On the Roots of Bessels Functions

MATHEMATICS: J. H. McDONALD

66

PROC. N. A. S.

put it, by (1 *. e) electrons in passing from the compressed to the uncompressed metal. Working from 1 atm. to 2000 kgm./cm.2 and from 00 C. to 1000 C., he found this absorption to be positive throughout in fourteen of the metals; negative throughout in three, cobalt, magnesium, and manganin; mixed, sometimes positive and sometimes negative, in -three, aluminium, iron, and tin. In terms of my theory a positive effect here can be accounted for by a decrease of (kf +. k) X under pressure, and a negative effect by an increase. It seems probable that (kf . k) is generally decreased by pressure, and that the increase of (kf . k) X indicated for certain cases by Bridgman's experiments is to be attributed to an increase of X sufficient to overbalance the decrease of (kf . k). A pridri one might expect X to decrease with increase of pressure, causing reduction of volume, since, according to the formula X' = X' + sRT, it increases with rise of temperature, causing expansion. But it is unsafe to assume that a contraction caused by pressure will have the same effect on the properties of a substance as a contraction caused by fall of temperature. Thus Bridgman says :2 "The volume of many metals at 00 C. and 12,000 kg. [per cm.2] is less than the volume at atmospheric pressure at 00 Abs. The resistance of most metals tends towards zero at 00 Abs., but at 00 C. at the same volume the resistance is only a few per cent less than under normal conditions." A change of about 8% in the value of (kf . k) X would account for the maximum Peltier effect between compressed and uncompressed bismuth, as observed by Bridgman, and a still smaller per cent change, in most cases much less than one per cent, would serve for the other metals dealt with in this paper. 1 These PROCSUDINGS, October, 1920, p. 613. Proc. Amer. Acad. Arts & Sci., 52, No. 9 (638).

2

ON THE ROOTS OF BESSEL'S FUNCTIONS By J. H. MCDONALD DVPARTMUNT OF MATHUMATICS, UNIVERSITY Or CALIFORNIA Communicated by E. H. Moore, December 13, 1920

The roots of the equation J(z) -= 0 are known to be all real if v < -1. The methods of Sturm when applied to the function J,(z) show that the roots are increasing functions of n if n> 0, that is to say, denoting by y6k(n) the kth positive root 4I,k(n')> i'k(n) if n'>n>0. In the following it will be shown that Ok(n')> Ok(n) if n'>n> -1 so that the inequality holds as well when -1
gr(z)

are to be considered.

They satisfy the recurren'ce

'67

MATHEMATICS: J. H. McDONALD

Vot. 7., 1921

formula g, + 1 = (n + v + 1)g, + zg1 i with initial values g..I = 0, g. = 1 and are connected with fn by the relation fn = g,f + + zg, - If +v + 1. Putting Av = g,g,' + l-g, + 1 g,' where differentiation is with respect to z it is known (Hurwitz, Math. Ann., 33, 1889, p. 246) that A, +2 = (n + v + 2)g2, + 1+z2A, I= n + 1, A2 = (n + 1)2(n + 2) = - and A1 = n + 1, it follows that v+lis an in and since-( dz \g, / gy21 creasing function of z if n + 1 > 0. If

differentiation

is taken with

respect

to

n

and

D,

= g, dn+

_ Do l D1 (n + 1)2-z. gy- ZD pl Pit isfound that D =, dn with use of the + For, from dg =g + (n + v + 1 dn dn dn . equation g, +1 = (n, + v + 1) g, + zg, -1, it follows that g,, d?t+ I dg,.. If a negative value isas= 2 -Z -( dgI g_ I dn dn dn signed to z D,> 0 and gv + 1 is an increasing function of n, with the con-

gp + l

dgn

-

9P dition as before n + 1 > 0. From these properties of g, + 1 it follows if z1 is a root of g, = 0 that

or <0 according as g,, changes from negative to positive or the reverse when x increases through z1. Denoting the dependence of g, on n by gn it follows if n' is slightly greater than n that g'(zi) >0 (Z,) in the first case and <0 in the second because in both cases gg,' >I( )

g,,- I (z1) >0

gP (zi)

= O. If

z,' denotes the root of g' (z) = 0 which differs slghtly

r- 1 (Z,)

IziJ.

The roots of fn = 0 from z1 it follows in both cases that Izl'I> = 0, v 1,2...., hence, are known to be the limits of the roots of gn (z) = 0 and Pk' of fn = 0, |Pk'| > if Pk denote the kth root of fn is the theorem stated at the This The equality can easily be excluded. z and if n + 1>0, all the roots that <0 beginning. It has been assumed z1 are <0 in fact. A formula of another kind may be obtained as follows: Differentiating the equation fA = g,fn + + zg, - lAf + +1 and equating the result to fA + 1 expressed in terms of fn + v + 1, fn + v + 2, and comparing coefficients it is found that ge+1 = gn + (n + v + 1)(g')' + (zgx- 1)' and gn +' gn = (gn+ 1)' An application of the second equation may be made to the calculation

|Pkl-

,

MATHEMATICS: J. H. McDONALD

68

PROC. N. A. S.

of the roots of f4 = 0. Assuming n + 1>0 as before, so that the roots of gn = 0 are all eal aand negative, let zi, Z2, 33 be the kth roots of g, + 1 g" and gn+l = 0, then z3>z2>z3 and the functions g all change in the same sense from negative to positive or the reverse when z passes through the corresponding root. If z' is the kth root of (g' + l)' = 0, from the above equation gn + ' (z') = gn (z') and it can be.seen that (g' + I(z2))> >0 or <0; according as g, + 1 changes from negative to positive or the reverse when z passes through zi. This allows a series of approximations to a root of fn = 0, say the first one, as follows: Let Z2 be the root of

Z2- g3 (2), Z4 = Z3- 4 (Z3) etc. These g4,(33), g9(Z2) z's converge to the first root of fn = 0. In the case of the first root the approximations are rational functions of n. g2

=

0; form the quantities X3

=