Yasuda, Ryo

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RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:A Theorem Relating to the Orders ofa Transcendental Integral Functionof Two Independent VariablesYasuda, RyoYasuda, Ryo. A Theorem Relating to the Orders of a Transcendental Integral Function ofTwo Independent Variables. Memoirs of the College of Science, Kyoto Imperial University.Series A 1924, 7(5): 345-3481924-10-20http://hdl.handle.net/2433/256681A Theorem Relating to the Orders of a Transcendental Integral Function of Two Independent Variables. By Ryo Yasuda. (Received February 19, 1924) Definition. Let f(z, z') be a transcendental integral function of z for any z' in a finite domain D(z'), and be a regular function of z' in D(z') for any finite z. If p be a finite number such that i) for any value of z' in D(z') and for any prescribed positive value e, there corresponds a positive value R=R (e,z') such that I I p +E f(z,z') L1 for all lzl=r::::,,.R, ii) for a certain value z' in D(z') and for any prescribed posi-tive value e, I I p- E f(z,z') )e7 for infinitely many values of I z I =r which diverges without limit. p is called the order off (z,z') in z for z' in D(z'). Similarly we may define the order of f (z ,z') in z'. Definition. Let E(z,z') be a field such that z' is any point in a finite domain D(z') while z is any finJ.te point in the whole Gauss plane, and f ( z ,z') be a regular function of z and z' in E(z ,z'). Tf f (z ,z') be a transcendental integral function of order p in z such that for any prescribed positive value e and for all z' in D(z'), there cor-346 Ryo Yasuda. responds a positive value R (independent of z') and we have for all izl =r~R, then f ( z ,z') is called a uniformly increasing function of the pth order in z for all z' in D(z'). Note. If f (z,z') be a uniformly increasing function of the pth order in z for all values of z' in the vicinity of any point in D(z'), it will also he true, by an extension of the Heine-Borel theorem\ for all z' in D(z'). Theorem. Let f (z,z') be a transcendental integral function of z and z' and be a uniformly inrreasing Junction of the p'" order in z for all z' in a finite domain D(z'). Let S={a1, a2 , a3 , •••••• } be a set of points in the z-plane. If f(ai, z')=P;(z'), (i=l, 2, 3, ........ ), where l\(z') is a polynomial of the n!" degree (ni< N: a finite number), then there will be no limiting point of S at finiteness and the exponent of convergence of I ai I, (£= 1, 2, 3, ....... }, will not be greater than p. A f( ') P(') c· 1 ) iJNf(aiz') of'. 1 s ai, z == i z , t= , 2, 3, ...... , i)z'N - or t== , 2, 3,. . . . . . . . Accordingly S has no limiting point at finiteness, unless f (z ,z') should he a mere polynomial of z' for any value of z, which is contrary to the assumption. Thus the first part of the theorem is proved. As f (z,z') is a uniformly increasing function of the p'" order in z for all z' in D(z'), we have p+a I iJf(z,z') 1-1 1 r f(z,C) d"I L.__!__ L. /+e' for all izl =r~R. i)z' - 21rJ-IJc(c--z')2" - ,,-e where C is a circle contained in D(z'), centre the point z' and radius equal to r'. For a fixed value of r', e' may be taken as small as we iJJ(z,z') please. Accordingly ~~-- is a transcendental integral function at i)z' 1 cf. E. W. Hobson: The theory of functions of a real varirble (1907), p. 88. A Theorem Relating to the Orders of a Transcendental Integral etc. 347 most of the p'" order in z for any fixed value z' in D(z') : and so also iJNJ(z,z') for iJzN As a.=l=aJ for i-=t=j, we may assume that la1 i < ja2 j < ja3I < .. • .1 • There are many transcendental integral functions of z, which have simple ze~o points at Sand no others. Let g(z) be one of the lowest order among them. Then the exponent of convergency of I a, I, (i= 1, 2, 3, ........ ), is equal to the order of g(z). As S is the set of the simple zero points of g(z), g'(a.) =t=O, (i= 1, 2, 3, ........ ). Put p -I ,, ( 1 1 . z z • ) P.(z') E.(z,z )== -+-+2 + ........ +- '( ) , ~~ ~ ~ ~ g~ and if a1=0, put P1(z') g'(o) If Max I P;(z') I in D(z') be Ai, we have z L 1 I zP• I• I A / for I z I < I a. I . -1-1 :t I aft+• g'(a,) Hence, we determine p, so as to satisfy for lzlL la.I - 2 , 00 where e, >O and ~e. is convergent. 00 i-1 Then ~E.(z,z') is absolutely and uniformly convergent in a field i-1 E(z,z') (such that z' is any point in D(z'), while z is any finite poii;it t If the exponent of convergence of la.ti, (i=1, 2, 3, ......... ), be transfinite, we take a subset S1={b1, b2, b3, ......... } of S, instead of S, for which [bil<[bJ/, (i<j), and the exponent of convergence of lb,I, (i=l, 2, 3, ...... ), is finjte and is ~reater than p, 348 Ryo Yasuda. in the whole Gauss plane) except the vicinity of z=ai, (i=1, 2, 3, .. . . . . . . ). 00 Put g(z) ~Ei(,~,z')=F(z,z'). i-1 Then F(z,z') is regular in R(z,z') and takes the value Plz') at z=ai, (i =1, 2, 3, ...... ). We now consider the function f(z,z')-F(z,z'). It is regular in E(z,z') and vanishes at S. Hence we may put /(z,z')-F(z,z')==g(z) •H(z,z') where H(z,z') is regular in E(z,z'), and is a transcendental integral function of z for any z' in D(z'). iJNf(z,z') . 1s, as before, a transcendental integral function at most of iJzlN the pth order in z for any fixed value z' in D(z'). As F(z,z') is a iJNF(z z') polynomial at most of the (N-tr degree in z', iJz'; 0. As H(z,z') is regular in E(z,z'), and is a transcendental integral function of iJNH(z z') iJNH(z z') z for any z' in D(z'), oz'; must be one also. For if oz';, iJN f(z z') =O · ' would be also identically zero, which is absurd. Thus - ' Oz1N from the identity o"f(z,z') . we infer that the order of g(z) can not exceed that of · , z. e., i)zlN the exponent of convergency of /a./, (i= 1, 2, 3, ........ ), can not exceed p. Q. E. D. The theorem may be easily extended to the case where p is transfinite but less than o>°'= SJ. 

A Theorem Relating to the Orders of a Transcendental Integral Function of Two Independent Variables

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A Theorem Relating to the Orders of a Transcendental Integral Function of Two Independent Variables

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