PERIODIC FUNCTIONS. The general consideration of periodio magnitude, that is, of magnitude which varies in such manner as to go through stated cycles of changes, each cycle befog a reiteration of the preceding one, is the subject of Temosourser. Though that science derives its name from the measurement of triangle. (as geometry does from that of the earth), its resources have been cultivated and its methods expanded until no definition short of the preceding will express its object or convey an Idea of its powers.
In the article cited will therefore be explained the mode of measure ment applied to periodic magnitude : the present one is intended for nothing more than to point out a peculiarity of some classes of algebraic) functions which has procured for them the name of periodic functions.
The calculus of FracrioNs considers only forms, and the operations neceasary to convert one given form into another, or to satisfy equations in which some forms of operation are unknown. Periodic functions are those which, performed any given number of times on a variable, reproduce the simple variable itself. Thus 1 - x and - x are periodic functions of the second order, since 1 - (1 - x) r, - (- x) x.
Again, 1 : (1 - x) is a periodic function of the third order, since, if we begin with x, and write I : (1 - x) for x three times in succession, we end with x also 1 x - 1 x, 1 Periodic functions are remarkable in the calculus of functions for the simplicity with which questions connected with them can be solved, when compared with the difficulty of solution of cases in whioh non-periodic functions enter. Their principal properties are here briefly pointed out • for further information, consult Babbage's Examples of Functional Equation.' (appended to Peacock's Examples), or the article on the Calculus of Functions contained in the Encyclopsedia 3letropolitana; where further references will be found.
Let tax be a function of x; abbreviate 4Asx) into ; into &c. : then, if es be a periodic function of the nth order, lox = x.
Let r be one of the nth ROOTS of unity, then rx is the simplest periodic function of the nth order ; but very simple ones of the form (1 + bx) : (c kx) may be obtained by making - 2 Locos° + k= —2(1 + cos 0) where 8 means the nth part of any multiple of four right angles. For
instance, if a = 4,8 may = 90", and cos 0 = 0, whence k = (b" +0) : 2; whence 2 + 26x 2c - (6' + is a periodic function of the fourth order.
Let ex be any function of x whatsoever, and its INVERSE function, so that = x; then if cox be a periodic function, is also periodic. Thus 1 - x being periodic of the second order, so are log (1 - e' V(1 - (1- sin x), &c. &c.
Let be called a derivative of ; then if cox and be two periodic functions of the nth order, either can in an infinite number of ways be made a derivative of the other. Thus one of the ways in which 1 : x is a derivative of - x is • . Let 1, r, be all the n roots of unity, let be any function of x and y, and lot r, be the same function of sox and 4'y, r, of ipx and 4.y, &c. From the equation r + r= P, + + = 0,find y in terms of x; say y= ex. Then will 41/=04s.r, or that is, Ox is as a derivative of spa-.
For example, let fx = - x, = 1 - x, be the periodic functions, of the second order ; then r = - I. Let = ex + by, then the pre ceding equation becomes ex + by + ( - 1) (- a.r + Cr=y) ei 0; - 2ax b - 26y or y = at 26 -, b - 2a - (6 - 26x) : 2a esse-tx =1 in 1 r.
26 • The periodic functions, as before observed, arc those whose relations are most easily obtained. For example, let v., be given functions of x, and 4sx an unknown function, to be determined by the equation cox = Ox + Txstetx.
If al" be not periodie, there are two difficulties in the way, each most reqoently insuperable : first, the determination of some one solution of this equation; secondly, the determination of the Isrvaatante function of this equation, or the solution of 4six = 4,r. But when ax is periodic, both difficulties can be overcome and a general solution given. Say a"x x, then Ox is any symmetrical function of x, ax, • .. • . ce"x; and if &c., be 0r, Bar, &a., and o„ fic., be yr, -rar, &o., the general solution divides into two cases, in the first of which the solution does not depend on the invariable function, The moat general case gives + a, + c„ c, + + 1- c, c, o, • .