…
►The only singularities are algebraic branch points, with and finite at these points.
…All real values of are normal values.
To 4D the first branch points between and are at
with , and between and they are at
with .
For real with , and are real-valued, whereas for real with , and are complex conjugates.
…
►Analogous statements hold for , , and , also for .
…
…
►When the series (16.2.1) converges for all finite values of and defines an entire function.
…
►The branch obtained by introducing a cut from to on the real axis, that is, the branch in the sector , is the principal branch (or principal
value) of ; compare §4.2(i).
Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at
, and .
…
►On the circle the series (16.2.1) is absolutely convergent if , convergent except at
if , and divergent if , where
…
►(However, except where indicated otherwise in the DLMF we assume that when
at least one of the is a nonpositive integer.)
…
…
►Here is an arbitrary point in the interval .
The integration path begins at
, encircles once in the positive sense, followed by once in the positive sense, and so on, returning finally to .
…The branches of the many-valued functions are continuous on the path, and assume their principal valuesat the beginning.
…
►The right-hand side may be evaluated at any convenient value, or limiting value, of in since it is independent of .
…
►and the integration paths , are Pochhammer double-loop contours encircling distinct pairs of singularities , , .
…
…
►►►Figure 20.3.2:
, , = 0.
…Here approximately, where corresponds to the maximum value of Dedekind’s eta function as depicted in Figure 23.16.1.
Magnify
…
►►►Figure 20.3.6:
, , = 0, 0.
…
Magnify
…
►In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase.
…
►In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase.
…
►►