818 U N F U N for glandular swellings, carbuncles, and abscesses, and was formerly valued in jaundice, and in cases of accidental swallowing of the beard of grain (see F. Porter Smith, Conlrib. towards the Mat. Medico, . . . of China, p. 99, 1871). The name fumitory, Latin fumus lerrce, has been supposed to be derived from the fact that its juice irritates the eyes like smoke (see Fuchs, De Historia Stirpium, p. 338, 1542); but The Grete Herball, cap. clxix., 1529, fol., following the De Simplici Medicina of Platearius, fo. xciii. (see in Nicolai Prccpositi Dispensatorium ad Aromatarios, 1536), says : " It is called Fumus terre. fume or smoke of the erthe bycause it is engendred of a cours fumosyte rys- ynge frome the erthe in grete quantyte lyke smoke : this grosse or cours fumosyte of the erthe wyndeth and wryeth out : and by workynge of the ayre and sonne it turneth into this herbe." For figures of various species of fumitory, see J. T. Boswell Syme, English Botany, vol. i., 1863. FUNCHAL. S38MADEIKA. FUNCTION". Functionality, in Analysis, is depend ence on a variable or variables ; in the case of a single variable u, it is the same thing to say that v depends upon it, or to say that v is a function of u, only in the latter form of expression the mode of dependence is embodied in the term "function." We have given or known functions such as it 2 or sin 11, and the general notation of the form tf>u, where the letter < is used as a functional symbol to denote a function of it, known or unknown as the case may be : in each case u is the independent variable or argu ment of the function, but it is to be observed that if v be a function of u, then v like u is a variable, the values of v regarded as known serve to determine those of u; that is, we may conversely regard u as a function of v. In the case of two or more independent variables, say when w depends on or is a function of it, v, &c., or w = <j)(u, v, . . ), then u, v, . . are the independent variables or arguments of the function; frequently when one of these variables, Bay it, 13 principally or alone attended to, it is regarded as the independent variable or argument of the function, and the other variables v, &c., are regarded as parameters, the values of which servo to complete the definition of the function. We may have a set of quantities u< } t, . . each of them a function of the same variables v, v, . .; and this relation may be expressed by means of a single functional symbol <, (w, t, . .) = <(, v . .); but, as to this, more hereafter. 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, x and y being real numbers, each susceptible of continuous variation between the limits - oo , + co , and i denoting */ - I. 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 n = x + i>j. To a given value x + iy of u there corresponds in general for v a value or values of the 1 ike form v x + iy t and we obtain a geometrical notion of the meaning of the functional relation v =fu by regarding a-, y as rectangular coordinates of a point P in a plane II, and x, y as rectangular coordinates of a point I" in a plane (for greater convenience a different plane) IT; 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 II, and corresponding hereto we have a point or points P in the plane IT, representing (if more than one, each of them) a value of the variable v = x + iij. 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. AYe have the series of powers w 2 , u 3 , . each calculable not only for a real but also for a complex value of n, (x + iy} 2 = x" - ?y 2 + 2ixy, (x + iy) 3 = x 3 - 3xif + t(3 2 - y 3 ), &c., nnd thence, if ,&,.. be real or complex numbers, th general form a + lu + cu 2 . . . + 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 11. 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 (m being a positive integer) that v m = u v is here the irrational function = u m . 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 i_ i)<i = u, which serves as the definition of u m ; 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 + lu + cv? . . . + &u*, 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, I, c . . may be such as to make the series always convergent ; for instance, this is the case for the exponential function,
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 determi nate calculable value. A function thus expressed by means of a convergent infinite scries 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 transcen dental 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 CXp U exp t = CXp ( + * ) i whence also exp x exp iy ="cxp (x + iij) , so that exp (x + iy} 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 where obviously each scries is convergent and actually calculable for any real value whatever of y , calling the two scries cosine y and sine y respectively, or in the ordinary abbreviated notation cos y and sin y, the equation is
