FUNCTIONS, a mathematical term. When two or more variables are combined with constants in an equation, and are such that a change of value of one implies a corre sponding change of value of one or more of the others, then such variables are said to depend on and to be F' of each other; and the expression of the mode of dependence is said to be aft/net/on of such variables. .1f such an expression involves but one variable, it is said to be a functiou of one variable: if two are involved, to be a function of two variables; and so on. Thus sin x, , tog. x, +/TV -e are F. of one variable—viz. of x; tan (ax±by), x3, are F' two variables, z and y; so zyz are F. of three variables, and soon. F. are denoted by the symbols F, f, ep, etc Thus 1'W means , a function of one variable, z combined with constants or not, as the case may be; .Czyz) a function of three variables. These functional symbols are general, and their specific forms are the particular F which arise from operations in algebra, trigonom etrv, etc.
F. are implicit or explicit. When one variable is expressed in terms of others, it is said to be an explicit function of them; but when all the variables remain involved in one expression, the function is said to be implicit. Thus, — 0 is an implicit function of two variables, but y = — xY is an explicit function of one variable. 'in explicit F., the variable which is expressed in terms of the others is called the dependent variable, and the others the independent variables. Explicit F. are usually written in the form z =Au); implicit in the form u = F (rip) = 0, F., again,. are algebraical or transcendental. Algebraical F. are those which involve tho operations of addition, etc., and of involution and evolution, Transcendental F. are those where the operations symbolized, arc such as cam, x, sin x, etc.—i.e., exponential, logarith mic, or circular. F., also., 'are simple or compound according as they invotve one or several operations. y = sin x is a. simple function; but y = log. sin x is compound.
Further, F. are divided into the continuous and the discontinuous, the circulating and the periodic. Continuous F. are such as arc subject to the following conditions: 1. As the variable gradually changes, the function must gradually change; 2. The law sym bolized by the functional character must not abruptly change. Circulating F. are those whose values lie' within certain limits for all values of the variables. y = sin x is an example at once of a continuous and of a circulating function. A function is said to be periodic when it takes the form f ° (x) = x, signifying that if on x the operation f be performed n times, the resulting value will be x. Thus, fix) = 1 xis a periodic func tion of the third order. For performing the operation indicated by f the second time on 1 1 as the variable, we have p(x)-..,.
x 1 — ' 1 1 x =-7 1 ; and the third time we have = 1— — = x. The functional calculus is a recent growth of the transcen dental analysis. The object of the differential calculus (q.v.) is generally to ascertain the changes iii F. arising from the continuous and infinitesimal variation of their subject variables. The object of the new functional calculus is, speaking generally, to investi gate the forms of F. and their growth, when they arc subject to a continuous and infini tesimal change as to form. According to Mr. Price (treatise on the infinitesimal calcu lus), as the differential calculus investigates, properties of continuous numbers, so does the new calculus the properties of continuous F.; and as there is an integral calculus of numbers, so there is an inverse calculus of functions. Of the new calculus, the calculus of variations (q.v.) may be considered the main branch. It includes„ of course, the sub ject of functional equations. 'Functional equations are those in which it is required to determine from equations the forms of F entering them: e.g., what is the function of x and y which satisfies the equation fix) xAy) =,Ax +y)? See article CALCULUS OP FUNCTIONS in the Encyclopcedia, ffetropolitana.