# general case

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##### 1: 16.18 Special Cases
###### §16.18 Special Cases
The ${{}_{1}F_{1}}$ and ${{}_{2}F_{1}}$ functions introduced in Chapters 13 and 15, as well as the more general ${{}_{p}F_{q}}$ functions introduced in the present chapter, are all special cases of the Meijer $G$-function. …
##### 4: Guide to Searching the DLMF
DLMF search is generally case-insensitive except when it is important to be case-sensitive, as when two different special functions have the same standard names but one name has a lower-case initial and the other is has an upper-case initial, such as si and Si, gamma and Gamma. …
##### 5: 32.8 Rational Solutions
In the general case assume $\gamma\delta\neq 0$, so that as in §32.2(ii) we may set $\gamma=1$ and $\delta=-1$. … In the general case assume $\delta\neq 0$, so that as in §32.2(ii) we may set $\delta=-\tfrac{1}{2}$. … In the general case, $\mbox{P}_{\mbox{\scriptsize VI}}$ has rational solutions if …
##### 8: 23.22 Methods of Computation
• (a)

In the general case, given by $cd\neq 0$, we compute the roots $\alpha$, $\beta$, $\gamma$, say, of the cubic equation $4t^{3}-ct-d=0$; see §1.11(iii). These roots are necessarily distinct and represent $e_{1}$, $e_{2}$, $e_{3}$ in some order.

If $c$ and $d$ are real, and the discriminant is positive, that is $c^{3}-27d^{2}>0$, then $e_{1}$, $e_{2}$, $e_{3}$ can be identified via (23.5.1), and $k^{2}$, ${k^{\prime}}^{2}$ obtained from (23.6.16).

If $c^{3}-27d^{2}<0$, or $c$ and $d$ are not both real, then we label $\alpha$, $\beta$, $\gamma$ so that the triangle with vertices $\alpha$, $\beta$, $\gamma$ is positively oriented and $[\alpha,\gamma]$ is its longest side (chosen arbitrarily if there is more than one). In particular, if $\alpha$, $\beta$, $\gamma$ are collinear, then we label them so that $\beta$ is on the line segment $(\alpha,\gamma)$. In consequence, $k^{2}=(\beta-\gamma)/(\alpha-\gamma)$, ${k^{\prime}}^{2}=(\alpha-\beta)/(\alpha-\gamma)$ satisfy $\Im k^{2}\geq 0\geq\Im{k^{\prime}}^{2}$ (with strict inequality unless $\alpha$, $\beta$, $\gamma$ are collinear); also $|k^{2}|$, $|{k^{\prime}}^{2}|\leq 1$.

Finally, on taking the principal square roots of $k^{2}$ and ${k^{\prime}}^{2}$ we obtain values for $k$ and $k^{\prime}$ that lie in the 1st and 4th quadrants, respectively, and $2\omega_{1}$, $2\omega_{3}$ are given by

where $M$ denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ($2\omega_{1}$, $2\omega_{3}$), corresponding to the 2 possible choices of the square root.

• (b)

If $d=0$, then

23.22.2 $2\omega_{1}=-2i\omega_{3}=\frac{\left(\Gamma\left(\frac{1}{4}\right)\right)^{2% }}{2\sqrt{\pi}c^{1/4}}.$

There are 4 possible pairs ($2\omega_{1}$, $2\omega_{3}$), corresponding to the 4 rotations of a square lattice. The lemniscatic case occurs when $c>0$ and $\omega_{1}>0$.

• (c)

If $c=0$, then

23.22.3 $2\omega_{1}=2e^{-\pi i/3}\omega_{3}=\frac{\left(\Gamma\left(\frac{1}{3}\right)% \right)^{3}}{2\pi d^{1/6}}.$

There are 6 possible pairs ($2\omega_{1}$, $2\omega_{3}$), corresponding to the 6 rotations of a lattice of equilateral triangles. The equianharmonic case occurs when $d>0$ and $\omega_{1}>0$.

• ##### 9: 25.16 Mathematical Applications
$H\left(s\right)$ is the special case $H\left(s,1\right)$ of the function …
##### 10: 16.8 Differential Equations
Analytical continuation formulas for ${{}_{q+1}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)$ near $z=1$ are given in Bühring (1987b) for the case $q=2$, and in Bühring (1992) for the general case. …