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Chap 1. Complex Numbers. 1. Sums and Products. Complex numbers can be defined as ordered pairs (x,y) of real numbers that are to be interpreted as points in the complex plane, with rectangular coordinates x and y, just as real numbers x are thought of as points on the real line. • (x,0) real number • (0, y) pure imaginary number • It is customary to denote a complex number (x,y) by z , so that z x, y x: real part of z Re z = x y: imaginary part of z Im z = y 1 • Sum and product of z1 x1 , y1 , z2 x2 , y2 are defined as z1 z2 x1 , y1 x2 , y2 x1 x2 , y1 y2 z1 z2 x1 , y1 x2 , y2 x1 x2 y1 y2 , y1 x2 x1 y2 z x, y x, 0 0, y 0,1 y, 0 0, y z x, 0 0,1 y, 0 let i 0,1 but Then z x iy i 2 (0,1)(0,1) 1 2 2. Algebraic Properties z1 z2 z2 z1 commutative law z1 z2 z2 z1 z1 z2 z3 z1 z2 z3 z1 z2 z3 z1 z2 z3 z z1 z2 zz1 zz2 associative law distributive law z+0=z 0=(0,0) additive identity z 1=z 1=(1,0) multiplicative identity For each z, there is a -z such that z+(-z)=0 additive Inverse 3 For any nonzero z=(x,y), multiplicative Inverse There is a z 1 such that zz 1 1 less obvious than additive inverse x, y u, v 1, 0 xu yv 1 x u x2 y 2 yu xv 0 v y x2 y 2 x y z 1 2 , , z0 2 2 2 x y x y • Division by a non-zero complex number z1 1 z1 z2 z2 z2 0 if z1 x1 , y1 , z2 x2 , y2 xx y y xx x y z1 z1 z21 1 22 1 2 2 , 1 22 1 2 2 z2 x2 y2 x2 y2 x1 x2 y1 y2 x1 x2 x1 y2 2 i 2 2 2 x2 y2 x2 y2 z1 x1 iy1 x2 iy2 z2 x2 iy2 x2 iy2 得到相同結果 4 Other Identities z1 z2 z1 z2 1 1 1 z1 z2 z1 z2 z3 z4 z3 z4 z1 0, z2 0 z3 0, z4 0 Example. 1 1 1 2 3i 1 i 2 3i 1 i 1 5i 5i 5 i 5 i 5 i 26 5 i 5 1 i 26 26 26 26 5 3. Moduli and conjugates It is natural to associate any nonzero complex number z=x+iy with the directed line segment, or vector, from the origin to the point (x,y) that represent z in complex plane. In fact, we often refer to z as the point z or the vector z. y z1 z2 x1 x2 , y1 y2 (x,y) (-2,1) -2+i x+iy z2 Z1 Z1+Z2 Z2 z1 0 x -Z2 Z1-Z2 -Z2 z1 z2 z1 z2 Although the product of two complex number z1 and z 2 is itself a complex number represented by a vector, that vector lies in the same plane as the vectors for z1 and z 2 . This product is neither the scalar nor the vector product used in vector analysis. 6 •The modulus, or absolute value, of a complex number z=x+iy is defined as z x2 y 2 length of the vector z. distance between point z and 0 the distance between two points z1 x1 iy1 z2 x2 iy2 is z z 1 2 z z0 R • x1 x2 y1 y2 2 a circle z Re z Im z 2 Z0 R 2 2 2 Re z Re z z Im z Im z z z x+iy • complex conjugate of z =x+iy is z x iy z x-iy 7 • If z1 x1 iy1 , z2 x2 iy2 z1 z2 x1 x2 i y1 y2 x1 iy1 x2 iy2 z1 z2 z1 z2 z1 z2 z1 z2 z1 z2 z1 z1 , z2 0 z2 z2 z z 2 Re z Re z zz 2 zz z z z 2i Im z Im z zz 2i 2 z1 z2 z1 z2 8 4. Triangle Inequality z1 z2 z1 z2 Equality holds when z1, z2 ,0 are colinear. z1 z2 z2 z1 geometrically 9 algebraically, z1 z2 z1 z2 z1 z2 z1 z2 z1 z2 2 z1 z1 z1 z2 z1 z2 z2 z2 z1 2 Re z1 z2 z2 2 z1 2 z1 z2 z2 2 2 z1 2 z1 z2 z2 2 z1 z2 2 2 2 z1 z2 z1 z2 Now z1 z1 z2 z2 z1 z2 z2 z1 z2 z2 z1 z2 z1 z2 similarly z1 z2 z2 z1 z1 z2 z1 z2 10 When z2 is replaced by z2, z1 z2 z1 z2 z1 z2 z1 z2 Example: z on unit circle z 1 or z 0 1 z3 2 z3 2 z 2 3 3 2 z3 2 z 2 1 3 The triangle inequality can be generalized by mathematical induction to sums … z1 z2 ... zn z1 z2 ... zn n 2,3,... 