FUNG T ION 819 cxp iy cos y + t sin y; and if we herein for y write c:, and multiply the two expressions together, observing that the product will be = exp i (y + z), we obtain the fundamental equations cos (y + ?) coa y cos z sin y sin * ; sin (y + 2) = sin y cos z + sin cos j/ , for the functions sine and cosine. Taking y as an angle, and defining as usual the sine and cosine as the ratios of the perpendicular and base respectively to the radius, the sine and cosine will be functions of y ; and we obtain geometrically the foregoing fundamental equations for the sine and cosine ; but in order to the truth of the foregoing equation exp iij = cos y + i sin if, it is further necessary that the angle should be measured in circular measure, that is by the ratio of the arc to the radius ; so that it denoting as usual the number 3 14159 . ., the measure of a right angle is =)fir. And this being so, the func tions sine and cosine, obtained as above by consideration of the exponential function, have their ordinary geometrical significa tions. 3. The foregoing investigation was given in detail iu order to the completion of the theory of the irrational function u m . We hence forth take the theory of the circular functions as known, and speak of tan x, &c. , as the occasion may arise. We have x + iy = r(coa + i sin 0) , where, writing Jx- + if to denote the positive value of the square root, we have r = V/.T* and therefore also cos0= /====-, 6in0= V* 2 + S tan = Treating x, y as the rectangular coordinates of a point P, r is the distance (regarded as positive) of this point from the origin, and is the inclination of r to the positive part of the axis of x ; to fix the ideas may be regarded as lying within the limits 0, IT, or 0,-ir, according as y is positive or negative ; is thus completely determinate, except in the case, x negative, y = 0, for which is = TT or- IT indifferently. And if u x + iy, we hence have = (x + iy) i V m - r m , + 2s + i sin + IS - 1air m ) where r^ is real and positive and s has any positive or negative integer value whatever, but we thus obtain for u* only the m values corresponding to the values 0, 1, 2 . . . m - 1 of s. More generally we may, instead of the index -, take the indjx to be any rational fraction -^. Supposing this to be in its least terms, and m to be positive, the number of distinct values is always = m. If instead of ? - we take the index to be the general real or complex quantity m, we have u m , no longer an algebraical function of u, and having in general an infinity of values. 4. The foregoing equation exp (x + y) = exp r . exp y is in fact the equation of indices, a x + y = a*. <ti> ; exp x is thus the same thing as c x , where c denotes a properly determined number, and putting c* equal to the series, and then writing x = 1, we have c = 1+ "i + r2 + nT3 + &c-> tllat is> e = 2 7128 --- But a 8 wcl1 theoretically as for convenience of printing, there is considerable advantage in the iise of the notation exp u. From the equation, exp iy = cos y + i sin y, we deduce exp ( - iy) = cos y - i sin y, and thence c 9 V= {exp (i + cxp (-iy) j- , If ") sin y = J exp (iy) cxp ( iij) y ; 2 I J if we write herein ix instead of y we have cos ix -jcxp x + cxp ( x) j- sin ix - ^ -jexp ,r cxp (- .r) | , Viz., these values are cos ix = l + T - + 1.3 1.2.3.4 i S " f2l each of them real when x is real ; the functions in question + . . and x + q-^-o 4- . . . , regarded as func- 1 4- - - a. 1.2 + 1.2.3.4 1.2.3 tions of x, are termed the hyperbolic cosine and sine, and are repre sented by the notations cosh x and sinh x respectively; and simi larly we have the hyperbolic tangent tanh x, &c. : although it is easy to remember that cos ix, - sin ix, are in fact real functions of a; i and to understand accordingly the formula; wherein they occur, yet the use of these notations of the hyperbolic functions is often con venient. 5. Writing u - exp v then v is conversely a function of u which is called the logarithm (hyperbolic logarithm, to distinguish it from the tabular or Briggian logarithm), and we write v = log u, or what is the same thing, we have u = exp (log u) (and it is clear that if u be real and positive there is always a real and positive value of log u, in particular the real logarithm of c is = 1) ; it is however to be observed that the logarithm is not a one-valued function, but has an infinity of values corresponding to the different integer values of a constant s ; in fact, if log u be any one of its values, then log u + Isiri is also a value, for we have exp (log u + 2sirt) exp log u exp 2siri, or since exp Zsxi is= 1, this is- u; that is, log u + 2sirt is a value of the logarithm of u. We have tu = cxp (logttt) = cxp log u . cxp logr, and hence the equation which is commonly written log p = log u + log r , but which requires the addition on one side of a term 25m . And reverting to the equation x + iy = r (cos 6 + i sin 0), or as it is con venient to write it, x + iy = r exp i0, we hence have log (x + iy) = log r + i(0 + 2s7r), where log r may be taken to denote the real logarithm of the real positive quantity r, and 6 the completely determinate angle defined as already mentioned. Reverting to the function w m , we have w = exp log u, and thence u m = exp (m log u), which, on account of the infinity of values of log u, has in general (as before remarked) an infinity of values ; if u = c, then e M , = cxp (m log c), has in general in like manner an infinity of values, but in regarding c" as identical with the one-valued function exp m, we take log c to be = its real value, 1 . The inverse functions cos 1 a-, sin- 1 ^, tan -1 x, are in fact logarith mic functions ; thus in the equation cxp ix = cos a; + i sin x, writing first cos x = u, the equation becomes exp i cos~ ] = u + ifl - u?, or we have cos~ J = --. log (u + ?Vl -tt 2 ), and from the same equa- 1 tion, writing secondly sin x = , we have sin"" 1 u --. log (fl - u 2 + iu). But the formula for tau~ 1 Jiis a more elegant one, as not involving the radical Vl - u* ; we have and thence that is, i tan x - cxp t - r ~ cx P (->) _cxp8t>-l cxp t jr-f cxp (->) exp2ix+l* 1 + i tan x or if tan x = u, then The logarithm (or inverse exponential function) and the inverse circular functions present themselves as the integrals of algebraic functions fdx whence also and >. =sin-i:r , G. Each of the functions exp u, sin u, cos u, tan u, &c., as a one- valued function of u, is in this respect analogous to a rational func tion of u ; and there are further analogies of exp u, sin u, cos u, to a rational and integral function ; and of tan u, sec u, &c., to a rational non-integral function. A rational and integral function has a certain number of roots, or zeros, each of a given multiplicity, and is completely deter mined (except as to a constant factor) when the several roots and the multiplicity of each of them is given ; i.e., if a, b, c . . . be the roots, p, q, r . . . their multiplicities, then the form is / ^* / A ( 1 - - ) (!- -) . . . ; a rational (non-integral) function has a certain number of infinities, or poles, each of them of a given multiplicity, viz., the infinities are the roots or zeros of the rational and integral function which is its denominator.
