1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/Applications Exercises 9.5 Show that: 1 (^00) dx 11"J3 (PV) _ a a = -3 2 (a > 0) 00 x +a a 1 (^00) d ...

706 Chapter^9 z =a+ Rei^0 , %7r ~ () ~ -%7r, for the case t < 0. When t = 0 it suffices to integrate along the vertiq.1,l seg ...

Singularities/Residues/ Applications 707 ~ b~ r+ b~ ;q A2 R x b~ Fig. 9.20 which shows that Similarly, lim J za f ( z) dz = ...

708 Chapter^9 Example To evaluate f 0 00 xa dx/(x + b), where -1 < a < 0 and b > 0. Here f(z) = 1/(z+b) has the simple ...

Singularities/Residues/Applications 709 -R x Fig. 9.21 Under the stated conditions the integrand za f(z) is a single-valued ana- ...

710 Chapter^9 lim jzaf(z)dz = 0 r-•0 (9.11-38) 'i't lim J za f(z) dz= -i?T Reszaf(z) 6-•0 .z=p "Y2 and lim J za f(z)dz = 0 R-+oo ...

Singularities/Residues/Applications 711 Next, in view of condition (5) there is an R 1 > r such that x ;:=: R 1 implies that ...

712 Chapter9 and there are no poles outside the real axis. Hence (9.11-42) yields '.. 100 xa dx 100 xa dx. a (cos a7r + i sma7r) ...

Singularities/Residues/ Applications 713 1 n:/2 9. 0 (tan or dO =^1 krrsec^1 / 2 mr, -1 < a< 1 (Hint: Let x = btanO in pro ...

714 + 1R f(x)(Inxt dx + j f(z)(logzt dz r+ m = 21l"i L !~~ f(z)(logzt k=l Now for z = rei^0 , 0 < r < 1, 1l" 2:: () 2:: 0, ...

Singularities/Residues/ Applications 715 On the other hand, if we let x = -x' (x' > 0) in the first integral of (9.11-43), we ...

716 Chapter^9 The recursive formula (9.11-47) can be applied in particular to rational functions f(x) = P(x)/Q(x), with P and Q ...

Singularities/Residues/ Applications 717 by using (9.11-51). Equating imaginary parts in (9.11-52) the result in (9.11-50) is ob ...

718 Chapter9 1 r sinha(x + i7r) d J sinhaz d 1-R sinha(x + i7r) d + R sm. h( x + i7l". ) x + -.-h-sm z z + -r sm. h( x + i7l". ) ...

Singularities/Residues/Applications On the other hand, from we get sinh a( x + i7l") = sinh ax cos a7!" + i cosh ax sin a7!" sin ...

720 Chapter9 made along the line Res = c where the arbitrary constant c is chosen so that all singularities of f ( s) lie to the ...

Singularities/Residues/ Applications provided that lira J estf(s)ds = 0 R->co r+ 721 (9.11-60) A sufficient condition for (9. ...

722 Chapter^9 (with b finite or oo ). In this case a cut of the C* plane is made joining the points a and b (either directly or ...

Singularities/Residues/Applications 723 y y z=iy -1 x (a) (b) Fig. 9.26 For instance, g 0 (i) = -.J2 while g 0 (-i) = +.J2. For ...

724 Chapter9 R x Fig. 9.2"f .+ J fo(z)dz+ 1=:+' fo(x)dx+ j fo(z)dz (9.11-62) 'Yt 'Yt But for z E r+ we have so that Also, lfo(z) ...