1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 645 where ai = 1 +a - c and c1 = 2 - c. Hence ( = (a1,c 1 ;z) = (l + a - c, 2 - c; z) a ...

646 Chapter^8 From Theorem 8.51 (3) and the four relations above, the remaining 10 contiguous functions relations can be obtaine ...

Sequences, Series, and Special Functions 647 Bibliography L. V. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, New York, 1979 ...

648 Chapter^8 J. L. W. Jensen, Recherches sur la theorie des equations, Acta Math., 36 (1913), 181-185. J. L. W. Jensen, Surles ...

Sequences, Series, and Special Functions 649 A. Tauber; Ein Satz aus der Theorie der unenlichen Reihen, Monatsh. Math. Phys., 8 ...

9 Singularities. The Calculus of Residues. Appli.catio1:1s 9.1 REGVLAR AND SINGULAR POINTS Definition 9.1 A function f is said t ...

Singularities/Residues/ Applications 651 Since g(z) is supposed to be regular at z = O, we must have g(z) = bo +biz+ b2z^2 + · · ...

652 Chapter9 However, g'(z) = -1/(c+dz)^2 , so that g'(O) = -l/c^2. Definition 9.3 A function f that is regular at oo is said to ...

Singularities/Residues/ Applications 653 Definition 9.5 Let f: D --+ W, and suppose that a E D, the point a being either finite ...

654 Chapter9 9.2 ISOLATED SINGULARITIES Definitions 9.8 If z = a i-oo is an isolated singularity of f, then in any circular ring ...

Singularities/Residues/ Applications 655 lzl > R, has the form +oo f( z ) = L B kZ k = ···+ B-n - +"·+-+ B-1 B o+B1z+···+Bnz ...

656 Chapter^9 Conversely, if f(z) can be expressed in the form (9.2-4) where h(z) is analytic at z = a, and h( a) -f:. 0, then t ...

Singularities/Residues/Applications 657 Theorem 9.3 ·Let z = a be an isolated singularity of f. Then a is a pole of f iff lira Z ...

658 Chapter^9 provided that 0 < lz -al< min(8i,82). Conversely, suppose that limz--+a IJ(z)I = oo. Then there exists a 8 ...

Singularities/Residues/ Applications 659 singularity a, the values f(z) become arbitrarily close to any given complex number b ( ...

660 Chapter^9 for some points z such that z E NHO) n Dg, where Dg = {z: l/z En,}. Also, lg(z)I >I< for some points z such ...

Singularities/Residues/ Applications v y Fig. 9.3 and there is also a point z 2 in N^1 (a) such· that lf(z2)I > k 661 u Let C ...

662 Chapter^9 The function g(z) = f(l/z) = csc'l!"z has poles at z = 0, ±1, ±2, .... It follows that z == oo is a cluster poin ...

Singularities/Residues/ Applications 9.6 CHARACTERIZATION OF SOME SIMPLE FUNCTIONS BY THE NATURE OF THEIR SINGULARITIES 663 Clas ...

664 Chapter^9 has a Taylor series representation f(z) = ao + aiz + · · · + anzn + · · · valid for every z EC. If f has a pole of ...