1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/Applications 0 .. x Fig. 9.11 = lim { r-f [_::i_ +g(x)] dx+jb [_::i_ +g(x)] dx} E-+O+ 1 a X - C c+E X - C ...

686 Chapter^9 (c') (PV)l::0J(x)dx = limR-->ool~Rf(x)dx The abbreviation (PV) before the integral is sometimes dropped when it ...

Singularities/Residues/ Applications 687 Improper integrals of mixed type. If the interval of integration is infinite and the i ...

688 Chapter 9 we look for a meromorphic (in particular, holomorphic) function F(z) in C, or at least on either Imz;:::: 0 or Imz ...

Singularities/Residues/ Applications 689 quarter-circles), say"(, and again the existence of lim j f(z)dz r-+O 7+ is needed for ...

690 Chapter^9 _L 1_b4~-·b_3-=-t-·~·b_2~~--,:-~ •b1 -R 0 Fig. 9.la Since f(z) is even, we have J~n J(x) dx = f 0 R f(x) dx and (9 ...

Singularities/Residues/Applications 691 Hence 3 For integrals of this type the following alternative method is available: Consid ...

692 Chapter9 More generally, for any integral of the form 1 (^00) xmdx o xn +an where m and n are positive integers with m :::; ...

Singularities/Residues/Applications applying (9.8-2), we have R z + 1 _ bk + 1 __ ~ (b 2 b ) z=1~ z^4 + 1 - 4bi - 4 k + k since ...

694 Chapter9 However, in this case the integral on the left of (9.11-11) converges in the ordinary sense. To see this, consider ...

Singularities/Residues/ Applications 695 (9.11-14) Hence by using the residue theorem and the estimates (9.11-12) to (9.11-14), ...

696 as R -t oo uniformly for all z E r+. Hence letting R -t oo in J R xeiax J zeiaz 1 2 b2 dx + 2 b2 dz = 27ri. -2 e -ab -RX + Z ...

Singularities/Residues/ Applications 697 1 (^00) x (^2) dx 10. -oo (x2 + 2x + 2)2 = 71' 11 loo dx = 71' _ 00 (x^2 - 4x + 8)^2 1 ...

698 Chapter9 L~ ~ . f+ b2 bj ~ r -R 0 p-r p p+r R x Fig. 9.16 By Lem.ma 9.5, lim {jp-r f(x)dx+ [R f(x)dx} = (PV)jR f(x)dx r-+O - ...

Singularities/Residues/ Applications 699 the real axis is regarded as belonging to the upper half-plane. Formula (9.11-21) appli ...

700 Chapter^9 Rb b+r'b, --\ 'Y2 ~ f+ \ b~ \ \ b-r'b ~ 'Y1~ A 0 x Fig. 9.JL7 r J dz 0 R::Uoo z^3 - 1 = r+ lim ( {1-r _.!!::: + lR ...

Singularities/Residues/Applications 701 line calls for an indentation '"ts· Proceeding as in types III and IV, we find that (PV) ...

702 x and equating real and imaginary parts, we find that 1 (^00) cos ax 7r • (PV) 2 2 dx = - - smap -oo x - p p (PV) loo sin ax ...

Singularities/Residues/ Applications r+ -R R x Fig. 9.19 By changing x into -x in the first integral above, we get fr e-ix dx = ...

704 Chapter9 get [ 00 sinx dx = [ 00 sinmu du= ~ lo x lo u 2 (9.11-28^1 ) In particular, for m = 2 we have or [ 00 sin2u du= 2 ( ...