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§16.18 Special Cases►The and functions introduced in Chapters 13 and 15, as well as the more general functions introduced in the present chapter, are all special cases of the Meijer -function. …
§19.14(ii) General Case…
In the general case, given by , we compute the roots , , , say, of the cubic equation ; see §1.11(iii). These roots are necessarily distinct and represent , , in some order.
If , or and are not both real, then we label , , so that the triangle with vertices , , is positively oriented and is its longest side (chosen arbitrarily if there is more than one). In particular, if , , are collinear, then we label them so that is on the line segment . In consequence, , satisfy (with strict inequality unless , , are collinear); also , .
Finally, on taking the principal square roots of and we obtain values for and that lie in the 1st and 4th quadrants, respectively, and , are given by
If , then
There are 4 possible pairs (, ), corresponding to the 4 rotations of a square lattice. The lemniscatic case occurs when and .
If , then
There are 6 possible pairs (, ), corresponding to the 6 rotations of a lattice of equilateral triangles. The equianharmonic case occurs when and .