The notion of a function is applicable in geometry and mechanics as well as in analysis; for instance, a point Q, the position of which depends upon that of a variable point P, may be regarded as a function of the point P; but here, substituting for the points themselves the coordinates (of any kind whatever) which determine their positions, we may say that the coordinates of Q are each of them a function of the coordinates of P, and we thus return to the analytical notion of a function. And in what follows a function is regarded exclusively in this point of view, viz., the variables are regarded as numbers; and we attend to the case of a function of one variable v=fu. But it has been remarked (see Equation) that it is not allowable to confine the attention to real numbers; a number u must in general be taken to be a complex number u=*x + iy, χ and y being real numbers, each susceptible of continuous variation between the limits -oo, 4-co, and i denoting s -1. In regard to any particular function, fu, although it may for some purposes be sufficient to know the value of the function for any real value whatever of u, yet to attend only to the real values of u is an essentially incomplete view of the question; to properly know the function it is necessary to consider u under the aforesaid imaginary or complex form u = x + iy.

To a given value x + iy of u there corresponds in general for υ a value or values of the like form υ≈χt + iy' i and we obtain a geometrical notion of the meaning of the functional relation v =fu by regarding x, y as rectangular coordinates of a point P in a plane ∏, and x', y' as rectangular coordinates of a point P' in a plane (for greater convenience a different plane) II'; P, P' are thus the geometrical representations, or representative points, of the variables u = x + iy and u = x' + iy ’ respectively; and, according to a locution above referred to, the point P' might be regarded as a function of the point P; a given value of u = x + iy is thus represented by a point P in the plane ∏, and corresponding hereto we have a point or points P ’ in the plane ∏', representing (if more than one, each of them) a value of the variable υ = x' + iy'. And, if we attend only to the values of u as corresponding to a series of positions of the representative point P, we have the notion of the “path” of a complex variable u = x + iy. Known Functions,

1. The most simple kind of function is the rational and integral function. We have the series of powers u 3 , u 3 , , each calculable not only for a real but also for a complex value of u, (χ + iy) 3 = x 3 - iy* + 2ixy, (x + iy) 3 = x 3 - 3xy* + t(3z 2 - y 3 ), &c., and thence, if a, b, . . be real or complex numbers, th general form a + bu + cu 3 . . . + ku m , of a rational and integral function of the order m. And taking two such functions, say of the orders m and n respectively, the quotient of one of these by the other represents the general form of a rational function of u.

The function which next presents itself is the algebraical function, and in particular the algebraical function expressible by radicals. To take the most simple case, suppose (wι being a positive integer) that v* = u; v is here the irrational function = u ni . Obviously, if u is real and positive, there is always a real and positive value of v, calculable to any extent of approximation from the equation v* = u, which serves as the definition of u* ; but it is known (see Equation) that as well in this case as in the general case where u is a complex number there are in fact m values of the function u m ; and that for their determination we require the theory of the so-called circular functions sine and cosine; and these depend on the exponential function exp u, or, as it is commonly written, c*, which has for its inverse the logarithmic function log u ; these are all of them transcendental functions.

2. In a rational and integral function a + bu + cu* . . . + ku* t , the number of terms is finite, and the coefficients a, b,: . . k may have any values whatever, but if we imagine a like series a + bu + cu* + ... extending to infinity, non constat that such an expression has any calculable value,—that is, any meaning at all; the coefficients a, b, c . . . must be such as, either for every value whatever of u (that is, for every finite value) or for values included within certain limits, to make the series convergent. It is easy to see that the values of a, b, c . . may be such as to make the series always convergent; for instance, this is the case for the exponential function, w .= 1+ . + g + ⅛ + * ti taking for the moment u to be real and positive, then it is evident that however large u may be, the successive terms will become ultimately smaller and smaller, and the series will have a determinate calculable value. A function thus expressed by means of a convergent infinite series is not in general algebraical, and when it is not so, it is said to be transcendental (but observe that it is in nowise true that we have thus the most general form of a transcendental function); in particular, the exponential function above written down is not an algebraical function.

