where A affects x only; and, assuming as initial conditions where A, λ and μ are constants (Cambridge Problems). B. F. D. 19 11. Given Ux+1, 9+1+(α-x-2y-2) ux, 1+1+(x + y) u with the conditions 0, 0 ux,-1=0, u。。 = 0, and Ux,x+1 = = 0, find ux, y = 0 [Cayley, Tortolini, Series II. Vol. II. p. 219.] 12. Ux,y=Ux-y,1+Ux-y, 2 + &c. +Ux-y, y° ... [De Morgan, Camb. Math. Jour. Vol. IV. p. 87.] CHAPTER XV. OF THE CALCULUS OF FUNCTIONS. 1. THE calculus of functions in its purest form is distinguished by this, viz. that it recognizes no other operations than those termed functional. In the state to which it has been brought more especially by the labours of Mr Babbage, it is much too extensive a branch of analysis to permit of our attempting here to give more than a general view of its objects and its methods. But it is proper that it should be noticed, 1st, because the Calculus of Finite Differences is but a particular form of the Calculus of Functions; 2ndly, because the methods of the more general Calculus are in part an application, in part an extension of those of the particular one. In the notation of the Calculus of Functions, {(x)} is usually expressed in the form pyx, brackets being omitted except when their use is indispensable. The expressions pox, 40px are, by the adoption of indices, abbreviated into px, p3x, &c. As a consequence of this notation we have φα =x independently of the form of 4. The inverse form 1 is, it must be remembered, defined by the equation -1 Hence may have different forms corresponding to the same form of 4. Thus if we have, putting px = t, and 4 has two forms. $x = x2+ax, a ∞ = $ ̃1. t = − a ± √ (a2 + 4t) The problems of the Calculus of Functions are of two kinds, viz. 1st. Those in which it is required to determine a functional form equivalent to some known combination of known forms; e.g. from the form of x to determine that of "x. This is exemplified in B, page 167. 2ndly. Those which involve the solution of functional equations, i.e. the determination of an unknown function from the conditions to which it is subject, not as in the previous case from the known mode of its composition. We may properly distinguish these problems as direct and inverse. Problems will of course present themselves in which the two characters meet. Direct Problems. 2. Given the form of x, required that of "x. There are cases in which this problem can be solved by successive substitution. Ex. 1. Thus, if x=x, we have Again, if on determining x, y3 as far as convenient it should appear that some one of these assumes the particular form x, all succeeding forms will be determined. Ex. 2. Thus if yx=1-x, we have Hence x = 1−x or x according as n is cdd or even. Functions of the above class are called periodic, and are distinguished in order according to the number of distinct forms to which "x gives rise for integer values of n. The function in Ex. 2 is of the second, that in Ex. 3 of the third, order. Theoretically the solution of the general problem may be made to depend upon that of a difference-equation of the first order by the converse of the process on page 167. For The arbitrary constant in the solution of this equation may be determined by the condition t1 = x, or by the still prior condition It will be more in analogy with the notation of the other chapters of this work if we present the problem in the form: Given t, required yt, thus making x the independent variable of the difference-equation. Ex. 4. Given t = a + bt, required yt. |