whittaker hypergeometric

8/6/2019 whittaker hypergeometric

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1 9 0 3 . ] G E N E R A L I Z E D H Y P E R G E O M E T R I C F U N CT IO N S . 1 2 5

A N E X P R E S S I O N O F C E R T A I N K N O W N F U N C T I O N S A S G E N E R A L I Z E D H Y P E R G E O M E T R I C F U N C T I O N S .BY MR. E. T . WH ITTAK ER.

( Rea d be fore th e Am er ican Ma them at ica l S oc ie ty , A ugus t 31 , 1903. ) 1. Introduction

It has long been known that many functions (e. g.y the logarith m and the Le ge nd re functions) are partic ular cases of thehypergeometric function, and that other functions (e. g., theexpo nential function and th e Bessel functions) are l imitingcases of the hyp erge om etric function. Th e object of the pre sent pape r is to show that the latt er class of functions is m uchlarge r tha n has h ither to been supposed ; th at, in fact, i t includes (in addition to those functions already know n to belongto it) the following five types of functions, namely :1. Th e functions which a rise in harmo nic analysis in connection with the parabolic cylinder.2 . T he err or function, which arises in connection with thetheories of pro bab ili ty, errors of observations, refraction, andconduction of heat.3 . Th e incomplete gamm a funct ion, s tudied by Leg endre ,Hocevar , Sch lmi lch , Prym, and Valuer .4 . T he logar i thm integral , d iscussed by Eu ler , Soldner ,G auss , Bessel, L ag ue rre , and Stielt jes, which is applied in theexpression of th e nu m ber of prime s less than a given nu m ber .5. The cosine integral, studied by Schlmilch and Besso.T he first result obtained in the pres ent pap er is tha t all thefunctions above named can be derived by specialization and transformation from a single new function.T his function is denoted by Wh^m(z), and the various functions above mentioned arise from it by taking special relationsbetween the parameters k and m, and transform ing the functions thus obtained : thus, the functions of the parabolic cylinders are der ived by takin g m = J , the incomplete gamm afunction is derived by tak ing h + m = J-, an d so on . I t isfurth er shown th at the Bessel functions are also cases of thesame function W k (z).


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1 2 6 G E N E R A L I Z E D H Y P E R G E O M E T R I C F U N CT IO N S . [ D e c ,It is evident, therefore, that if a general theory can be foundfor the function W k m(z), then all the pro pertie s of th e specialfunction s in question can be deduced as m ere corollaries. T hi sis effected in 3 of th e pres ent pap er, where it is sh own t ha tthe function Wk^m(z) is really a limiting case of the hypergeometricfunction (some of the exponents of the hypergeometric functionbecom ing infinite in a ce rtain wa y), and tha t it possesses atheory which can be derived from that of the hyp ergeo m etricfunction. I t appe ars, how ever, tha t the function W k m(z ) has,

in the course of the limit process by which it is obtained, acquired certain new prop erties which are n ot possessed by theparent hypergeometric function ; these properties are given.The general family of functions Wk,Jz) includes other classesof functions besides the five typ es of k now n functions alrea dymentioned, and attention is drawn to some of these classeswhich have not hitherto been investigated. 2 . Expression of the Parabolic Cylinder Functions, ErrorFunction, Incomp lete Gam ma Functions, LogarithmIntegral, and Cosine Integral by meansof a Single Function.

