ths in the region
considered described about [Sigma], and therefore, by Laurent's
Theorem (see FUNCTION), capable of expression in the annular region
about this point by a series of positive and negative integral powers
of x - [xi], which in general may contain an infinite number of
negative powers; there is, however, no reason to suppose r1 to be an
integer, or even real. Thus, if all the roots of the determinantal
equation in [mu] are different, we obtain n integrals of the forms (x
-[xi])^r1 phi1, ..., (x - [xi])^rn [phi]_n. In general we obtain as
many integrals of this form as there are really different roots; and
the problem arises to discover, in case a root be k times repeated, k
- 1 equations of as simple a form as possible to replace the k - 1
equations of the form y^0 + ... + y^0_n-1 v_n-1 = [mu](y^0 + ... + y^0_n-1
u_n-1) which would have existed had the roots been different. The most
natural method of obtaining a suggestion lies probably in remarking
that if r2 = r1 + h, there is an integral [(x - [xi])^(r1 + h) [phi]2
- (x -[xi])^r1 [phi]1]/h, where the coefficients in [phi]2 are the
same functions of r1 + h as are the coefficients in [phi]1 of r1; when
h vanishes, this integral takes the form
_ _
| d[phi]1 |
(x - [xi])^r1 | ------- + [phi]1 log (x - [xi])|,
|_ dr1 _|

or say (x-[xi])^r1 [[phi]1 + [psi]1 log (x - [xi])];

denoting this by 2[pi]i[mu]1K, and (x-[xi])^r1 [phi]1 by H, a circuit
of the point [xi] changes K into

1
K' = ----------- [e^(2[pi]ir1) (x - [xi])^r1 [psi]1 + e^(2[pi]ir1) (x - [xi])^r1 [phi]1 (2[pi]i + log(x - [xi]))]
2[pi]i[mu]1

= [mu]1K + H.

A similar artifice suggests itself when three of the roots of the
determinantal equation are the same, and so on. We are thus led to the
result, which is justified by an examination of the algebraic
conditions, that whatever may be the circumstances as to the roots of
the determinantal equation, n integrals exist, breaking up into
batches, the values of the constituents H1, H2, ... of a batch after
circuit about x = [xi] being H1' = [mu]1H1, H2' = [mu]1H2 + H1, H3' =
[mu]1H3 + H2, and so on. And this is found to lead to the forms (x -
[xi])^r1 [phi]1, (x - [xi])^r1 [[psi]1 + [phi]1 log (x - [xi])], (x -