Introductions to Floor

Transcription

1 Itroductios to Floor Itroductio to the roudig d cogruece fuctios Geerl The roudig d cogruece fuctios hve log history tht is closely relted to the history of umber theory. My clcultios use roudig of the flotig-poit d rtiol umbers to the closest smller or lrger itegers. J. Nemorrius (37) ws oe of the first mthemticis to use the quotiet of two umbers m d i moder sese, but the word quotiet ppered for the first time roud 50 i the writigs of Meister Gerdus. Specil ottios for roudig d cogruece fuctios were itroduced much lter. C. F. Guss (80) suggested the symbol mod ( m mod ) for the ottio of the property tht the rtio m is iteger. He observed tht d m re the cogruet modulo. The umber is clled modulus. C. F. Guss (808) d J. Liouville (838) widely used the floor d roud fuctios i their ivestigtios. They d other mthemticis used differet d sometimes cofusig ottios for those fuctios. The moder ottios of z d z for floor d ceilig fuctios, respectively, were suggested by K. E. Iverso (96). The ottio z for the roudig fuctio ws proposed by J. Hstd (988). Defiitios of the roudig d cogruece fuctios The roudig d cogruece fuctios iclude seve bsic fuctios. They ll del with the seprtio of iteger or frctiol prts from rel d complex umbers: the floor fuctio (etire prt fuctio) z, the erest iteger fuctio (roud) z, the ceilig fuctio (lest iteger) z, the iteger prt itz, the frctiol prt frcz, the modulo fuctio (cogruece) m mod, d the iteger prt of the quotiet (quotiet or iteger divisio) quotietm,. The floor fuctio (etire fuctio) z c be cosidered s the bsic fuctio of this group. The other six fuctios c be uiquely defied through the floor fuctio. Floor For rel z, the floor fuctio z is the gretest iteger less th or equl to z. For rbitrry complex z, the fuctio z c be described (or defied) by the followig formuls: x ; x x z Rez Imz. Exmples: 3. 3, 3 3, 0.,.3 3, 3 0, Π, 5, 3 5, 7 3. Roud

2 For rel z, the roudig fuctio z is the iteger closest to z (if z ±, ± 3, ). For rbitrry z, the roud fuctio z c be described (or defied) by the followig formuls: x ; x x z Rez Imz ; ;. Exmples: 3. 3, 3 3, 0. 0,.3, 3, Π 3, 5 3, 5, 7. Ceilig For rel z, the ceilig fuctio z is the smllest iteger greter th or equl to z. For rbitrry z, the fuctio z c be described (or defied) by the followig formuls: x ; x x z Rez Imz. Exmples: 3., 3 3, 0. 0,.3, 3, Π 3, 5 3, 5 3, 7. Iteger prt For rel z, the fuctio iteger prt itz is the iteger prt of z. For rbitrry z, the fuctio itz c be described (or defied) by the followig formuls: itx ; x 0 sgx x x 0 itz itrez itimz. Exmples: it3. 3, it3 3, it0. 0, it.3, it 0, 3 itπ 3, it 5 3, it 5, it 7 3. Frctiol prt For rel z, the fuctio frctiol prt frcz is the frctiol prt of z. For rbitrry z, the fuctio frcz c be described (or defied) by the followig formuls: frcx x ; x 0 sgx x x 0

