The Sgm Summto Notto #8 of Gottschlk's Gestlts A Seres Illustrtg Iovtve Forms of the Orgzto & Exposto of Mthemtcs by Wlter Gottschlk Ifte Vsts Press PVD RI 00 GG8- (8)
00 Wlter Gottschlk 500 Agell St #44 Provdece RI 0906 permsso s grted wthout chrge to reproduce & dstrbute ths tem t cost for eductol purposes; ttrbuto requested; o wrrty of fllblty s posted GG8-
N. fles the fle of legth ( ŒP) = the - fle = d = fle = the set of the frst postve tegers df = {,,, } = P[, ] the fle of legth zero = the 0 - fle = 0 d = zero fle = the empty set df thus 0 = = {} = {, } 3 = {,, 3} etc GG8-3
N. the bsc cptl sgm summto otto the sum + + + = plus plus plus of elemets = the (termwse) sum of the oered - tuple (,,, ) = the (termwse) sum of the -fmly ( Œ) = (,,, ),,, ( ŒP) of ddtve semgroup d = summto / sum of (,,, ) = ( Œ) = summto / sum of ( Œ) = = summto / sum for of = d d d = summto / sum from equls to of Œ = wh Œvr or t lest s teger - vlued vrble tht cots ts rge GG8-4
N. th esgm summto otto = s boud = dummy = umbrl vrble, sclled ' the summto dex (vrble)', the set of vlues of used the summto (vz here) s clled ' the summto rge', ( Π) s clled ' the th summd' or ' the th dded' ; the summto dex vrble s ofte tke to be becuse s the tl letter of ' dex' d of ' teger' ; summto dex vrbles re frequetly tke from the lphbetc ru h,, j, k d other good cddtes re m,, r C. 755 Euler troduced cptl sgm S to deote cotued sums; the dcl otto ws dded lter by others; the Greek letter sgm Ss correspods phoetclly & tr slterto to the Lt - Eglsh letter ess Ss w the tl letter of summ (Lt) = sum GG8-5
N. exteded cptl sgm summto otto = summto / sum from equls m to of = + + + m+ wh m, Œ Z st m & Œvr Z[ m, ] or s vrble whose vlues re tegers d whose rge cludes Z[m, ] & =m df m m,,, m+ re elemets of ddtve semgroup GG8-6
the (termwse) sum of ( ŒI) = ( ŒI) = summto / sum of ( = ŒI) = summto / sum for I of = d d df wh jj ( ŒI) s oempty fte fmly ŒI jœ ddtve commuttve semgroup & = c I & j: Æ I Œ bjecto ote: f I s lso totlly oered set, the j s uquely defble s oer - somorphsm & the commuttvty of the semgroup my be dropped GG8-7
the (elemetwse) sum of A = A d = summto / sum of A = ( Œ A) = df ŒA wh Asoempty fte subset of ddtve commuttve semgroup GG8-8
N. the sgm summto otto for seres the sum of the rght seres + + = m+ = summto / sum from equls m to fty of = lm ( Œvr Z[m, Æ)) e m = d =m df wh m ŒZ & m Æ =m,, m+ =m re elemets of ddtve topologcl semgroup GG8-9
the sum of the left seres + + = d = lm ( m Œvr Z(, ] ) e, = = summto / sum from equls to mus fty of df wh ŒZ &, - = - mæ- =m - = re elemets of ddtve topologcl semgroup GG8-0
the sum of the bseres + + + + = = d = summto / sum from equls mus fty to equls (plus) fty = lm ( m, Œ vr Z & m ) e df wh - 0 = = - Æ mæ- =m ŒZ, -, 0,, re elemets of ddtve topologcl semgroup GG8-
N. the sgm summto otto wth two or more summto dexes = dces; here re three exmples, j= = summto / sum from, j equl to of = sum of summds from to = summto / sum from, j, k equl to of = sum of summds from to j jk, j,k=, j= <j j 3 = summto / sum from, j equl wth less th j to of j = sum of C = ( - ) summds from to t s uderstood tht s postve teger & tht the ' s re from ddtve commuttve semgroup GG8 - -, j jk
D lws for elemets ( Œvr P) of y ddtve semgroup specl rge lws 0 = Œ = = 3 = etc = = = 0 e = = + = + + 3 df GG8-3
D lw for elemet of y ddtve group the rght dstrbutve lw for multples Ê Á Ë = ˆ = ( ŒP & ŒZ for Œ) = GG8-4
