1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 505 ( ( z( ) = ( - z + R^2 - z( d1/J ( ( z ) = (-z + (-z d1f-i R2 -r2 = I( - zl2 d1/J so that (7.28-12) becomes u(re ...

506 Chapter^7 Now, by equating real parts in (7.28-17), we obtain ( ) = I_ 1+00 yu(t, 0) dt = I_ 1+00 yu(t, 0) dt ux,y 7r -oo 1t ...

Integration 507 9. (a) If f is an entire function such that Ref(z) ~ M (M :'.:'.: 0) for all z, show that f is a constant functi ...

508 Chapter^7 *19. (a) With the same notation as in Theorem 7.38, show that ( + Z = 1+2 ~ (!:...)n ein(O-.P) (-z 6 R (1) n=l (b) ...

Integration 509 Let r = (x1,x 2 ,x3) and v = (v 1 ,v 2 ,v 3 ). The fluid motion is said to be two-dimensional or plane-parallel ...

510 Chapter^7 y 0 x Fig. 7.30 the integral j Wr ds = J u dx + v dy (7.29-3) c+ c+ exists and it is called the circulation around ...

Integration 511 and F'(z)=u+iv=w The function F is called the complex velocity potential of the fl.ow, and IF'(z)I = lwl is term ...

512 Chapter^7 rotw = 'V X w = k(v., - uy) where T7 8. {), {)k v=-1+-J+- OX ay {)z Hence the conditions div w = 0 and rot w = 0 a ...

Integration y vu (^0) x Fig. 7.31 is called the vorticity (at each point), and C = k\i'^2 V is the vorticity vector. 513 The fl. ...

514 Chapter^7 and sources or sinks correspond to isolated singular points of the complex potential F( z ). If there are isolated ...

Integration 515 y arg z = c' x Fig. 7.32 are rays emanating from the origin (Fig. 7.32). The velocity at each point z E D is giv ...

516 Chapter^7 On the other hand, the complex potential m z-a F( z) = 2 7r log z _ b with m > 0 corresponds to a fl.ow with a ...

Integration 517 We may consider also the fl.ow determined by a complex potential of the form F(z) = _J_ logz ·27f where q = m + ...

518 Chapter 7 A. L. Cauchy, Memoire sur les integrales definies prises entre des limites imag- inaires, Bure Freres, Paris, 182 ...

Integration 519 J. Plemelj, Ein Erganzungssatz zur Cauchyschen Integraldarstellung analytis- chen Funktionen, Randwerte betreff ...

8 Sequences and Series of Functions. Series Representations. Some Special Functions In this chapter we proceed further with the ...

Sequences, Series, and Special Functions 521 By hypothesis 2 to every f > 0 there corresponds an integer N, such that n > ...

522 Chapters Since the f n are analytic in R and C is homotopic to a point in R ( R being simply connected), we have J fn(z) dz= ...

Sequences, Series, and Special Functions 523 Proof We know (Theorem 4.19) that the series converges uniformly to F(z) on lzl ::; ...

524 Chapter 8 Since the integrand in (8.1-2) is a continuous function of ( along C, by applying Theorem 8.1 we get lim F(k)(z) = ...