W e now proceed to define a function in term s of w hich th evarious kno wn functions alread y men tioned can be expressed.The function in question will be denoted by W k m(z), and will bedefined for a ll v alu es of z (real or com plex) by the eq uationTSt +ir^ r * + ^ * f ( - f)-*-x+ f 1 + -- \~AJrme-fdt,

where the path of integration begins at t = + oo, and after en circ l ing the point t = 0 in th e counter-clockwise direction retu rns to t = + oo aga in. T h e -plane is supposed to be di ssected by a cu t from t =* 0 to t = + oo, and the express ion( t)-k-yz+m is defined to meane(-k-yz+m) log (-*)

w her e th e rea l v alu e of lo g (- t) is taken when t is real andnega t ive .When the real par t of k - \ + m is posit ive, the path ofinteg ration can be deformed so as to coincide with the real axisin the -plane, and then (as is easily seen) the complex integralcan be replaced by the real integral


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1 9 0 3 . ] G E N E R A L I ZE D H Y P ER G E OM E T R IC F U N CT IO N S . 1 2 7W k> m{z) =

= r * e~y** t-*-*+m ( 1 + - ) er 'cK.r(A + + m) j 0 v ZJWe shall now show that the functions of the paraboliccylinder, error function, etc., can be expressed in terms of thisfunction WhiJz).1. The functions of the parabolic cylinder.The standard function associated with the parabolic cylinder

in harmonic analysis is * ~ r ( " - i )wh ere the integra tion is take n along a pa th which begins andends at infinity in the direction which ma kes zH positive, andwhich encircles the point t^= 1. Th is can be w rit tenz^D^H(V2z)

and on writ ing t = 1 + 2u/z in the integral and comparing withthe definition of the function Whm(z), we have

It follows that the parab olic cylinder f unctions are expressedin terms of the function Wfctfn(z) by this equation, and in factcorrespond to the particular case m = J .2 . The error function.T he erro r function is defined by the int eg ral

' e~ t2dt .Th is function can be expressed in term s of the function

Wkjm(z). For from the definition of W k% m (z) we have*dt_Ht *(*) * * r * ( 1 + ~ y A e ~ * c

= 2z^e~y^ H (f^-^dv, where v2 = 1 + l2 >*Cf . a pape r by the author in Proc. Lond. Math. Soc, vol. 35, p. 417.


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1 2 8 G E N E R A L I Z E D H Y P E R G E O M E T R I C F U NC T IO N S. [ D e c ,

X CK > e~ s2ds, where s = vz.It follows that the error function

Ce-*dt*J z

can be expressed in terms of the function W k m(z) in the form

Ce rtain other integra ls discussed by Riem ann in connectionwith the conduction of heat, e. g.} the integral b e-*-*Vt"dt

can be evaluated in term s of the error function, and so in term sof the function Wk>m(V).3 . The incomplete gamm a function.The incomplete gamma function is denoted by y(n, z)} and isdefined by the equation

y(n, z) = f t"-le-*dt.I n orde r to expres s this in term s of the function W k m(z)9 weobserve that from the definit ion of the latter function we have

= zm-w f V - V d , where s = t + z,*Jz= 2;(i-)e*{r(n) y(n} z)}.

It follows that the incomplete gamm a function can be expressedin terms of the function W k m(z) by the relationy(n, z) = T{n) - * * - - * ' W ^ , , , *(*).

4. The logarithm integral.The logar i thm in tegra l li{z) is defined by the equation


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1 9 0 3 .] GENERALIZED HYPERGEOMETRIC FUNCTIONS. 1 2 9The name was given by Soldner, and tables of the functionwere given by Bessel and Gauss.W r i t i n g z = e~* and i = e~u, we have

'* e~uduu

B u t= i n + j J e~scf, where s = u f.

Thereforeor Ae logarithm integral is expressed in terms of the functionWkm(z) by the relation

li(z) = ( log z)~*z* W_iAf0(- log z) .5. T/ie cosine integral.T h e cosine int eg ral C(s) is defined by the equation

*cos tCS ( * ) = f' dt.V

It can be expressed in terms of th e function W k Jz) in thefollowing way. W e haveCi(z) = i + J

P u t t i n g t = is in the first of these integrals, and t = is inthe second, and remembering thatr.s zero when the pa th of integ ration is w holly at infinity, wehave 'e-'ds r~ ize~ sdsC K , = i f r ^ + i f : s= iK(") + iU{e-^)= \z-^e^+i^W_y^(- iz) + \z-y^-y^-^W_y^(iz\


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13 0 GENERALIZED HYPERGEOMETRIC FUNCTIONS. [ D e c ,which exp resses the cosine integral in term s of the functionWKm(z).6. I t m ay be added t ha t th e Bessel functions can w ithoutmuch difficulty be shown to be likewise expressible in terms ofthe function W k m(z). The actual relation will be found to beJJz) = (2irz)-*{^x**W0tJ2k) + e-(m+y*v/2W0>m(-- 2iz)}.

