1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 425 Also, by Theorem 6.38, for a given 'f/ > 0 there exists a 6 > 0 such that for any partition P with IPI < ...

426 Chapter^7 = J f(g(())g'(() d( "'!' Theorem 7.4 Let t = h( T) be a continuous one-to-one mapping of the interval [a', ,8'] on ...

Integration 427 so that j f(z) dz= j f(z) dz 'l'l '/'2 Conversely, suppose that 1 is a closed rectifiable curve. Then 1 and -1 h ...

428 Chapter^7 where r+D.z .6.s = (-y) Jz Id(! Since J(((r)) is a continuous function of r, it follows that M(.6.t) ---t 0 as .6. ...

Integration 429 Putting back this value in (7.8-8) and letting z = z 1 , we obtain ('Y) 1: 1 f(()d(=1/;(z1) -1/J(zo) (7.8-9) Not ...

430 Chapter 7 that F'(z) = f(z) for every z E R. More generally, the property holds if 'Y is a rectifiable arc in R. Proof. (a) ...

Integration 431 and it follows that 1 z+h F(z+h)-F(z)=(-y') z J(()d( 1 z+h =hf(z)+(-y') z [f(()-J(z)]d( and F(z + h) - F(z) = f( ...

432 Chapter^7 From (7.8-10) we have F'((k-1)((k - (k-1) = F((k) - F((k-1) -11k((k - (k-1) so that n n n l:F'((k-1)((k - (k-1) = ...

Integration 433 F(z) = (z - a)m+ljm + 1 is still analytic in the region C - {a}, and F'(z) = (z-a)m. However, form= -1, (7.8-13) ...

434 Chapter^7 If Po - {a = t 0 , t 1 , ••• , in = ,8} is a partition of the interval [a, ,8] and tk-1 $ 'Tk $ tk, !Pol = max(tk ...

Integration · [x(tk) - x(tk-1) + i(y(tk) - y(tk-1))] = if3 [u(x(t), y(t)) dx(t) - v(x(t), y(t)) dy(t)] i 1: [v(x(t), y(t)) dx(t ...

436 Chapter 7 Fig. 7.5 a proof of the theorem can be given that does not make use of the con- tinuity of the derivative. This so ...

Integration 437 Since f is analytic in D, we have Ux = Vy and Uy = -vx. Hence each of the double integrals on the right of (7.9- ...

438 Chapter 7 Examples: J 0 ez sin z dz = 0 for any simple closed curve C with graph in the complex plane, since f(z) = ez sinz ...

Integration 2+i (a) 1 (t^2 + 1 + it)dt (b) 1i"'(et + i cost) dt Evaluate. (a) J lzl dz, where 1: z = t +it, 0 :::; t :::; 1 'Y ...

440 Chapter 7 (b) J Log z z dz 1f ( R 1/ ) h it 1/ 2 :::; R Log +^2 7f , w ere 7: z = Re , -^2 7r :::; t :::; 27fe-aR < R , w ...

Integration 441 12. In Green's formula (7.9-2) let P = -GFy and Q = GFx, where Fis of class c<^2 >(D) and G is of class C( ...

442 Chapter^7 y d ---·-- R R4 r l R3 A - B R1 i! R2 G c - - - 0 a b x Fig. 7.6. where fJR is the boundary of R described ...

Integration 443 is satisfied, although it may be satisfied by more than one rectangle. If there is just one, call it Ri, and if ...

444 Chapter^7 whenever lz - z 01 < 8, and f(z) = f(zo) - zof'(zo) + zf'(zo) + (z - zo)11(z) (7.10-7) again for lz - zol < ...