1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/Applications 725 Hence 1 1 lfo(z)J = JI+ z2JJl + zJlf2JI - zJl/2 < el/2 which gives ase-tO Similarly, ...

726 z - b = r2ei62, A= IAleilT Then !( ) z -_ vi Ah r2 e (1/2)i(^111 +o^2 -2w+1!"+2k1r) , r from which we select fo(z) = vJAhr2 ...

Singularities/Residues/ Applications 727 y f+ Fig. 9.28 1 a+E J + b-, fo(x)dx + fo(z)dz (9.11-63) 'Yt But J f^0 (z) dz= - 27l"i ...

728 Chapter9 so that J 27rB fo(z)dz = J=A r+ Also, J fo(z) dz= 27ri Res fo(z) = 27ri lim J Az^2 + 2Bz + C z=O z->0 '"ft since ...

Singularities/Residues/ Applications 729 replace the circle 'Yt of that figure (see Fig. 9.29). Then we have J fo(z) dz= 1-S fo( ...

730 while lim J fo(z)dz = i7rVC 6-+0 '"Yo '+ lim J fo(z) dz= i7r(-VC) 6-+0 -rt Chapter9 since limz-+O zf 0 (z) equals VG or -VG ...

Singularities/Residues/Applications 731 1 (^00) xa ln x 7rba --b-dx = -.- 2 -(7rcosa7r -lnbsina7r), -1 <a< 0, b > O o ...

732 Chapter9 (^0) -- R x Fig. 9.30 24 [1 _ dx = 2rr · Jo ·{/x^2 (1-x) V'3 f 1 {/x^2 (1-x) dx = 2 :, Jo 9v3 26.1= .xa-l cos bx ...

Singularities/Residues/Applications 733 c;t; is the positively oriented boundary of the rectangle with vertices at (b + m, -m) ...

734 Chapter^9 provided that mis large enough so that m >Mand b/m < 1. Thus if we take limits in (9.12-1) as m -t oo, we ge ...

Singularities/Residues/Applications 735 or changing ( into z, 1 00 2z 7r cot7rz = - + '\"' 2 2 z n=l L..!z -n 1 00 --+ ( --+--^1 ...

736 Chapter^9 while on y = ±m, 1 1 1 I csc 7l"ZI = < < - < 1 Vsin^2 7rX + sinh^2 7rm sinh7rm - 7rm so that lf(z)I = 171 ...

Singularities/Residues/Applications 737 corresponding Bernoulli number is known. For instance, since B 2 = 1/ 6 we have rr^2 1 1 ...

738 Chapter^9 Another expression for the B 2 n may be obtained by adding (9.12-6) to (9.12-9). This gives ~ (22n -1) 2n (-l)n-1 ...

Singularities/Residues/ Applications 739 and on the horizontal sides y = ±m, I I I I sec 7rzl = :::; "nh :::; -;--h < O.I V c ...

7 40 Chapter 9 9.13 The Logarithmic Derivative Definition 9.13 The logarithmic derivative of J(z) is defined to be d f'(z) dz lo ...

Singularities/Residues/ Applications 741 in some deleted neighborhood of a. Hence a is a simple pole of f' / f with residue -a. ...

742 Chapter^9 0 :::; t ~ 2rr, is given by 1 J f'(z) d - Res f'(z) =Res...!:_ !'(1/z) N = 2rri f(z) z = z=oo f(z) z=O z^2 J(l/z) ...

Singularities/Residues/Applications where the last term is analytic at ak. It follows that h(z) f'(z) = akh(z) + h(z) g'(z) f(z) ...

744 Chapter9 where, as before, N denotes the number of zeros and P the number of poles off inside C (counting multiplicities). T ...