3 3 frcz frcrez frcimz. Exmples: frc3. 0., frc3 0, frc0. 0., frc.3 0.3, frc 3 3, frcπ 3 Π, frc 5 3 3, frc 5, frc 7. Mod For complex d m, the mod fuctio m mod is the remider of the divisio of m by. The sig of m mod for rel m, is lwys the sme s the sig of. The mod fuctio m mod c be described (or defied) by the followig formul: m mod m m. The fuctiol property m mod m mod m m mes the behvior of m mod similr to the behvior of m. Exmples: 5 mod, 8 mod 3, 5 mod 3, 7 Π mod 3 7 Π, 7 3 mod 3, frcπ 3 Π,.7 3 mod 5.7. Quotiet For complex d m, the iteger prt of the quotiet (quotiet) fuctio quotietm, is the iteger quotiet of m d. The quotiet fuctio quotietm, c be described (or defied) by the followig formul: quotietm, m. Exmples: quotiet5,, quotiet3, 3, quotiet, 3, quotietπ,, quotiet7 3, 5 5, quotietπ,, quotiet.7 3, 5. Coectios withi the group of roudig d cogruece fuctios d with other fuctio groups Represettios through relted fuctios The roudig d cogruece fuctios z, z, z, itz, frcz, m mod, d quotietm, hve umerous represettios through relted fuctios, which re show i the followig tbles, where the symbol Χ mes the chrcteristic fuctio of set (hvig the vlue whe its rgumet is elemet of the specified set, d vlue of 0 otherwise):

4 x ; x or m, ; m Floor Roud Floor x x x ; x x x ; x x Χ x Roud x x x ; x x x ; x x Χ x Ceilig x x ; x x x ; x x x ΘΧ x x x x ; x x x ; x x Χ x ItegerPrt FrctiolPrt itx x ; x 0 x itx x ; x 0 x itx x sgχ x Θx frcx x x ; x 0 x frcx x x ; x 0 x frcx x x sgχ x Θx itx x x ; x 0 itx x x ; x 0 itx x Χ x sgχ x Θx frcx x x x ; x 0 frcx x x x ; x 0 frcx x x Χ x sgχ x Θx Mod m mod m m m mod m m m ; m mod m m m ; m mod m Χ m m Quotiet quotietm, m quotietm, m m ; quotietm, m m ; quotietm, m Χ m

5 5 x ; x or m, ; m Mod Quotiet Floor x x x mod x quotietx, Roud x x frcx x x frcx ; x ; x x x x mod Χ x x quotietx x, ; x quotietx x, ; x quotietx, Χ x Ceilig x x x mod x quotietx, ItegerPrt itx x x mod ; x 0 x itx x x mod ; x 0 x itx x x mod sgχ x Θx itx quotietx, ; x 0 x itx quotietx, ; x 0 x itx quotietx, sgχ x Θx FrctiolPrt Mod Quotiet frcx x mod ; x 0 x frcx x mod ; x 0 x frcx x mod sgχ x Θx quotietm, mm mod frcx x quotietx, ; x 0 x frcx x quotietx, ; x 0 x frcx x quotietx, sgχ x Θx m mod m quotietm,

6 6 z or m, Floor Roud Ceilig Floor z Χ Rez Χ Imz z z ΘΧ Rez Θ Χ Imz z z Roud z z z z Χ Rez Χ Imz Χ Imz Χ Rez Ceilig z z ΘΧ Rez Θ Χ Imz z z z z Χ Rez Χ Imz ItegerPrt itz z sgχ Rez ΘRez sgχ Imz ΘImz itz z Χ Rez Χ Imz sgχ Rez ΘRez sgχ Imz ΘImz itz z sgχ Rez ΘRez sgχ Imz ΘImz Θ Χ Imz ΘΧ Rez FrctiolPrt frcz z z sgχ Rez ΘRez sgχ Imz ΘImz frcz z z Χ Rez Χ Imz sgχ Rez ΘRez sgχ Imz ΘImz frcz z z sgχ Rez ΘRez sgχ Imz ΘImz Θ Χ Imz ΘΧ Rez Mod m mod m m m mod m m Quotiet quotietm, m quotietm, m Χ Re m Χ Im m Χ Re m Χ Im m m mod m m ΘΧ Re m Θ Χ Im m m mod m m quotietm, m ΘΧ Re m Θ Χ Im m quotietm, m z or m, Mod Quotiet Floor z z z mod z quotietz, Roud z z z mod Χ Imz Χ Rez Ceilig z z z mod z quotietz, ItegerPrt itz z z mod sgχ Imz ΘImz sgχ Rez ΘRez z quotietz, Χ Rez Χ itz quotietz, sgχ Imz ΘImz sgχ Rez ΘRez FrctiolPrt Mod Quotiet frcz z mod sgχ Imz ΘImz sgχ Rez ΘRez quotietm, mm mod frcz z quotietz, sgχ Im sgχ Rez ΘRez m mod m quotietm, The roudig d cogruece fuctios z, z, itz, frcz, m mod, d quotietm, c lso be represeted through elemetry fuctios by the followig formuls:

