1.Complex numbers field,complex plane and sphere1.1 Introduction to complex numbers As early as the sixteenth century Ceronimo Cardano considered quadratic(and cubic)equations such as ,which is satisfied by no real number ,for example .Cardano noticed that if these“complex numbers”were treated as ordinary numbers with the added rule that ,they did indeed solve the equations.The important expression is now given the widely accepted designation .It is customary to denote a complex number:The real numbers and are known as the real and imaginary parts of ,respectively,and we write Two complex numbers are equal whenever they have the same real parts and the same imaginary parts,i.e.and .In what sense are these complex numbers an extension of the reals?We have already said that if is a real we also write to stand for a .In other words,we are this regarding the real numbers as those complex numbers ,where .If,in the expression the term .We call a pure imaginary number.Formally,the system of complex numbers is an example of a field.The addition and multiplication of complex numbers are the same as for real numbers.If1.2 Four fundamental operations The crucial rules for a field,stated here for reference only,are:Additively Rules:i.;ii.;iii.;iv.Multiplication Rules:i.;ii.;iii.;iv.for .Distributive Law:Theorem 1.The complex numbers form a field.If the usual ordering properties for reals are to hold,then such an ordering is impossible.1.3 Properties of complex numbers A complex number may be thought of geometrically as a(two-dimensional)vector and pictured as an arrow from the origin to the point in given by the complex number.Because the points correspond to real numbers,the horizontal or axis is called the real axis the vertical axis(the axis)is called the imaginary axis.Figure 1.1 Vector representation of complex numbers The length of the vector is defined as and suppose that the vector makes an angle with the positive direction of the real axis,where .Thus .Since and ,we thus have This way is writing the complex number is called the polar coordinate(triangle)representation.Figure 1.2 Polar coordinate representation of complex numbers The length of the vector is denotedand is called the norm,or modulus,or absolute value of .The angle is called the argument or amplitude of the complex numbers and is denoted .It is called the principal value of the argument.We have Polar representation of complex numbers simplifies the task of describing geometrically the product of two complex numbers.Let and .Then Theorem 3.and As a result of the preceding discussion,the second equality in Th3 should be written as .“”meaning that the left and right sides of the equation agree after addition of a multiple of to the right side.Theorem 4.(de Moivres Formula).If and is a positive integer,then .Theorem 5.Let be a given(nonzero)complex number with polar representation ,Thenthe th roots of are given by the complex numbers Example 1.Solve for .Solution:If ,then ,the complex conjugate of ,is defined by .Figure 1.3 Complex conjugationTheorem 6.i.iv.and hence is ,we have .ii.iii.forvii.vi.and v.if and only if is realTheorem 7.i.vii.vi.v.iv.that is,and .iii.and ;ii.If ,then Figure 1.4 Triangle inequality1.4 Riemann sphere For some purposes it is convenient to introduce a point “”in addition to the points .Figure 1.5 Complex sphere Formally we add a symbol“”to to obtain the extended complex plane and define operations with by the“rules”2.Complex numbers sets Functions of complex variable2.1Fundamental concepts(1)neighborhood of a point :(2)A deleted neighborhood of a point :(3)A point is said to be an interior point of .If there exists .(4)A set is open iff for each ,is an interior point of .2.2 Domain Curve An open set is connected if each pair of points and in it can be joined by a polygonal line,consisting of a finite number of line segments joined end to end,that lies entirely in .An open set that is connected is called a domain.A curve,if ,then is continuous and ifthen is called a simple curve.If and is called a smooth curve(a piecewise smooth curve).A domain is called the simply connected iff,for every simply closed curve in ,the inside of also lies in ,or else it is called the multiple connected domain.2.3 Mappings and continuity Let be a set.We recall that a mapping is merely an assignment of a specific point to each ,being the domain of .When the domain is a set in and when the range(the set of values assumes)consists of complex numbers,we speak of as a complex function of a complex variable.We can think of as a map ;therefore becomes a vector-valued function of two real variables.Thus and are merely the components of thought of as a vector function.Hence we may write uniquely ,where and are real-valued functions defined on .For ,we can let and define and .Def 1.Let be defined on a deleted neighborhood of .Themeans that for every ,there is a such that ,and imply that .We also define,for example,to mean that for any ,there is an such that implies that .Figure 1.6 is close to when is close to The limit as is taken for an arbitrary approaching but not along any particular direction.ii.The limit is unique.The following properties of limits hold:If and ,theni.iv.iii.if .Also,if is defined at the points and ,then Th1.Let then andProof:It is easy by using the following inequalities Def 2.Let be an open set and let be a given function.We say is continuous at iffand is continuous on is is continuous at each .From(i),(ii),and(iii)we can immediately deduce that if and are continuous on ,then so are the sum and the product ,and so is if for all .Also if is defined and continuous on the range of ,then the composition ,defined by ,is continuous by(iv).Theorem 2.Let is continuous at and are continuous at .EX1.If ,the limit does not exist.Figure 1.7 For,if it did exist,it could be found by letting the point approach the origin in any manner.But when is a nonzero point on the real axis(Fig 1.7),and when is a nonzero point on the imaginary axis,Thus,by letting approach the origin along the real axis,we would find that the desired limit is .As approach along the imaginary axis would,on the other hand,yield the limit .Since a limit is unique,we must conclude that does not exist.EX2.Find that (1);(2);(3)Solution:(1),;(2);(3).
展开阅读全文