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Functions of Complex Variable and Integral Transforms Department of Mathematics Harbin Institutes of Technology Gai Yunying Preface 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. 1 Chapter 1 Complex Numbers and Functions of Complex Variable 1. Complex numbers field, complex plane and sphere 1.1 Introduction to complex numbers As early as the sixteenth century Ceronimo Cardano considered quadratic (and cubic) equations such as x 2 2 x 2 0, which is satisfied by no real number x , for example 1 1 . Cardano noticed that if these “complex numbers” were treated as ordinary numbers with the added rule that 1 1 1 , they did indeed solve the equations. 1 is now given the widely accepted designation i 1 . The important expression It is customary to denote a complex number: z x iy The real numbers x and y are known as the real and imaginary parts of z , respectively, and we write Re z x, Im z y Two complex numbers are equal whenever they have the same real parts and the same imaginary parts, i.e. z1 z2 x1 x2 and y1 y2 . In what sense are these complex numbers an extension of the reals? We have already said that if a is a real we also write to stand for a a 0i . In other words, we are this regarding the real numbers as those complex numbers a bi , where b 0. If, in the expression a bi the term a 0 . We call a pure imaginary number. 1.2 Four fundamental operations The addition and multiplication of complex numbers are the same as for real numbers. ( x1 iy1 ) ( x2 iy2 ) ( x1 x1 ) i( y1 y2 ) ( x1 iy1 )( x2 iy2 ) ( x1x2 y1 y2 ) i( x1 y2 x2 y1 ) If x2 iy2 0, x1 iy1 ( x1 iy1 )( x2 iy2 ) ( x1 x2 y1 y2 ) i( x2 y1 x1 y2 ) x2 iy2 ( x1 iy2 )( x2 iy2 ) x22 y22 Formally, the system of complex numbers is an example of a field. The crucial rules for a field, stated here for reference only, are: Additively Rules: i. z w w z ; ii. z ( w s ) ( z w) s ; iii. z 0 z ; iv. z ( z ) 0. Multiplication Rules: i. zw wz ; ii. ( zw) s z ( ws ) ; iii. 1 z z ; iv. z ( z 1 ) 1 for z 0 . Distributive Law: z ( w s ) zw zs Theorem 1. The complex numbers field. form a 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 2 given by the complex number. Because the points ( x,0) 2 correspond to real numbers, the horizontal or x axis is called the real axis the vertical axis (the y axis) is called the imaginary axis. Imaginary axis y axis z a ib 0 Real axis x axis Figure 1.1 Vector representation of complex numbers The length of the vector (a, b) a ib is defined as r a 2 b 2 and suppose that the vector makes an angle with the positive direction of the real axis, where . Thus tan b / a . Since a r cos and y b r sin , we thus have a ib a bi r cos (r sin )i r (cos isin ) This way is writing the complex number is called the polar coordinate( triangle ) representation. r 0 r sin x r cos Figure 1.2 Polar coordinate representation of complex numbers The length of the vector z a ib is denoted | z | and is called the norm, or modulus, or absolute value of z . The angle is called the argument or amplitude of the complex numbers and is denoted arg z . arg z Argz arg z 2k k 0, 1, 2, It is called the principal value of the argument. We have y z I or IV arctan x y arg z arctan z II x y z III arctan x Polar representation of complex numbers simplifies the task of describing geometrically the product of two complex numbers. Let z1 r1 (cos1 isin 1 ) and z2 r2 (cos 2 isin 2 ) . Then z1 z2 r1r2 ([cos1 cos2 sin 1 sin 2 ] i[cos1 sin 2 cos 2 sin 1 ]) r1r2 [cos(1 2 ) isin(1 2 )] Theorem 3. | z1 z2 || z1 | | z2 | and arg( z1z2 ) arg z1 arg z2 As a result of the preceding discussion, the second equality in Th3 should be written as arg z1z2 arg z1 arg z2 (mod 2 ) . “ mod 2 ” meaning that the left and right sides of the equation agree after addition of a multiple of 2 to the right side. Theorem 4. (de Moivre’s Formula). If z r (cos isin ) and n is a positive integer, then z n r n (cos n isin n ) . Theorem 5. Let w be a given (nonzero) complex number with polar representation w r (cos isin ), Then the n th roots of w are given by the n complex numbers 2k zk r cos n n n 2k isin n n , k 0,1, , n 1. Example 1. Solve z 3 1 for z . Solution: 2k 2k z 1 | 1| cos isin 3 3 3 3 1 3 i cos 3 isin 3 2 2 cos isin 1 5 5 1 isin cos 3 i 3 3 2 2 k 0 k 1 k 2 If z a ib , then z , the complex conjugate of z , is defined by z a ib . Theorem 6. i. z z z z ii. zz z z y z a ib 0 x z a ib Figure 1.3 Complex conjugation iii. z / z z / z for z 0 iv. zz | z |2 and hence is z 0 , we have v. z z if and only if z is real vi. Re z z z / 2 vii. z z . and Im z z z / 2i z 1 z / | z |2 . Theorem 7. i. | zz || z | | z | ii. If z 0 , then | z / z || z | / | z | iii. | z | Re z | z | and | z | Im z | z |; that is, | Re z || z | and | Im z || z | . y z2 iv. | z || z | z1 v. | z z || z | | z | z1 z2 x 0 vi. | z z | | z | | z | vii. | z1w1 Figure 1.4 Triangle inequality zn wn | | z1 | | zn | | w1 | | w0 | 2 2 2 2 1.4 Riemann sphere For some purposes it is convenient to introduce a point “ ” in addition to the points z . N P Q 0 P Figure 1.5 x y Q Complex sphere Formally we add a symbol “ ” to extended complex plane by the “rules” z z to obtain the and define operations with 2. Complex numbers sets Functions of complex variable 2.1Fundamental concepts (1) A neighborhood of a point z0: N {z | z z0 | }. (2) A deleted neighborhood of a point z0: {z 0 | z z0 | }. (3) A point If there exists z0 is said to be an interior point of E. N ( z0 ) E . (4) A set E is open iff for each z0 E , z0 is an interior point of E . 2.2 Domain Curve An open set S is connected if each pair of points z1 and z2 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 S . An open set that is connected is called a domain. if : z z (t ) x(t ) iy(t ) ( t ) x(t ), y(t ) C[ , ] , then is continuous and ift1 t2 z (t1 ) z (t0 ) then is called a simple curve. A curve, If z(t ) x(t ) iy(t ) 0 and x(t ), y(t ) C[ , ]. is called a smooth curve (a piecewise smooth curve). A domain D is called the simply connected iff, for every simply closed curve in ,Dthe inside of alsolies in , or else D it is called the multiple connected domain. 2.3 Mappings and continuity Let G be a set. We recall that a mapping f : G is merely an assignment of a specific point to each z G , G being the domain of f ( z) and when the range f . When the domain is a set in (the set of values f assumes) consists of complex numbers, we speak of f as a complex function of a complex variable. f :G 2 2 ; therefore f becomes a vector-valued function of two real variables. We can think of f as a map For f : G , we can let z x iy ( x, y ) and define u ( x, y ) Re f ( z ) and v( x, y ) Im f ( z ) . Thus u and v are merely the components of f thought of as a vector function. Hence we may write uniquely f ( x iy ) u ( x, y ) iv( x, y ) , where u and v are real- valued functions defined on G . Def 1. Let f be defined on a deleted neighborhood of z0 . The limit f ( z ) A z z0 means that for every 0 , there is a 0 such that z D( z0 , r ), 0 | z z0 | , z z0 , and | z z0 | imply that | f ( z ) A | . We also define, for example, lim f ( z ) A to z mean that for any 0 , there is an R such that | z | R implies that | f ( z ) A | . The limit as z z0 is taken for an arbitrary z approaching z0 but not along any particular direction. v y z0 ( x0 , y0 ) 0 Figure 1.6 A w f ( z) x 0 f ( z ) is close to A when z is close to z0 u The limit A is unique. The following properties of limits hold: If lim f ( z ) A and z z0 lim g ( z ) B , then z z0 i. lim[ f ( z ) g ( z )] A B z z0 ii. lim[ f ( z ) g ( z )] AB z z0 iii. lim[ f ( z ) / g ( z )] A / B if B 0 . z z0 Also, if h is defined at the points lim h( w) c , then wa iv. lim h( f ( z )) c z z0 f ( z ) and Th1. Let f ( z ) u ( x, y) iv( x, y), A a ib then lim f ( z ) A lim u ( x, y ) a and z z0 x x0 y y0 lim v( x, y ) b. x x 0 y y0 Proof: It is easy by using the following inequalities | u a |,| v b | (u a) 2 (v b) | u a | | v b | Def 2. Let A be an open set and let f : A be a given function. We say f is continuous at z0 A iff lim f ( z ) f ( z0 ) z z0 and f is continuous on A is f is continuous at each z0 A . From (i), (ii), and (iii) we can immediately deduce that if f and g are continuous on A , then so are the sum f g and the product fg , and so is f / g if g ( z0 ) 0 for all z0 A . Also if h is defined and continuous on the range of f , then the composition h f , defined by h f ( z ) h( f ( z )) , is continuous by (iv). Theorem 2. Let f ( z ) u ( x, y ) iv( x, y ) is continuous at z0 x0 iy0 u ( x, y ) and v( x, y ) are continuous at ( x0 , y0 ) . z EX1. If f ( z ) , the limit lim f ( z ) does not exist. z 0 z For, if it did exist, it y could be found by letting z (0, y ) the point z ( x, y ) approach the origin in any manner. But when z ( x,0) is a nonzero point on the real axis (Fig 1.7), f ( z ) x i0 1; x i0 0 z ( x,0) Figure 1.7 x and when z (0, y ) imaginary axis, is a nonzero point on the 0 iy f ( z) 1. 0 iy Thus, by letting z approach the origin along the real axis, we would find that the desired limit is 1 . As approach along the imaginary axis would, on the other hand, yield the limit 1 . Since a limit is unique, we must conclude that lim f ( z ) does not exist. z 0 iz 3 2z i EX2. Find that (1) lim ; (2) lim ; z 1 z 1 z z 1 iz 3 1 (3) lim . z i z i Solution: z 1 iz 3 (1) lim 0 , lim ; z 1 iz 3 z 1 z 1 2/ z i 2 iz (2) lim lim 2 ; z 0 1/ z 1 z 0 1 z iz 3 1 0. (3) lim z i z i