7 7 z z t cotπ z ; z z Π z z t cotπ z ; z z Π itz z t cotπ z sgθz ; z z Π frcz t cotπ z sgθz ; z z Π m mod Π t cot Π m ; m m quotietm, m Π t cot Π m ; m m. The best-ow properties d formuls of the umber theory fuctios Simple vlues t zero The roudig d cogruece fuctios z, z, z, itz, d frcz hve zero vlues t zero: it0 0 frc0 0. Specific vlues for specilized vribles The vlues of five roudig d cogruece fuctios z, z, z, itz, d frcz t some fixed poits or for specilized vribles d ifiities re show i the followig tble:

8 8 z z z z itz frcz Π Π , frc 0, frc, 0 frc 0, frc, 0 frc 0, ; it frc 0 ; it frc 0 x y ; x y x y x y x y x y x y x y itx y itx ity 3 0 frcx y frcx frcy frc 0, frc, 0 frc 0, frc, 0 frc 0, The vlues of mod fuctio m mod, d quotietm, t some fixed poits or for specilized vribles re show here:

9 9 m \ mod 0 ; 0 0 mod ; mod ; 0 0 mod ; mod 3 ; mod ; mod 5 ; mod 6 ; mod 7 ; mod 8 ; mod 9 ; mod 0 ; 0 m m mod 0 ; m mod 0 m mod 0 m \ quotiet0, 0 ; quotiet, ; 0 quotiet, 0 ; quotiet, 0 ; quotiet3, 0 ; quotiet, 0 ; quotiet5, 0 ; quotiet6, 0 ; quotiet7, 0 ; quotiet8, 0 ; quotiet9, 0 ; quotiet0, 0 ; 0 m quotietm, m ; m quotiet, m quotiet, m mod m ; m m m mod m ; m m m mod m ; m m p mod p p ; p p p mod p 3 ; p p 3 B mod mod,0 3 Χ Χ mod quotietm, 0 ; m m

10 0 quotietm, ; m m quotietm, ; m m p p quotietp, p ; p p quotiet p p, p 3 p 3 p p ; p p 3. Alyticity All seve roudig d cogruece fuctios (floor fuctio z, roud fuctio z, ceilig fuctio z, iteger prt itz, frctiol prt frcz, mod fuctio m mod, d the quotiet fuctio quotietm, ) re ot lyticl fuctios. They re defied for ll complex vlues of their rgumets z d m,. The fuctios z, z, z, d itz re piecewise costt fuctios d the fuctios frcz, m mod, d quotietm, re piecewise cotiuous fuctios. Periodicity The roudig d cogruece fuctios z, z, z, itz, frcz, d quotietm, re ot periodic fuctios. m mod is periodic fuctio with respect to m with period : m mod m mod m mod m mod ;. Prity d symmetry Four roudig d cogruece fuctios (roud fuctio z, iteger prt itz, frctiol prt frcz, d mod fuctio m mod ) re odd fuctios. The quotiet fuctio quotietm, is eve fuctio: z z itz itz frcz frcz m mod m mod quotietm, quotietm,. The roudig d cogruece fuctios z, z, z, itz, d frcz hve the followig mirror symmetry: z z Χ Imz z z z z Χ Imz itz itz frcz frcz.