11 5. Polar coordinates and Euler’s Formula Let r, and be polar coordinates of the point (x,y) that corresponds to a non-zero complex number z=x+iy. since x r cos , y r sin , z r cos i sin if z=0, the coordinate is undefined. r z the length of the radius vector for z. has an possible values. Each value of is called an argument of z. and the set of all such values is arg z The principal value of arg z, Arg z, is that unique arg z Arg z 2n using Euler’s formula then , s.t. n 0, 1, 2,... ei cos i sin z rei 12 • Two non-zero complex numbers z1 r1ei1 are equal iff z2 r2ei2 r1 r2 and 1 2 2n , z Rei z R n 0, 1, 2,... z z0 Rei z z0 R 0 2 6. Product and Quotients in Exponential From ei1 ei2 cos 1 i sin 1 cos 2 i sin 2 i cos 1 2 i sin 1 2 e If i 1 2 z1 r1ei1 , z2 r2ei2 z1 z2 r1r2 e 1 i 2 (1) 13 z1 r1 ei1 e i2 r1 i1 2 e i 2 i 2 z2 r2 e e r2 1 1 i z e z r z n r n ein 1 if e i r 1, n 0, 1, 2,... n ein cos i sin Ex. Find 3 i 64 7 3 i n n 1, 2,... cos n i sin n Moivre’s formula 7 7 i 7 2e 6 27 e 6 26 ei 2ei 6 i 3i 14 Z1Z2 argument of product arg z1 z2 arg z1 arg z2 Z2 (7) 1 2 Z1 1 If we know two of these, can find the third. A. If arg z1 1 arg z2 2 B. If 1 2 From Expression (1) is a value of arg z1 z2 arg z1 z2 1 2 2n arg z1 1 2n1 If we choose arg z2 2 2 n n1 (7) is satisfied. 15 C. Similarly for Z1Z2 arg z1 z2 1 2 2n Z2 1 2 Z1 arg z2 2 2n2 1 Then choose arg z1 1 2 n n2 Finally Ex: arg arg z arg z Z1 Z2 1 Arg 2 arg z1 i z2 1 z1 z2 i 16 7. Roots of Complex Numbers Suppose z is nth root of a nonzero number z0 . z n z0 or r n e jn r0 ei0 n 0 2k r n r0 and r n r0 k 0, 1, 2,... 0 2k n 0 2k k 0, 1, 2,... n n 2 k z n r0 exp i 0 k 0, 1, 2,... n n are the nth root of z0 These roots are on the circle z n r0 and are equally spaced every 2 n 17 All of the distinct roots are obtained when k = 0,1,2,…,n-1 Let Ck n r0 exp i 0 2k n and z0 1 n n k 0,1, 2,..., n 1 denote the set of nth roots of denote these distinct roots z0 z0is a positive real number r0 then r0 1n denotes the entire set of roots. (іі) if 0in (1) is the principal value of arg z0 (і) if 0 C0 n r0 exp i 0 n is referred to as the principal root. 18 Ex. nth roots of unity 1 1exp i 0 2k k 0, 1, 2,..., n 1 1 0 2k 1 n n 1 exp i n n 2k exp i k 0,1, 2,..., n 1 n n 2 : 1 n3 : 1 n=4 : 1 19 2 Let n exp i n 2 k k then n exp i n and 1 n 1, n , n ,..., n 1 Ex.2. Find 8i 2 1 n 1 3 8i 8 i 8exp i 2k 2 2k Ck 2 exp i 3 6 k 0,1, 2 C0 2 exp i 3 i 6 C1 2i C03 C2 3 i C03 k 0, 1, 2,... c2 c1 2 c0 2 20 • Regions in the complex Plane closeness of points to one another • -neighborhood or neighborhood z z0 of a given point z0 • Deleted neighborhood Z Z0 0 z z0 • Interior point z0 is said to be an interior point of a set S whenever there is some neighborhood of z0 that contains only points of S. A point Z0 S • Exterior point when there exists a neighborhood of z0 containing no points of S 21 • Boundary point all of whose neighborhoods contain points is S and points not in S Boundary = { all boundary points } Ex. z 1 is the boundary of z 1 and z 1 • A set is open if it contains none of its boundary points • A set is closed if it contains all of its boundary points. • The closure of a set S is the closed set consisting of all points in S together with the boundary of S z 1 is open z 1 is closed and closure of - z 1, z 1 0 z 1 neither open nor closed. 22 - The set of all complex number is both open and closed since it has no boundary points. • An open set S is connected if each pair of points joined by a polygonal line that lies entirely in S. z1 and z2 in it can be 1 z 2 open, connected Z1 Z2 1 2 • An open set that is connected is called a domain. (any neighborhood is a domain ) • A domain together with some, none, or all of its boundary points is a region. • A set S is bounded if every point of S lies inside some circle Z R; otherwise it is unbounded. 23 • A point z0 is said to be an accumulation point of a set S if each deleted neighborhood of z0 contains at least one point of S. - If a set S is closed, then it contain s each of its accumulation points. pf: If an accumulation point z0 were not in S, it would be a boundary point of S; (can not be exterior points) but this contradicts the fact that a closed set contains all of its boundary points. i z n 1, 2,... Ex: For the set n n the origin is the only accumulation point. 24