# §14.24 Analytic Continuation

Let be an arbitrary integer, and and denote the branches obtained from the principal branches by making circuits, in the positive sense, of the ellipse having as foci and passing through . Then

the limiting value being taken in (14.24.1) when is an odd integer.

Next, let and denote the branches obtained from the principal branches by encircling the branch point 1 (but not the branch point −1) times in the positive sense. Then

the limiting value being taken in (14.24.4) when .

For fixed , other than or , each branch of and is an entire function of each parameter and .

The behavior of and as from the left on the upper or lower side of the cut from to 1 can be deduced from (14.8.7)–(14.8.11), combined with (14.24.1) and (14.24.2) with .