3 . Properties of the Function W^m(z).W e shall now establish the principal prop erties of the function WktJz).1. The differential equa tion satisfied by the function Wkm(z).I f we wri te / f\k-%+m

t- k~x+ ml 1 + - 1 e-'dt,where th e integratio n is tak en along a path wh ich begins att =s oo, encircles the point t = 0, and re turns to t = oowe haved2v (2k \dv | ~ m 2 + (&-l)

+ k \ m+ (1 2m)-l


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1 9 0 3 . ] GENERALIZED HYPERGEOMETRIC FUNCTIONS. 1 3 1and on subst i tut ing this value for v in the differential equationwe see that Whm{z) satisfies the equation

~a + (k I m 2

) w=o.It may be noted that any differential equation of the form

dz2'

can be bro ug ht to this form, and therefore solved in term s ofthe functions Wkj7n(z).2 . Reduction of the function W k m(z) to a lim iting ease ofthe hypergeometric function.The functions W kt m(z) have in general an irregu lar singu larity (using the word " irre gu lar " in the sense cu stoma ry inthe the ory of linear differential equ ations) at s = oo, an d aretherefore not m em bers of the family of hype rgeom etric func

tions, which are characterized by the fact that their singularit iesare all reg ular . I n spite of this, how ever, the functions Wkm(z)can be deriv ed from the hyperg eom etric function by a l imitprocess, name ly, mak ing the expon ents at two of i ts singularit iesinfinite, and making the two singularities in question to coalesceat infinity.F o r the differential equation co rrespond ing to the hy per geometric function0 c

c kk

is easily found to ten d, for c = oo, to the limi ting formd2l + d?L +dz2 dz

and on pu t t ing y = e~y*zv in this, we obtaindhdz2 + [- i + - +zk J m 2! . . 0,


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1 3 2 G E N E R A L I Z E D H Y P E R G E O M E T R I C F U N CT IO N S , [ D e c , ,which is the differential equation satisfied by the functionWktm{z).3 . Othe r solutions of the differential equa tion of the functionWhiJz).I t is clear tha t the differential equation of the functionWkm(z) is unch anged if m be replaced by m ; it is also unchanged if k be replaced by Jc , provided z be replaced by z at the same t ime. H enc e the f our functions

W ki Jz), W K _m(z), W _kt m( - z) , W__k^ _m( - z) ,are solutions of the differential equa tiond2y ( h i m 2 \

Cases of this resu lt are the well-kno wn theorems tha t Jn(z)and J_ n(z) each satisfy BessePs equ ation , an d th at DJz) andD_n_x{iz) each satisfy the equation of the parabolic cylinder.4 . The asymptotic expansion of W k m(z).We haveWhtm(z) =

2TT / A &-#+and it is easily seen that the divergent series which is obtainedby exp and ing the qua nt i ty (1 -f t/z)k~1/2+m by the binomialtheorem, and then integrat ing term by term, is the asymptot icexpansion of the integral for large posit ive values of z. Thi sseries is , m 2 - ( ^ - J ) 2^ 1! z

m' - ( * _ ! ) } { m * - ( * - ! ) }+ 2 ! z2 ^This is therefore the asymptotic expa nsion of the f unction Wktm(z).The wel l -known asymptot ic expansions of the logar i thminte gra l, erro r function and Bessel functions, and the asym ptotic expansion of the parabolic cylinder functions given bythe author in Proceedings of the London Ma thematical Society,volum e 35 , page 4 17 , are cases of this expan sion.


