has been called Bernoulli’s function of the mth order by J. L. Raabe (Crelle’s J. f. Math. Bd. xlii., 1851). Bernoulli’s numbers and functions are of especial importance in the calculus of finite differences (see the article by D. Seliwanoff in Ency. d. math. Wiss. Bd. i., E., 1901).
When x is given in terms of y by means of a power series of the form
x = y (C0 + C1y + C2y2 + ...) (C0 ≠ 0) = yƒ0(y), say,
there arises the problem of expressing y as a power series in x. This problem is that of reversion of series. It can be shown that provided the absolute value of x is not too great,
| y = | x | + Σn=∞n=2 [ | xn | · | dn−1 | 1 | ]y=0 | |
| ƒ0(0) | n! | dyn−1 | {ƒ0(y)}n |
To this problem is reducible that of expanding y in powers of x when x and y are connected by an equation of the form
y = a + xƒ(y),
for which problem Lagrange (1770) obtained the formula
| y = a + xƒ(a) + Σn=∞n=2 [ | n | · | dn−1 | {ƒ(a)}n ]. |
| n! | dan−1 |
For the history of the problem and the generalizations of Lagrange’s result reference may be made to O. Stolz, Grundzüge d. Diff. u. Int. Rechnung, T. 2 (Leipzig, 1896).