An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

368

Expansion of ambiguous analytic functions.

Bk. IV. ch. III.

II. f() 2ixn" J ft d +u +) ( - c f)dL1 (X - *.) - ) (Z - a) +
+ (~_ t~)Taf f (2) Co ~ }
( -a) m J
Each integral in this series refers to the closed curve taken
positively in all the leaves round the point a; in the various leaves
rn-1
f(z) and (z - a) m assume their prescribed values.
The root (u - )m and its powers assume their different values
according as the value of f(zu) is to be determined in one or other
of the m leaves.
The statement of equation II. in words is:
When the branching point in which m values of the function are
cyclically connected is not also a singular point, each of the m values of the
function in its neighourhood can be expanded in an ascending series of
1
positive integer powers of (u- a)"m. This neighbourhood is coextensive
with a domain wherein there is neither a singular point nor another
branching point of the function.
The significance of the coefficients in II. can be shown otherwise.
Putting u = a, we have:
f (a) / g fm J d - a
If further we differentiate the equation with respect to u, which
in case of a series of powers is done by differentiating term by term, we
m-1
obtain, on multiplying both sides by (u - a) m and putting u a:
mn - 1)
Lim f'(u) (u - C)r 2 i7rn2f f(2)d
By the same process is found:
2(nm —1) nm —2
Lim jf"(u) (u - a)m + - f)(u - a)_ (2-y) I r d (z etc.
- m2 2 i - e m +2 X..
(J z - )?n
Writing series II. briefly in the form:
1 2
f(u) =-, -+ (u - C) a', + (u - a)m a2 +