1550251515-Classical_Complex_Analysis__Gonzalez_

3 Functions. Limits and Continuity. Arcs and Curves 3.1 Complex Functions Mappings and, in particular, functions have been defin ...

126 Chapter^3 At times we shall also consider functions from en (n ~ 2) into C. These are called complex functions of several co ...

Functions. Limits and Co11tinuity. Arcs and Curves 127 If we let z = rei^8 and w = f(z) = u +iv, the components u and v become f ...

128 Chapter^3 To sidestep such inconvenience, a two-dimensional surface (usually, the Gaussian plane or the Riemann sphere) is u ...

Functions. Limits and Continuity. Arcs and Curves 129 v y x 4 u Fig. 3.2 In recent years several techniques of computer graphin ...

130 Chapter 3 M. Murrill [12] has introduced a so-called complex hyperanalytic geometry of four dimensions with a frame of refe ...

Functions. Limits and Continuity. Arcs and Curves y v 1.:-- - - ...-w2 _....-f* ,,,,,.,:: .::: - - --f - - - - - - z~-=--.:---- ...

132 Chapter 3 Example If f(z) = z^2 + i, D = {z: Rez > O}, then 1 F(z) = z2 + i G(z)= 1 z^2 +i z2 M(z)= l+iz2 { on D- 1-i} ./ ...

Functions. Limits and Continuity. Arcs and Curves 133 Note in the definition above that the domain D can be any infinite set of ...

134 v. LI Fig. 3.6 Alternatively, if for every e > 0 there is a N' ( oo) such that zEN'(oo)nD (Fig. 3.6). We write Z->00 l ...

Functions. Limits and Continuity. Arcs and Curves 135 Example If f(z) = l/z^2 ,D = <C-{O}, then limz-+oof(z) = 0. In fact, gi ...

136 Chapter^3 or (6) By properties 2 and 5 we have lim [f(z) + J(z)] = L + L, z-+a lim[2Ref(z)] = 2ReL, z-+a which gives lim Ref ...

Functions. Limits and Continuity. Arcs and Curves that ArgL-a < Argf(z) < ArgL+a or I Argf(z) -ArgLI <a whenever z E N6 ...

138 Chapter^3 limz_,a(f /g)(z) = oo 4. limz-->a (g / !)( Z) = 0 limz-->af(z) = 00 limz->a IJ(z)I = 00 Proofs (1) By ...

Functions. Limits and Continuity. Arcs and Curves 139 Notation We write lim f(z) = oo z-+oo or f(z) -too as z -too Example For f ...

140 Chapter3 3. The infinitesimal J(z) = z^3 + 3z^4 is of higher order than g(z) = z^2 + z (as z ---t 0) since limz_,.O f(z)/g(z ...

Functions. Limits and Continuity. Arcs and Curves 141 Examples f(z) = o {z^2 } as z ~ 0 means that f(z) is an infinitesimal of ...

142 Chapter^3 6. If J(z) and g(z) are infinitesimals, f(z)-:/= g(z), and if J(z),...., g(z), show that J(z) -g(z) is of an order ...

Fm1ctions. Limits and Continuity. Arcs and Curves 143 ----.. a / ' 0 / z I / ' \ I a I I \ D I \ I ' .... __ / / (a) (b) (c) Fig ...

144 Chapter^3 Example The function f(z) = l/z defined from (<C,d) - {O} into (<C,d) is not continuous at z = 0, since f(z) ...