1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/ Applications 745 tion, and not passing through any of the zeros or poles off, by applying formula (9.14- ...

746 From'(9.15-5) we hav~ 4t '.'. - f(z)F(z) = f(z) + g(z) so that !le arg f(z) + b.e arg F(z) =!le arg[f(z) + g(z)] or !le arg ...

Singularities/Residues/ Applications and let f(z) = zn, g(z) = a1zn-l +···+an Choose a circle C: lzl = R such that R >. max(l ...

7 48 Chapter 9 for all n :> N and all z E 'Y· Let Yn(z) = fn(z) - f(z). Then combining (9.15-8) and (9.15-9), we have IYn(z)I ...

Singularities/Residues/Applications (a) z^4 + 5z + 1 = 0 (b) z^7 - 4z^4 + z^3 + 1 = 0 ( c) z^6 - 8z^2 + 3 = 0 (d) z^9 +z^5 +7z^4 ...

750 Chapter9 9.16 MAPPING PROPERTIES OF ANALYTIC FUNCTIONS. INVERSE FUNCTION THEOREMS Theorem 9.23 If the nonconstant function f ...

Singularities/Residues/Applications 751 has a zero other than a in iz - al :::; 6'. Thus if z 1 E N 6 1(a) is such that f ( z1) ...

752 Chapter9 Proof Since l'(z 0 ) -:/:- 0 the function F(z) = l(z) - wo has a simple zero at zo. As in Theorem 9.23, we can find ...

Singularities/Residues/Applications 753 valid at least for lw - w 01 < E. That is, the series (9.16-2) can be reversed in a s ...

754 Chapter^9 we assumed that z 0 = w 0 = 0). There we saw that the coefficients b 1 , b 2 , ... can be obtained successively in ...

Singularities/Residues/ Applications Hence, by Exercises 4.2(26), we get the special limit Consider the function nn n-+oo lira ...

756 Chapter9 z in N 6 (0) satisfying w = f(z). Each of these points lies in an analytic single-valued branch of z = 1-^1 (w). Pr ...

Singularities/Residues/Applications 757 where 0 ::=; Argwl/m < 27r/m How~ver, to w = 0 'there corresponds only one inverse im ...

758 Chapter 9 T. M. Macrobert, Functions of a Complex Variable, 4th, ed., Macmillan, London, 1958. A. I. Markushevich, Theory o ...

Index Abelian theorem, 562 Abel's limit theorem, 557-560 Abel summable, 561 Absolute value, 19 Accumulation point, 97 Algebra: f ...

760 Borel-Caratheodory theorem, 586-587 ~ Branch, 5, 2~6 ." . .---· algebraic, 287:2~6::.:.::c::::--.. I (^1) ogant.. 'h rmc,~. ...

Index Continuity of the first derivative, 385-387 Convergence: absolute, 186, 201 almost uniform, 522 conditional, 189 of integr ...

762 Factorial function, 630-632 Finite increments formula, 355 Finite intersection property, 109 Fixed points, 225 Fluid: irrota ...

Index Improper real integrals, 684-687 evaluation by residues, 687-732 Incomplete gamma function, 646 Index (see Winding number) ...

764 Metric spaces, 90-93 Minimum modulus theorem, 384, 577, 580 Mobius transformation (see Bilinear function) Module of periodic ...