By forming the expression of exp v, and multiplying together the two series, we derive the fundamental property exp u exp v = exp (« + ·); wheµce also exp x exp iy ='exp (x 4 iy), so that exp (x ÷ if) is given as the product of the two series exp x and exp iy. As regards this last, if in place of u we actually write the value iy, we find

-p⅛=(>- 1 ⅛÷rfi s -∙)+<(i -⅛÷-).

where obviously each series is convergent and actually calculable for any real value whatever of y ; calling the two series cosine y and sine y respectively, or in the ordinary abbreviated notation cos y and sin y, the equation is exp iy = cos V + t sin y∙, and if we herein for y write z, and multiply the two expressions together, observing that the product will be ≈ exp i (y + z), we obtain the fundamental equations cos (y +1) = cos y cos z — sin y ein t; sin (y + z) ~ ∙i∏ y co ∙ z + sin z cos y, for the functions sine and cosine.

Taking y as an angle, and defining as usual the sine and cosine ns 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 iy — cos y + i sin y, 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 π denoting as usual the number 3 , 14159 . ., the measure of a right angle is = ⅜τr. And this being so, the functions sine and cosine, obtained as above by consideration of the exponential function, have their ordinary geometrical significations.

3. The foregoing investigation was given in detail in order to the completion of the theory of the irrational function u m . We henceforth 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(cos θ + i sin θ) , where, writing shx t + y' i to denote the positive value of the square root, we have r = √⅛* + j∣*, cosθ = ∕ Λ~ , βinfl= ,^⅛=,

, , , √∙r 2 + ⅛ 2 √x 2 + v 2 and therefore also tanθ = ^∙

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 0 is the inclination of r to the positive part of the axis of x ; to fix the ideas 0 may be regarded as lying within the limits 0, π, or 0,-τ, according as y is positive or negative; 0 is thus completely determinate, except in the case, x negative, y = 0, for which θ is = π or-π indifferently.

And if u = x + iy, we hence have

.L<.+U-Λ~⅛P+'* 4 ⅛≡)∙

1

where r m 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 index 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 ω∙, no longer an algebraical function of u, and having in general an infinity of values.

4. The loregoing equation exp (x + y) = exp x . exp y is in fact the equation of indices, a x +v = a x . al ; exp x is thus the same thing as c x , where e denotes a properly determined number, and putting c x equal to the series, and then writing x — 1, we have β = l + γ + + &c., that is, β = 2∙7128... But as well

1 1.2 1.2 o theoretically as for convenience of printing, there is considerable advantage in the use of the notation exp u.

From the equation, exp iy ≈ cos y +ι sin y, we deduce exp ( - iy) = cosy - i sin y, and thence c ° 3 V= I {exp (⅛) + exp (-ty) j∙ , ei“V = ~ {exp (íy) - exp ( - tj∕)∣; if we write herein ix instead of y we have cos ix = 1 ∙^exp x + exp (— x) ∣ ein ix - 1 sexp x — exp (− x) j∙, Viz., these values are

"∙ t '= 1 + ra + i⅛ + ∙ 1 ,ι^<*=≈ + 1 ⅛+∙∙

each of them real when ae is real; the functions in question 1 + L2 + L 2⅛4 + ∙ ∙ and íc + j-j3 + . . ., regarded as fαnc-tions of x, are termed the hyperbolic cosine and sine, and are represented by the notations cosh x and sinh x respectively; and similarly we have the hyperbolic tangent tanh x, &c.: although it is easy to remember that cos ix, sin ix, are in fact real functions oîx, and to understand accordingly the formulae wherein they occur, yet the use of these notations of the hyperbolic functions is often convenient.

β. 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 « = exp (log«) (and it is clear that if u be real and positive there is always a real and positive value of lθg w, 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 + 2sιri is also a value, for we have exp (log u + 2sιri) — ex P log u exp 2sιrι, or since exp2sτiis = l, this is — u; that is, log u + 2sιri is a value of the logarithm of u.

