1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences and Series 205 The number R is called the radius of convergence of the power series, and the circle jzj = R is called ...

206 Chapter4 term anzn is a continuous function everywhere (in particular, at zo), the continuity of f(z) at z 0 is a consequenc ...

Sequences and Series 207 superior and L2 the limit inferior of lan+1l/lanl, as stated in the following theorem, due to S. Pinche ...

208 Chapter4 so that series converges absolutely if lzl < {IL;, and diverges if lzl > {IL;. It follows that the radius of ...

Sequences and Series 209 Consider the series 2 2 1 2 1 3 1 3m + 3 Z + 3 Z + 22. 3 z + ... + 2sm-13m z 1 z3m+l 1 z3m+2 + 2sm-l3 ...

210 Chapter4 for every n, i.e., the two power series have the same coefficients and, as a consequence, R 1 = R2. Proof Since f(z ...

Sequences and Series 00 (a) 'Lsinhn 1 oo (in)P ( c) L - 1 - (p an integer) n=l n. 00 (b) f)-lr1·3···(2n-1) n=l^2 · 4· · · (2n) o ...

212 oo I (e) "'"'2 ;_,J !::_Zn nn n=l (^00) ( I )2 (g) L (~~ )! Zn n=l ( i) ~ 22.42···(2n)2 Zn ~ 32 · 52 · · · (2n + 1)^2 00 0 ( ...

Sequences and Series 213 00 (b) The radius of convergence of L anbnzn is R;::: R1R2. 0 21. Show that :L:=l (1/n^2 )zn converges ...

5 Elementary Functions In this chapter we discuss some elementary functions of a complex variable, namely certain of the most si ...

Elementary Functions 215 y W=z+b (^0) x Fig. 5.1 transformed into straight lines, and circles into congruent circles. Angles bet ...

216 Chapter^5 We note that if lal = 1, so that a = eic>, the transformation reduces to a pure rotation (of amplitude a). In p ...

Elementary Functions y (^0) x Fig. 5.3 and arg(wa -w1)-arg(wa -w2) = arg(za - z1)-arg(z 3 - z 2 ) (see Fig 5.3) 217 The converse ...

218 Chapter5 5.4 The Reciprocal Function w = 1/ z This function is defined in C-{O}. Between moduli and arguments of the variabl ...

Elementary Functions 219 Circles through the origin are mapped into straight lines not containing the origin. By considering th ...

220 Chapter 5 5.5 The Bilinear Function w = (az + b)/(cz + d) This function is also known as the bilinear transformation, the li ...

Elementary Functions 221 Theorem 5.4 The bilinear transformation has the following properties: Circles are transformed into cir ...

222 Chapter 5 and then defining w(oo) = W(O) = a/c (5.5-3) Thus if c f 0, the image of oo is the finite point a/ c, and if c = 0 ...

Elementary Functions 223 we see that the simple ratio is a bilinear function of the third point. Thus, in accordance with the de ...

224 Chapter 5 If z 2 = oo, then T is given by Z -Z3 Tz = ---= (z,oo,z3,z4) = (z3,z4,z) Z-Z4 Similarly, if z 3 = oo, then and if ...