By Mario Gonzalez

Textual content at the conception of capabilities of 1 advanced variable includes, with many embellishments, the topic of the classes and seminars provided by way of the writer over a interval of forty years, and will be thought of a resource from which a number of classes might be drawn. as well as the elemental themes within the cl

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**Example text**

6-1) defines a function f : C onto JR+. 3 extends to the complex number system the concept of the absolute value of a real number x. 3 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. = Vx'i Proofs {xx if x > 0 if x < 0 The following properties hold: 0 iff z lzl > 0 and lzl lzl 1-zl lzl 2 zz lzl lzl < lxl + IYI lxl < lzl and IYI < lzl lz1llz2I lz1z2I lz1 ±z2I < lz1I + lz2I lz1 ±z2 I > llz1I - lz2II lzi/z2l lz1l/lz2 l, z2 -:f. 3, and we omit them. 2, and 3 follow easily from 20 Chapter 1 Property 4 can be written as Vx2 + Y2 < lxl + IYI which is equivalent to x2 + y2 < x2 + 2lxl IYI + y2 , since both sides of the inequality are nonnegative real numbers.

For every x+ 8=x for all x E X there exists an element, denoted = 8. x E X and every a E F, the product ax -x, x E X. such that is uniquely defined E X. and ax a(f3x) =(af3)x. (a+f3)x=ax+f3x. 9. a(x+ y)=ax +ay. 10. µx=x, where µ is the 7. 8. unit of F. It is clear that the complex number system C satisfies all of conditions to 10, with F 1 the commutative field of the reals. + O'nXn=0 O'k E F, implies that O'k=0 (k= 1, 2, . . , n ) . X is of dimension n ( n > 1) if there are in X n linearly independent elements x1, ...

12. For the complex number z = 0 the argument is not defined, so that argz is not continuous in the whole plane either. From trigonometry we have, for z 'f:. 7-5) Chapter 1 30 Since the trigonometric functions cos (J and sin (J can be defined analytically by means of the power series expansions, cos 8 = 1 sin 8 =(} 82 2! 84 4! 3! 5! - + - - ()3 ()5 - + - - - . - · . · . 7-5). In particular, 80 = Argz is that solution of (I. 7-5) which lies in the interval [0,27r). That the formulas there is just one solution in each case is clear from the discussion of the real functions sin(} and cos(} in trigonometry, or in introductory courses on real variables.