1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 405 J, Horvath, A generalization of the Cauchy-Riemann equations, Contrib. Differential Equations Univ. Md., 1 ...

406 Chapter6 H. Looman, Uber the Gauchy-Riemannschen Differentialgleichungen, Nach- richten Gesellschaft Wissenschaft, Gottinge ...

Differentiation 407 H. Rademacher, Uber streckentrene und winkeltrene Abbildung, Math. Z., 4 (1919), 131-133. H. Rademacher, Be ...

408 Chapter6 A. Terracini, A first contribution to the geometry ofmonodriffic polynomials, Actas Acad. Nae. Ci. Lima, 8 (1945), ...

7 Integration 7.1 INTRODUCTION In this chapter we consider the definition and properties of the integrals of continuous complex ...

410 Chapter^7 1. Ref: f(t)dt = J:Ref(t)dt, Imf: f(t)dt = J:Imf(t)dt If f 1 and h are continuous on [a, b], then J:[f1(t) + f2(t ...

Integration 411 also holds for k complex, let k = k 1 + ik 2 (k 2 I= 0). Then we have k lb f(t) dt = (k1 + ik 2 ) [lb u(t) dt + ...

412 Chapter^7 B ----------- b g a ----- A 0 c d T Fig. 7.1 lb v(t)dt = 1d v[g(r)]g'(r)dr By adding to the first equation the sec ...

Integration 413 we have g(t) = lt u(r)dr, h(t) = lt v(r)dr so that g'(t) = u(t), h'(t) = v(t) and G'(t) = g'(t) + ih'(t) = u(t) ...

414 Chapter^7 complex function of the real variable t over the interval [a, ,B]. The integral defined in (7.3-1) is also called ...

" Integration y B (2+i) 0 A^3 x Fig. 7.2 7 .4 INTEGRAL OF A COMPLEX FUNCTION OF A COMPLEX VARIABLE ALONG A PIECEWISE DIFFERENTIA ...

416 Chapter^7 The following additional types of complex integrals are defined by the corresponding right-hand sides: J f(z)ds = ...

Integration 417 The sum Sp is called the variation of z(t) over the interval [a, ,B] corre- sponding to the partition P, sometim ...

418 Chapter 7 y ~ zo" z, (^0) Tk x to t, tk - 1 tk In Fig. 7.3 J f(z)dz = 1: f(z(t))dz(t) (7.6-2) n = lim L f(zk)(zk - Zk-1) IPl ...

Integration 419 where tk-1 < rf. < tk and tk-1 < rf < tk. Hence n n L f(z(rk))[z(tk)-z(tk-1)] = L f(z(rk))x'(rf.)(tk ...

420 Chapter 7 Definition 7. 7 With the assumptions and notations of Definition 7 .6, we also define (7.6-4) The limit on the rig ...

Integration 421 the integral on the left-hand side exists. With the notation of Definition 7.6, we have J[kd1(z) + kzf2(z)] dz " ...

422 Chapter^7 = J f(z)dz+ J f(z)dz /1 /2 If 71 and 72 are not in juxtaposition, then we define J /1 + /2 f(z) dz to be J n f(z)d ...

Integration :::; 1: Jf(z(t))JJz'(t)J dt = J Jf(z)JJdzJ '"'( If maxzE'"'f JJ(z)J :::; Mand L = L('Y) = J'"'f JdzJ, it follows th ...

424 Chapter^7 Proof ( a)Rectifiable arc. A partition P = { t 0 , t 1 , ••• , tn} of the interval [a, ,8] induces a subdivision { ...