11 Sets of discotiuity The floor d ceilig fuctios z d z re piecewise costt fuctios with uit jumps o the lies Rez Imz l ;, l. The fuctios z (d z) re cotiuous from the right (from the left) o the itervls,,, d from bove (from below) o the itervls,,. The fuctio z is piecewise costt fuctio with uit jumps o the lies Rez Imz l ;, l. The fuctio z is cotiuous from the right o the itervls,,, d from the left o the itervls,,. The fuctio z is cotiuous from bove o the itervls,,, d from below o the itervls,,. The fuctio itz (d frcz) is piecewise costt (cotiuous) fuctio with uit jumps o the lies Rez Imz l ;, l, 0, l 0. The fuctios itz d frcz re cotiuous from the right o the itervls,,, d from the left o the itervls,,. The fuctios itz d frcz re cotiuous from bove o the itervls,,, d from below o the itervls,,. The fuctios m mod d quotietm, re piecewise cotiuous fuctios with jumps o the curves Re m Im m l ;, l. The fuctiol properties m mod m mod m m d quotietm, quotiet m, m me the behvior of tht fuctios similr to the behvior of floor fuctio m. The previous described properties c be described i more detil by the formuls from the followig tble:

12 Floor Roud z±ε or m ± Ε ; Ε 0 lim Ε0 z ± Ε z Rez lim Ε0 z ± Ε z ; Rez ± lim lim Ε0 z ± Ε z ± ; Re z Ceilig lim Ε0 z ± Ε z ± Rez ItegerPrt FrctiolPrt Mod Quotiet lim Ε0 itz ± Ε itz ; ±Rez lim Ε0 itz ± Ε itz ± ; Rez lim Ε0 frcz ± Ε frcz ; ±Rez lim Ε0 frcz ± Ε frcz ; Rez lim Ε0 m ± Ε mod m mod lim Re m 0 lim Ε0 quotietm ± Ε, quotietm, lim Re m 0 ; ; ; ; z± Ε or m ± Ε ; Ε 0 lim Ε0 z ± Ε z ; Imz Ε0 z ± Ε z ; Imz ± lim Ε0 z ± Ε z ± ; Imz lim Ε0 z ± Ε z ± ; Imz lim Ε0 itz ± Ε itz ; ±Imz lim Ε0 itz ± Ε itz ± ; Imz lim Ε0 frcz ± Ε frcz ; ±Imz lim Ε0 frcz ± Ε frcz ; Imz Ε0 m ± Ε mod m mod ; Im m 0 Ε0 quotietm ± Ε, quotietm, ; Im m 0 Series represettios The roudig d cogruece fuctios z, z, z, itz, frcz, m mod, d quotietm, hve the followig series represettios: Floor Roud Ceilig ItegerPrt FrctiolPrt Mod Quotiet x x x Π x x Π x x Π itx x Π frcx Π si Π x si Π x si Π x si Π x si Π x ; x x m m m si Π m ; x x m m ; x x m m si Π m si Π m Θx ; x x it m m sg m Θx ; x x frc m sg m m mod Π m mod cot Π ; m cot Π ; m cot Π ; m si Π m si Π m si Π m si Π m quotietm, m Π quotietm, m cot Π cot Π ; m ; m m cot Π ; m si Π m si Π m ; m cot Π Trsformtios d rgumet simplifictios (rgumets ivolvig bsic rithmetic opertios) The vlues of roudig d cogruece fuctios z, z, z, itz, d frcz t the poits z, ± z, z ; c lso be represeted by the followig formuls:

13 3 z z z z z z z z ; Rez Imz z z sgrez sgimz ; Rez Imz z z Χ z ; z z z Χ ImzsgImz Χ RezsgRez z z Χ Imz z Imz Rez z z Χ Rez z Imz Rez z z z z z z z z z z z ; Rez Imz z z sgimz sgrez ; Rez Imz z z Χ Imz sgimz Χ Rez sgrez z z Χ Imz z Imz Rez z z Χ Rez z Imz Rez itz itz itz it z itz it z itz it z Re z frcz frcz frcz frc z frcz frc z frcz frc The vlues of the fuctios m mod d quotietm, t the poits ±m,, m,, m,, ± m,, m, ±, d m, hve the followig represettios: m, m mod quotietm, m, m mod m mod quotietm, quotietm, m, m, m mod m mod Χ m ; m m mod m mod Χ Re m sgre m Χ Im m sgim m m mod m mod ; m m m mod Χ m m mod ; m m mod m mod Χ Re m sgre m Χ Im m sgim m quotietm, quotietm, Χ m ; m quotietm, quotietm, Χ Re m Χ Im m sgim m quotietm, Χ m quotietm, ; m quotietm, quotietm, Χ Re m Χ Im m sgim m m, m mod m mod quotiet m, quotietm, m, m mod Χ Im m m mod quotiet m, quotietm, Χ Im m m, m mod Χ Re m m mod quotiet m, quotietm, Χ Re m m, m mod m mod Χ Re m quotietm, quotietm, Χ Re m m, m mod m mod Χ Im m quotietm, Χ Im m quotietm, m, m m mod mod quotiet m, quotietm, Trsformtios d rgumet simplifictios (rgumets ivolvig relted fuctios)

