THE SINE INTEGRAL As one learns in elemenary calculus, he limi of sin(/ as vanishes is uniy. Furhermore he funcion is even and has an infinie number of zeros locaed a ±n for n1,,3 Is plo looks like his- Funcions of his ype are encounered in a variey of areas including and mos imporanly in diffracion heory for elecromagneic waves. Ofen one wishes o find he oal area under his curve. This value can be obained by use of he Sine Inegral- sin( ) by leing approach infiniy and muliplying by wo. The eplici value of he area under he enire curve is simples o arrive by comple variable mehods. One has- Area Im + z ep( iz) dz z 1 ep( iz) Im{ i res } z z
To find ha par of he area from o one needs o evaluae he above sine inegral. This evaluaion is no possible in erms of elemenary funcions bu i is raher simple o deermine using a series epansion found by inerchanging summaion and inegraion evaluaion order. One finds - Si ( n n+ 1 n n+ 1 ( 1) ( 1) n (n+ 1)! n (n+ 1)(n+ 1)! Alhough his infine sum is valid for any, he sum converges slowly when >>1. To find an asympoic epansion for when >>1 one can use he following approach. Rewrie as- sin( ) and hen inegrae by pars several imes o yield- cos( sin( + sin( ) 3 From his we have he asympoic form- cos( sin( valid for large. A graph of he funcions and is approimaions for small and large follows-
One sees ha he asympoic form and he cubic approimaion for work very well in heir respecive regions of. The greaes uncerainy occurs near 1 bu his does no pose a problem since he given infinie series epansion for converges quie rapidly here. We can also consruc an inegral represenaion for valid over he enire range of. This inegral is developed as follows. Sar wih- and se ep(iθ). This yields- Si ( Im ep( i ln( ) / Cons cos( cos( θ ))ep( sin( θ )) dθ θ
The consan can be evaluaed a o yield Cons1. Hence we have a convenien inegral represenaion for given by- and valid for <<. / e θ sin( θ ) The differeniaying his resul once, we find- cos[ cos( θ )] dθ d d Re / θ d [ep (sin( θ ) + i cos( θ ))] dθ d / sin( θ + cosθ )ep sinθdθ θ Bu his resul mus also mach sin(/. Hence as, we have he slope- dsi d ε ) ε / ( ) lim sin( sin( θ ) dθ cos() cos( / ) 1 θ ( ) ε To work ou he higher derivaives of i is simples o sar wih- d y d sin( ' Then- y cos( sin( 1 and y'' cos( [ ]sin( ' so ha- y '' + y' + y
This differenial equaion is recognized as a spherical Bessel equaion of zeroh order. I means ha one has he soluion- y( j ( J1/ ( sin( Finally we look a an approimaion for valid for he full range of. Using he inegral represenaion of he sine inegral as a saring poin, we have- Si ( Re / ep [sin( ) i cos( )] And we ne le T(n,) be an approimaion for an() valid in he range <</. See one of our earlier noes which shows how o ge good quoien approimaions for he angen funcion. In erms of such an approimaion, we have- Re / ep [ 1+ T ( n, ) + I T ( n, ) ] Using one of he simples of such approimaions, namely- we ge he plo- 15 T (1, ) 15 6 3
This is seen o be in ecellen agreemen wih he eac values of shown in red. Oc.11