820 FUNCTION The function cxp u has no finite roots, but an infinity of roots / i each = - oo ; this appears from the equation cxp (! where n is indefinitely large and positive. The function sin u has the roots sv where s is any positive or negative integer zero included ; or what is the same thing, its roots are and sir s now denoting any positive integer from 1 to co ; each f ifi of these is a simple root, and we in fact have sin M - n( 1 - -^
5 " 7r
Similarly the roots of cos u are (s + i) v, s denoting any positive or negative integer, zero included, or what is the same thing, they are (s + i) T, s now denoting any positive integer from to oo ; (It^ 1 .- jTo-i 2 I (S+ 2J" 71 "/ the product each root is simple, and we have cos u n . * - . 2 2
V s 2 J" 71 ",
Obviously tan it, as the quotient sin -r-cos u, has both roots and infinities, its roots being the roots of sin u, its infinities the roots of cos u ; sec u as the reciprocal of cos u has infinities only, these being the roots of cos u, &c. f 7" In the foregoing expression sin u = uU f 1 - -^ must be understood to mean the limit of n"( 1 -^ ) f r an i n de- V s " 7r / finitely large positive integer value of n, viz., the product is first to be formed for the values s=l, 2, 3 . . . up to a determinate number n, and then n is to be taken indefinitely large. If, separating the positive and the negative values of 5, we consider the product uU n (l 4- ) n* ( 1 - ) , (where in the first pro-
i TT/ S T* )
duct a has all the positive integer values from 1 to m, and in the second product s has all the positive integer values from 1 to n), then by making m and n each of them indefinitely large, the function does iwt approximate to sin u, unless m : n bo a ratio of equality. 1 And similarly as regards cos u, the product n"( 1 + - - ) n ( - )> m an d n indefinitely large, does not approximate, to cos u, unless m : n be a ratio of equality. 7. The functions sin u, cos u, are periodic, having the period 2ir ) 111 (u + 2ir = S1U (u; and the half-period ir, S11 * (u + *} Ln u ; the periodicity may be verified by means of the fore going fractional forms, but some attention is required ; thus writing, as we may do, sin it = , where s extends from - n to n, n ultimately infinite, if for u we write U + TT, each factor of tlc numerator is changed into the following one, and the numerator is unaltered, save only that there is an introduced factor n + (n+ l)ir at the superior limit, and an omitted factor u-mr&t the inferior limit ; the ratio of these (u + n+ lir)- (u -nir), for n infinite is - 1, and we thus have, as we should have, sin (w-t- IT) = sin u. The most general periodic function having no infinities, and each root a simple root, and having a given period a, has the form A sin -^- + B cos - , or, what is the same thing, L sm f -^ + A |. a a cos a J 8. We come now to the Elliptic Functions ; these arose from the consideration of the integral l , where R is a rational function J vx of x, and X is the general rational and integral quartic function ax 4 + )S.c 3 -f yj? + Sx + f ; a form arrived at was r dx _ r rf< J VfT^jn:^ J A/i_A2srn4 on putting therein x = sin <j>, and this last integral was represented by F<t>, and called the elliptic integral of the first kind. In the /fly, - = sin- 1 :? and it thus /I - x- appears that Ffy is of the nature of an inverse function ; for passing to the ^direct functions we write F<p = u, and consider (f> as hereby determined as a function of u, $ - amplitude of , or for shortness am u. And the functions sin $, cos <j>, and ^1 - Jfc* sin 8 ^ were then considered as functions of the amplitude, and written sin am u, cos am u, A am u ; these were afterwards written sn u, en u, dn M, which may be regarded either as mere abbreviations of the former functional symbols, or (in a different point of view) as functions, no longer of am u, but of u itself as the argument of the functions ; .MI is thus a function in some respects analogous to a sine, and en and dn functions analogous to a cosine ; they have the corrcspond- 1 Tlie value of the function in question n"(l + ) n m (l + ), A"JT ] * Sir when m, n are each indefinitely large, but -not" 1, is -/-V sin u. n m ing property that the three functions of u + v are expressible in terms of the functions of u and of v. The following formula; may be mentioned : en 2 a = 1 - sn 1 J/, dn- u 1 - A 2 sn 2 u, sn u = en u dn M, en u - sn u dn v, dn u = - k- sn u en u, (where the accent denotes differentiation in regard to u) , and the addition-formulae : sn (w+r) = sn en u dn v + sn f en u dn a, ( ) en (u+r) = en v en r sn u dn u sn v dn r, ( ) dn(tt-fr) = dn u dn v k 2 sn u en u sn v en r, ( ) each of the expressions on the right-hand side being the numerator of a fraction of which Denom. = 1 -K- sn 2 u sn 2 r. It may be remarked that any one of the fractional expressions, differentiated in regard to u and to v respectively, gives the same result; such expression is therefore a function of u + r, and the addition-formulae can 1)3 thus directly verified. 