14 Compositios of roudig d cogruece fuctios z, z, z, itz, frcz, m mod, d quotietm, with the roudig d cogruece fuctios i my cses led to very simple zero results: z Floor Roud Ceilig ItegerPrt Floor z z z z 0 Roud z z z z z z 0 z z z z itz itz z z Ceilig z z z z z z z z 0 itz itz itz itz ItegerPrt itz z itz z itz z ititz itz itz itz 0 FrctiolPrt frcz 0 frcz 0 frcz 0 frcitz 0 Mod Quotiet m mod m m m mod 0 quotietm, m quotietm, m m mod m m m mod 0 quotietm, m quotietm, m m mod m m m mod 0 quotietm, m quotietm, m itm mod itm itm mod 0 quotietitm, it quotietitm, itm x x x ; x x ; x. m, Mod Quotiet Floor m mod m m m mod 0 quotietm, m m quotietm, 0 Roud m mod m m quotietm, m Ceilig m mod m m quotietm, m ItegerPrt itm mod itm m itquotietm, m FrctiolPrt frcm mod frcm m frcquotietm, 0 Mod Quotiet m mod mod m mod m mod mod m m quotietm mod, m mod quotietm mod, 0 m quotietm, mod m quotietm, mod 0 quotietquotietm,, m quotietquotietm,, m Additio formuls The roudig d cogruece fuctios z, z, z, itz, d frcz stisfy the followig dditio formuls:

15 5 z z z Floor z z ; z z z z z z z z Roud z z ; Rez Imz Ceilig z z ; z z z z z z z z ItegerPrt itz itz Θz Θz ; z FrctiolPrt frcz frcz Θz Θz ; z m mod m mod ; quotietm, quotietm, ;. Multiple rgumets The roudig d cogruece fuctios z, z, itz, frcz, m mod, d quotietm, hve the followig reltios for multiple rgumets: Floor Roud Ceilig ItegerPrt FrctiolPrt Mod Quotiet z or m z z 0 Θz mod Θz mod ; z z z 0 Θ z mod Θ z mod ; z it z itz sgχ z Θz sgχ z Θz 0 Θz mod Θz mod ; z frc z frcz sgχ z Θz sgχ z Θz 0 Θz mod Θz mod ; z m mod m mod j 0 j Θ m j quotietm, Θ m j quotietm, ; m quotiet m, quotietm, j 0 j Θ m mod j Θ m j mod ; m Sums of the direct fuctio Sums of the floor d ceilig fuctios z d z stisfy the followig reltios: z z z z z z z z 0 m x m 0 x m ; x m z z z z z z z z 0 x m m 0 x m ; x m. Idetities All roudig d cogruece fuctios stisfy umerous idetities, for exmple:

16 ; ; ; c mod b d mod ; mod b mod c mod d mod b c d. Complex chrcteristics Complex chrcteristics (rel d imgiry prts Rez d Imz, bsolute vlue z, rgumet Argz, complex cojugte z, d sigum sgz) of the roudig d cogruece fuctios c be represeted i the forms show i the followig tbles: z Floor Roud Ceilig ItegerPrt Re Rez Rez Rez Rez Rez Rez Reitz itrez Im Imz Imz Imz Imz Imz Imz Imitz itimz Abs z Imz Rez z z itz Rez Imz Imz Rez itimz itrez Arg Argz Argz Argz Argitz t Rez, Imz t Rez, Imz t Rez, Imz t itrez, itimz Cojugte z Rez Imz z Rez Imz z Rez Imz itz itrez itimz Sig sgz z z sgz z z sgz z z sgitz itz itz x y ; x y Floor Roud Ceilig ItegerPrt Re Rex y x Rex y x Rex y x Reitx y itx Im Imx y y Imx y y Imx y y Imitx y ity Abs x y x y x y x y x y x y itx y itx Arg Argx y Argx y Argx y Argitx y t x, y t x, y t x, y t itx, ity Cojugte x y x y x y x y x y x y itx y itx Sig sgx y x y x y sgx y x y x y sgx y x y x y sgitx y it i

17 7 m, Mod Quotiet Re Im Abs Rem mod Rem Imm ImRem Re Im Re Imm mod Imm Imm ReIm Rem Im Re m mod Imm Imm ReIm Rem Re Im Re Imm ImRem Re Re Im Re Imm ImRem Re Im Re Im Im Im Requotietm, Rem ReImm Im Im Re Imquotietm, Imm ReIm Rem Im Re quotietm, Arg Cojugte Sig Imm ReIm Rem Rem Im Re Re Imm ImRem Re Im Re Argm mod t Rem Imm ImRem Re Im Re Imm ReIm Rem Im Re m mod Rem Imm sgm mod Imm Imm Imm ReIm Rem Re Imm ReIm Rem Im Re Re, Imm Re Imm ReIm Rem Im Re Imm ImRem Re Im Re Imm ReIm Rem Im Re Imm ImRem Re Im Re Imm ImRem Re Im Re Imm ReIm Rem Im Re Im sgm mod sg ; m Im Re Im Imm ImRem Re Im Im Re Im Im Imm ImRem Re Im Re Imm ReIm Rem Im Re Im Rem Im Im Im Rem Re Re Imm ImRem Re Im Re Imm ReIm Rem Im Re Imm ReIm Rem Im Re Imm ImRem Re Im Re Re Re Re Re Rem ReImm Im Im Re Argquotietm, t quotietm, Imm ImRem Re Im Re Imm ImRem Re Im Re sgquotietm, I Im Im sgquotietm, sg m Differetitio Derivtives of the roudig d cogruece fuctios z, z, z, itz, frcz, m mod, d quotietm, c be evluted i the clssicl d distributiol sese. I the lst cse, ll vribles should be rel d results iclude the Dirc delt fuctio. All roudig d cogruece fuctios lso hve frctiol derivtives. All these derivtives c be represeted s show i the followig tbles: i clssicl sese z i distributiol sese for rel x x Α z Α i frctiol sese Floor z z 0 x x x Α z z Α z zα Α Roud z z 0 x x x Α z z Α zzα Α Ceilig z z 0 x x x Α z z Α z zα Α ItegerPrt itz z 0 itx x,0 x Α itz z Α itz zα Α FrctiolPrt frcx x frcx x x,0 x Α frcz z Α Α zα frcz zα Α Α

18 8 m i clssicl sese i clssicl sese m i distributiol sese for rel m, i distributiol sese for rel m, Α m Α i frctiol sese Mod Quotiet m mod m quotietm, m 0 m mod m quotietm, 0 m mod m quotietm, m m m sg m m mod quotietm, sgm m sg it m m,0 m,0 m Α m mod m Α Α m Α m mod m Α Α Α quotietm, m Α quotietm, m Α Α Idefiite itegrtio Simple idefiite itegrls of the roudig d cogruece fuctios z, z, z, itz, frcz, m mod, d quotietm, hve the followig represettios: Floor Roud Ceilig ItegerPrt f z z z z z z z z z z z z z z itz z z itz FrctiolPrt frcz z z frcz z Mod z mod z z z mod z Quotiet m mod z z z m m mod z quotietz, z z quotietz, quotietm, z z z quotietm, z Defiite itegrtio Some defiite itegrls of the roudig d cogruece fuctios z, z, z, itz, frcz, m mod, d quotietm, c be evluted d re show i the followig tble:

19 9 0 f t t ; 0 f t t 0 t Α f t t Floor 0 tt 0 tt 0 t Α tt Α Α ΖΑ ΖΑ, Roud 0 tt 0 tt 0 t Α tt Α Α Α ΖΑ Ζ Ceilig 0 tt 0 tt 0 t Α tt Α ΖΑΖΑ Α ItegerPrt 0 ittt ; 0 itt t it it 0 t Α ittt it Α ΖΑΑ FrctiolPrt 0 frctt ; 0 frct t frc frc 0 t Α frctt Α Α Α ΖΑ ΖΑ, ReΑ Mod 0 t mod t mod mod 0 m mod tt Ψ mm mod m m m mod Quotiet 0 quotiett, t quotiet, quotiet, 0 t Α t mod t Α Α Α ReΑ Α ΖΑ Α 0 t Α m mod tt Α Α m m Α Α ΖΑ, m 0 t Α quotiett, t Α q Α ΖΑ, quotiet, 0 t Α quotietm, tt quotietm, Α m Α ΖΑ,quotietm ReΑ Α Itegrl trsforms All Fourier trsforms of the roudig d cogruece fuctios z, z, z, itz, frcz, m mod, d quotietm, c be evluted i distributiol sese d iclude the Dirc delt fuctio:

20 0 F t f tz Fc t f tz Fs t f tz Floor Π z Π Π z Roud t tz Π Π z Ceilig t tz Π Π ItegerPrt t ittz Π FrctiolPrt t frctz Mod Π t t mod z Π z Πz Π z Π z Πz Π z Πz Π z Πz Π z Π z Πz Π z Π z Π z Π z Π z Π z c t tz c t tz Π z Π z cot z Π csc z c t tz Π z c t ittz Π z Π z Π z cot z z s ttz cot z c t frctz z cot z Π z Π z c t t mod z z cot z Π z Π z Π s ttz s ttz Π s t ittz Π s t frctz Π s t t mod Π Quotiet t quotiett, z Π z Π Π z Π z Π z c t quotiett, z Π z Π z cot z s t quotiet Π Lplce d Melli itegrl trsforms of the roudig d cogruece fuctios z, z, z, itz, frcz, m mod, d quotietm, c be evluted i the clssicl sese: L t f tz Floor t tz ; Rez 0 z z Roud t tz z ; Rez 0 z z Ceilig t tz z ; Rez 0 z z ItegerPrt t ittz ; Rez 0 z z M t f tz ttz Ζz z ; Rez ttz z Ζz, 3 z ; Rez tittz Ζz z ; Rez FrctiolPrt t frctz z ; Rez 0 z z tfrctz z ; Rez 0 z z Mod t t mod z z ; Re z 0 z z Quotiet t quotiett, z ; Re z 0 z z tt mod z z Ζz z t m mod tz mz Ζz z tquotiett, z z Ζz z t quotietm, tz mz Ζz z ; Rez 0 ; Rez 0 ; Rez ; Rez

21 Summtio Sometimes fiite d ifiite sums icludig roudig d cogruece fuctios hve rther simple represettios, for exmple: y 0 x x y y y x ; x y y 0 y p q p p q ; p q gcdp, q m m cot Π cot m Π ; m gcdm, p p l l p p p ; p p q p q p q p q ; p q gcdp, q y 0 x y x y x y y ; x y 0 x 0 y 0 p m p m m m p m p m ; m p m m ; m. Zeros Zeros of roudig d cogruece fuctios re give s follows: z 0 ; 0 Rez 0 Imz z 0 ; Rez Imz z 0 ; Rez 0 Imz 0 itz 0 ; Rez Imz frcz 0 ; Rez Imz m mod 0 ; m 0 0 quotietm, 0 ; 0 Re m 0 Im m.

22 Applictios of the roudig d cogruece fuctios All roudig d cogruece fuctios re used throughout mthemtics, the exct scieces, d egieerig.

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