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1 9 0 3 . ] G E N E R A L I Z E D H Y P E K G E O M E T R I C F U N C TI ON S . 1 3 35. Further prop erties of the f unctions Wkm(z).The remaining principal properties of functions of the typeW k m(z ) will be stated without proof, as the reader will withoutdifficulty be able to sup ply dem onstration s.(a) A second definite integral expression for Wk^ Jz) is

the inte gration being taken as before along a pa th which beginsand ends a t t = + oo an d encircles t = 0 .(6) The function W kf m(z) degenerates into an elementary function when k J db m is a positive integ er or zero ; the theore mthat the Bessel function degenerates into an elementary functionwhen its orde r is half an odd integ er is evid ently a partic ularcase of th is re sult.(c) T he function Wkm(z) degenerates into the indefinite integralof an elementary function when k J m is a negative integer ;the incomplete gamma function, logarithm integral, error function, and cosine integral, are all instances of this theorem.(d) Considering the case in which Wkm(z) degenerates intoan elementary function, if Whl,Jz) and Wk^m(z) are two functions of this k ind , with different para m eters k but the sameparameter m, the integral property

Jo zcan be established.(e) Recurrence formulas also exist : if there are a ny threefunctions W ki m(z ) such th at the correspon ding values of k differby integ ers, and the values of m also differ by intege rs, thenthe three functions are connected by a l inear relation, the coef-ficients in which are polynomials in z.

6. New classes of functions.A s has been seen abo ve, a nu m be r of functions which arereally members of the family of functions W ki m(z) have in thepast been separate ly discovered and investiga ted. Th ere are

other mem bers of this family wh ich have not hithe rto been no ticed,bu t which give promise of interes t ing proper t ies . Am ong thesem ay be m entioned th e families of functions for wh ich m = 0,


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1 3 4 T H E F A C TO R IN G O F L A R G E N U M B E RS . [ D e c ,an d those for which m = J. W ith regard to the former ofthese classes, for example, i t may be shown that if an arbitraryfunction (#) be expan ded in the form

(s) = a0 W 1/2, of + aLW3/2>of + a2 W w% 0(z) + -,then the coefficients are given by the relation

TRINITY COLLEGE, CAMBRIDGE,July, 1903.

O N T H E F A C T O R I N G O F L A K G E N U M B E R S .B Y P R O F E S S O R F . N . C O L E .

(Read be fore the Amer ican Mathemat ica l Soc ie ty , Oc tobe r 31 , 1903. )1. IN resolving a large n um ber i^ in to i ts pr ime factors , atable of quadratic remainders of N can be made to render efficient service in several different ways. F o r a twenty-tw oplace N, the remainders of the table may be restricted to products of about seventy of the smallest prime numbers available.If, for example, Nis always expressed in the form N= x2 =h a,with var iable x and a, the rem ainde rs =p a will contain onlythose prime s of which N i s quad rat ic remainder . By a

gra du al elim ination of common factors from the rem ainders =p a,we finally obtain a tab le of remainders not ad m itt ing furtherreduc tion. O the r forms for expressing N } such as N= 2x l 6,are of course often adva ntage ous, according to circumstances.I f the final table of remaind ers consists en tirely of the in dividual primes employed, each with the proper sign + or , JVis und oub tedly a prime num ber * ; in fact a mu ch sm aller sequen ce of prim e rem aind ers wo uld suffice to justify thi s conclusion, the wide range specified above being required only toensu re the success of the pre lim inary elimination process. ' If ,on the other han d, i t is found impossible, on repeated attem pt,

* F o r an inte res t in g discuss ion of t he use of the tab le of rem aind ers see P.Seelhoff, Zeitschrift fr Mathem atikund.Physik, vol. 31 (1886), pp. 166, 174, 306.

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whittaker hypergeometric