9. The existence of a denominator in the addition-formulae sug gests that sn, en, dn are not, like the sine and cosine, functions having zeros only, without infinities ; they arc in fact functions, having each its own zeros, but having a common set of infinities ; moreover, the zeios and the infinities are simple zeros and infinities respectively. And this further suggests, what in fact is the case, that the three functions are quotients having each its own numerator but a common denominator, say they are the quotients of four 0-functions, each of them having zeros only (and these simple zeros) but no infinities. The functions sn, en, dn, but not the 6-functions, are moreover doubly periodic ; that is, thereexist values 2w, 2v ( = 4A"aml4(AT+7 A") in the ordinary notation), such that the sn, en, or dn of u + 2u, and the sn, en, and dn of u + 2, are equal to the sn, en, and dn respectively of u ; or say that <p(u + 2i*.<} = (t)(u + 2v) = <t>u, where ty is any one of the three functions. As regards this double periodicity, it is to be observed that the equa tions <p(u + 2o>) = <pu, <p(u + 2u) = <pu, imply <$>(u + 2mw + 2nv) = $u, and hence it easily follows that if w, v were commensurable, say if they were multiples of some quantity a, we should have 0(w + 2a) = (pu, an equation which would replace the original two equations <p(u + 2w) = <pu, (t>(u + 2v) = <t>u, or there would in this case be only the single period a ; a> and v must therefore be incom mensurable. And this being so, they cannot have a real ratio, for if they had, the integer values m, n could be taken such as to make Zmw + Znv == k times a given real or imaginary value, k as small as ice please ; the ratio u : v must be therefore imaginary (as is in fact the case when the values are 4ATand 4(AT+rA")). 10. The function sn u has the zero and the zeros mu + ni,, m and n any positive or negative integers whatever, and this suggests that the numerator of sn u is equal to a doubly infinite product (Cayley, "On the Inverse Elliptic Functions," Camb. Math. Jour., t. iv., 1845; and " Memoire sur les fonctions double- ment periodiques, " Liounllc, t. x., 1845). The numerator is equal to m and n having any positive or negative integer values whatever, including zero, except that m, n must not be simultaneously =0, these values being taken account of in the factor u outside the product. But until further defined, such a product has no definite value, nor consequently any meaning whatever. Imagine in, n to be coordinates, and suppose that we have surrounding the origin a closed curve having the origin for its centre (i.e., the curve is such that if a, j8 be the coordinates of a point thereof, then - a, - are also the coordinates of a point thereof) ; suppose further that the form of the curve is given, but that its magnitude depends upon a parameter h, and that the curve is such that, when h is indefinitely large, each point of the curve is at an indefinitely large distance from the origin ; for instance, the curve might be a circle or ellipse, or a parallelogram, the origin being in each case the centre. Then if in the double product, taking the value of h as given, we first give to m, n all the positive or negative integer values (the simultaneous values 0, excluded) which correspond to points within the curve, and then make h indefinitely large, the product will thus have a definite value ; but this value itill still be dependent on the form of the, curve. Moreover, varying in any manner the form of the curve, the ratio of the two values of the double product will be = exp j8it 2 , where /3 is a determinate value depending only on the forms of the two curves ; or what is the same thing, if we first give to the curve a certain form, say we take it to be a circle, and then give it any other form, the product in the latter case is equal to its former value into exp. /Sit 2 , where /3 depends only upon the form of the curve in the latter case. Considering the form of the bounding curve as given, and writing the double product in the form nn /f^^ ,
