We shall evaluate here Dirichlet~'> i11tegrat ~ sinx ;r

Chapter 0, Problem 7.26

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We shall evaluate here Dirichlet~'> i11tegrat ~ sinx ;r --dx = - ( .r 2 hy intcgrali ng ei.: /;.around the simple dosed concou r shown in Fig. I 08. In that figure. p and R denote positive real numbers. where p < R: and L 1 and L2 represent the imervalsp ~ .r ~ Rand-R ~.r ~ -p.respcctively.on1herealaxis.Thesemicircles C,, and CR arc as shown in the figure. The semicircle C,, is introduced here in order 10 avoid passing through the singuhuity of the quoiielll ,i; /:.. -R -(> 0 R .\ FIGURE 108 The Cauchy-Goursal theorem tells us lhal r e~: d:. + r, ('~: d:. + J e~: d:. + r_ ei.: d:. = 0. J1., . ./cs:.. I.:.. Jc. or (5) J ei: f ,i: 1 ei: - 1 ei: - ~ d:. + ~ d:. = - - d' - - d ... J., I.: C,. ;. Cs: - TI1i:-. integrnl i:-. impor~111t in applied 111atl1ematic~ and. in particular. tl1c: theory of Fourier integral!'>. Sec Ilic authors' Fourier Serie.1 cu1d Bowulan \!alue Problem.I'. Xth ed .. pp. 163 165. 2012. where it i~ evaluated in a rnmplctcly different way. SEC. 90 Al'\ l!'\DENTATIOl'\ AROl'l\D .\ BRAl'\Cll POINT 277 Moreover. since the legs L 1 and -L2 have parametric represelllalions (6) : = re;o = r (p :::=- r :::=- R) and : = n,i:r = -r (p :::=- r :::=- R). respectively. the left-hand side or equation (5) can be wrillen j e': I e;~ 1R eir ;Re ir 1H eir - e ;,. -d: - -d: = -dr - --dr = dr . , I.~ : ti r I' r I' r !. H cir - e ;,- !.H sin,. = 2i dr = 2i --dr. ")' fJ _,,. /! ,. Consequently. equation (5) reduces lo !.H sin r 1 e;~ ; e;: 2i --dr = - -- d: - ~d:. ,. . ri C. ,, . C>1 (7) No~ from the Laurem se1ies represelllalion ei: l [ (i:) (i:)2 (i:.)' l I i i 2 i' , -=-I+-+--+--+ =-;-.+-I,_+,.,_,_:+_,.,,_:-+ : : I ~ 2 ~ 3 ~ _ .1 ({) < 1:1 < Xl). it is dear that ci: /: has a simple pole al the origin. with residue unity. So. according to the theorem al the beginning of this section. f ('i.~ Jim - d: = -ni. ti O. C, ;, Also. since when: is a poim on CR, \\1C know from Jordans lemma in Sec. 88 that j . ei.~ Jim -d: =0. R '- C1; - Thus. hy lening p tend to 0 in equation (7) and then letting R tend to oo. we aiTivc at the result [. " sin r 2i -- dr = ni. . ,. which is. in facl. the same as equation (4).