D lws for elemets ( Œvr P) of y ddtve commuttve group ( ŒP) the egto lw - = - = = the left dstrbutve lw for multples = ( ŒZ) = = GG8-5
D lws for elemets (, jœvr P) of y commuttve rg ( ŒP) the frst squre - of - sum lw Ê Á Ë ˆ = + = = j. = < j j the seco d squre - of - sum lw Ê Á Ë = ˆ = (- ) + ( =, j= <j + j ) the th squre - of - sum lw Ê Á Ë ˆ = - ( -j) = =,j= <j GG8-6
D lw for elemets, b ( Œvr P) of y ddtve commuttve semgroup ( ŒP) the ddtve lw = = = ( + b ) = + b GG8-7
D lws for elemets, b ( Œvr P) of y ddtve commuttve group ( ŒP) the subtrctve lw = = = ( -b ) = - b the ler lw = = = ( + bb ) = + b b (, bœz) GG8-8
D the boml theorem y commuttve rg ( ŒP) frst form ( + b) = =0 Ê Á ˆ Ë - b secod form ( + b) = =0 Ê Á ˆ Ë - b th form ( + b) =, j=0 +j=! b! j! j ote: the th form suggests the troml theorem whch hs three dexes & geerl the multoml theorem whch hs sy r dexes GG8-9
D. the Beroull umbers B 0, B, B, re defble by geertg fucto x (x Œrel vr) x e - s follows: e x x - =0 x = B (for x er 0)! whece B, B =, B = 0 = -, B3 = 0, B 4 = - 6 30 B5 = 0, B 6 =, 4 B 7 = 0, B 8 = - 30, B 9 = 0, 5 B 0 =, 66 GG8-0
D the complex fuctos expoetl, se, cose re defble s everywhere - coverget power seres s follows: z z z exp z = = + z + + + (" z ŒC)!! 3! s z = =0 (-) =0 + 3 3 5 7 z z z z = z - + - + (" z ŒC) ( + )! 3! 5! 7! 4 6 z z z z cos z = ( -) = - + - + (" z ŒC) ( )!! 4! 6! =0 GG8-
D exmple; the Luret seres of complex fucto tht s lytc o the puctured ple d tht hs essetl sgulrty t the org (z Πvr C) exp z + exp z = e = + z z z + + + + + + + + 3!! 3!! z! z 3! z = = = = z + e z 3 3 z z z + + + + + + + + 3 3! z! z! z!! 3! z!!z =0 =0 =0 - z! + z! =0 = - - GG8- + z! ++ z! =
N. the cptl p producto otto = = for the product of the elemets,,, ( ŒP) of multplctve semgroup s logous to the cptl sgm summto otto = = + + + for the sum of the elemets,,, ( ŒP) of ddtve semgroup eg the fctorl of ( ŒP) = fctorl = d =! fctorl = bg = r = df r= - r=0 ( - r) = 3 = (-)( -) = the product of the frst postve tegers GG8-3
ote: just s the sum of the empty set of ddtve semgroup elemets s defed to be the ddtve detty elemet zero e e Π= 0 e so logously the product of the empty set of multplctve semgroup elemets s defed to be the multplctve detty elemet uty e e Π= df df e GG8-4
C. the otto! for fctorl ws troduced 808 by Chrst Krmp of Strsbourg, Frce; 8 Guss troduced cptl p P to deote cotued products; the dcl otto ws dded lter by others; the Greek letter p Pp correspods phoetclly & trslterto to the Lt - Eglsh letter pee P p whch s the tl letter of productum (Lt) = product GG8-5
D. two dul kds of fctorl powers of x where x s elemet of commuttve utl rg & ŒP the rsg th fctorl power of x = x = x rsg (fctorl) = d df - r=0 ( x+ r) = x(x+)(x + ) ( x+ -) whch hs exctly fctors the fllg th fctorl power of x = x = x fllg (fctorl) = d df - r=0 ( x- r) = x(x-)(x-) ( x- + ) whch hs exctly fctors GG8-6
D the three boml formuls / theorems for three kds of powers commuttve utl rg ( ŒP) the boml formul / theorem for ory powers Ê ( + b) = Á ˆ Ë r r= 0 - r b r the boml formul / theorem for rsg fctorl powers Ê ( + b) = Á ˆ Ë r r= 0 - r b r the boml formul / theorem for fllg fctorl powers Ê ( + b) = Á ˆ Ë r r= 0 - r b r GG8-7
D three exmples of fte products Ê - ˆ = Ë = Ê ˆ Á - = Ë ( ) p = Ê ˆ p Á - = Ë ( + ) = 4 ote: the frst product s the product of the secod product d the th product GG8-8