By Vito Volterra

ISBN-10: 1161723854

ISBN-13: 9781161723854

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Additional resources for Theory of Permutable Functions

Example text

If F \$ must be of order m-n. Moretwo cases may arise according as the order n, then over, functions F and are or are not pennutable Clearly In the latter case, \b with one another. cannot be pennutable with F, otherwise the two would But if F and \$ resultant of the composition of the be permutable with either. * be permutable with JF are pennutable, will and i/f'? We shall prove that this property realized. In fact, F^F = ijf is actually we have F, FF = Hence and since this integral equation has but one solution, <\$>F and the theorem II.

If we write we have (16) where \$(x,y) is = 6(v) - an arbitrary function and v ~~ = %-ry 9 The solution of the equation (16) is obtained the method of successive approximations. by Applications of the above will be brought out in the next lecture. LECTURE III Ill I. We of the We shall begin work developed with some applications in the last lecture. /) which satisfies the equation function x (# JV(s, |) x &#) & +/*Xto ^ * *'^ /7* = ( on the hypothesis that order. l) Then F = -F(x,i/) . the condition <(#, #) + jFfo x) = clearly satisfied, and hence we shall be able to calculate all of the functions x (^ y) which is satisfy the relation in other words, all of the functions which have permutability of type one with a given function.

In mind, let us write the function We can see at once what the relation (8) Indeed, we have only to replace becomes. multiplication therefore have that by composition. We shall is In other words, the theorem of algebraic ad- LECTURE 39 II dition for the exponential function becomes for this new function a theorem of integral addi- we have tion as 7. * To go from the particular case to the general involves no difficulty. Consequently, we may state the theorem: To every theorem of algebraic addition, there corresponds a the- orem of integral addition, Thus, for example,, if we consider elliptic functions, we can pass from these to entire functions by the process of preceding lecture.