1550251515-Classical_Complex_Analysis__Gonzalez_

Introduction 5 have h( x) = Yo for all x E X, the function h : X -+ Y is called a constant function in X. If to each singleton { ...

6 Introduction one y. ~onversely, given such a subset H of X x Y, the single-vafued function f : X ---? Y is then well defined ( ...

1 Complex Numbers 1.1 The Complex Number System Definitions 1.1 Let R = {a, b, c, ... } be the set of real numbers, and let R^2 ...

8 Chapter^1 z2 or z1 -/:-z2; also, z1 = z1 (reflexive property), z1 = z2 implies that z2 = z1 (symmetric property), and z1 = z2, ...

Complex Numbers 9 We note that (e, d) - (a, b) = (e, d) +(-a, -b) (1.1-4) so the difference between ( e, d) and (a, b) may be ob ...

10 Chapter 1 in (1.1-7), (c,d) is the dividend and (a,b) the divisor. The opera- tions involved in (1.1-3) g,nd (1.1-7) are call ...

Complex Numbers 11 1.2 REAL AND IMAGINARY COMPLEX NUMBERS. THE COMPLEX UNITS Complex numbers of the special form (a, 0) behave a ...

12 Chapter^1 we see that any complex number can be expressed as a linear combination, with real coefficients, of the complex uni ...

Complex Numbers 13 7. z 1 ± Zz = .Z1 ± Zz. ZiZz = z1z2. 9. (zi/z2) = zifzz. 10. F(zi, z2, ... , Zn) = F(z1, z2, ... , .Zn), wh ...

14 ·Chapter^1 Letting a+ bi= A, 2c = C, the foregoing equation can be written as Az+Az+C= o where A i= 0 is a complex constant a ...

Complex Numbers Prove the following cancellation laws. (a) If Z1 + Zz = Z1 + Z3, then Zz = Z3. (b) If Z1Z2 = Z1Z3 and Z1 -=!= ...

16 Chapter^1 We recall that by a linearly ordered field (briefly, an ordered field) is meant a field F that contains a nonempty ...

Complex Numbers 17 In that case we shall say that f introduces a two-order in S (the one- order being the linear order). An orde ...

18 Chapter^1 the reals. We recall that, in general, by a linear system (or vector space) over a commutative field F = {a, (3, .. ...

Complex Numbers 19 same as the vector space addition, and the relations 1( ab) = ( 1a )b = a( 1b) for all / E F and all a, b E A ...

20 Chapter^1 Property 4 can be written as J x^2 + Y^2 :S !xi + IYI which is equivalent to x^2 + y^2 :::; x^2 + 2JxllYI + y^2 , s ...

Complex Numbers 21 can be written as lz 212 (zi/z 2 ) 2:: 0, or simply as zif z 2 2:: 0. Hence, under the assumption z 2 -:/= 0, ...

22 Chapter^1 Theorem 1.4 If n complex numbers zi, z 2 , ... , Zn (n ~ 2) are given arbitrarily, then (1.6-9) and the equality si ...

Complex Numbers 23 Theorem 1.5 (Lagrange's Identity). If {z 1 , z 2 , ... , Zn} and {w1, w2, ... , wn} are two sets of n complex ...

24 Chapter^1 Corollary 1.1 (Cauchy-Schwarz Inequality). If {zi, ... ,zn} and {w 1 , ... , wn} are two sets of complex numbers (n ...