We have uυ = exp (log ut) = exp log u. exp log ¢, and hence the equation which is commonly written log uv = log « + log r, but which requires the addition on one side of a term 2sτrt. And reverting to the equation x + iy = τ (cos 0 + i sin 0), or as it is convenient to write it, x+iy=r exp iθ, we hence have log (z + iy) = log r + i(θ + 2*ιr), where log r may be taken to denote the real logarithm of the real positive quantity r, and θ the completely determinate angle defined as already mentioned.

Reverting to the function u m , we have w=exp log w, 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 = e, then c n , = exp (m log c), has in general in like manner an infinity of values, but in regarding c m as identical with the one-valued function exp m, we take log e to be = its real value, 1.

The inverse functions cos -1 #, sin -1 x, tan -1 «, are in fact logarithmic functions; thus in the equation exp ix = cos x + i sin x, writing first cos x≈u, the equation becomes exp i cos - ht = u + í√l - u 2 , or we have cos* 1 m = -τ log (u + í√l - u*), and from the same equation,writing secondly sin x = u, we have sin~ 1 u≡ v log (√1 - + iu}. But the formula for tan -1 « is a more elegant one, as not involving the radical √1 - u 2 ; we have i tan χ — cχp ér-exp (-t-r) _ exp8⅛-l exp ix⅛exp (-iz) ’ exp2ta+l ’ and thence o . 1 + í tan x exp 2tz= 1 - ftanχ .

that is,

1 , l÷ítanx ≈ — 2í 10g l-itaax or if tan x≈u, then tan - ¼- 2i k >βf∑rru∙

The logarithm (or inverse exponential function) and the inverse circular functions present themselves as the integrals of algebraic functions

∕-d⅛ , s F = 10 « x ’ whence also f — - -− lo¢ 1 -±⅛= Un- 1 x,

∕ i+i≡~2i 1 β l-<x and f ⅛- =sin-ιx.

J √l-x«

6. Each of the functions exp u, sin u, cos u, tan u, kc., as a onevalued function of u, is in this respect analogous to a rational function 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 ot roots, or zeros, each of a given multiplicity, and is completely determined (except as to a constant factor) when.the several roots and the multiplicity of each of them is given; i.c., if ash, c . . . be the roots, p, q, r . . . their multiplicities, then the form is A Ç1 - -^ p ζl - . .; 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.

The function exp u has no finite roots, but an infinity of roots each - - ∞; this appears from the equation exp «=(1 + , where n is indefinitely large and positive. The function sin u has the roots sir where s is any positive or negative integer zero included; or what is the same thing, its roots are 0 and ± sir, s now denoting any positive integer from 1 to ∞; each of these is a simple root, and we in fact have sin u ≈ u ∏^l - Similarly the roots of cos u are (s + ⅛) π, s denoting any positive or negative integer, zero included, or what is the same thing, they are ± (s + ⅛) π, s now denoting any positive integer from 0 to ∞; each root is simple, and we have cos u = ∏( 1 — + *y ∙ t '∙ ) , Obviously tan u, as the quotient sin u — 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 «, &c.

In the foregoing expression sin u = uïl 1 - the product must be understood to mean the limit of ∏ ’ 1 l ( 1 — —, ) for an inde-

1 ∖ , s 1 ιr 1 J 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 s, we consider the product u∏"(l + J") π ι - » (w siere in the first pro duct s 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 wι and n each of them indefinitely large, the function does not approximate to sin u, unless m: n be a ratio of equality. And similarly as regards cos u, the product ∏o"(l + ∏ ( _y m and 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 ι>eriodic, having the period 2π t cos(tt + 2,r )"∙cos(2i )5 and the half -P eriod ιr > c 1 os(w + ,r ) — - sθ 11 u ; the periodicity may be verified by means of the foregoing fractional forms, but some attention is required; thus j ∙ π ( u + sιr )...