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11: 4.45 Methods of Computation

… ►Then we take square roots repeatedly until $|y|$ is sufficiently small, where … ►As an example, take $x=9.47376$. …

12: 8.13 Zeros

… ►When $-5\le a\le 4$ the behavior of the $x$-zeros as functions of $a$ can be seen by taking the slice ${\gamma }^{\ast }(a,x)=0$ of the surface depicted in Figure 8.3.6. …

13: 31.11 Expansions in Series of Hypergeometric Functions

… ►Taking $P_j=P_j^5$ or $P_j=P_j^6$ the coefficients $c_j$ satisfy the equations …where we take $c_0=1$ and where …

14: 2.11 Remainder Terms; Stokes Phenomenon

… ►Taking $m=10$ in (2.11.2), the first three terms give us the approximation … ►Optimum truncation in (2.11.6) takes place at $s=n-1$, with $|p+n-1|=|z|$, approximately. … ►Where should the change-over take place? Can it be accomplished smoothly? … ►In particular, on the ray $\theta =\pi$ greatest accuracy is achieved by (a) taking the average of the expansions (2.11.6) and (2.11.7), followed by (b) taking account of the exponentially-small contributions arising from the terms involving $h_{2s}(\theta ,\alpha )$ in (2.11.15). … ►Taking $x=5$ and rounding to 5D, we obtain …

15: 17.4 Basic Hypergeometric Functions

… ►Here and elsewhere it is assumed that the $b_j$ do not take any of the values $q^{-n}$. … ►Here and elsewhere the $b_j$ must not take any of the values $q^{-n}$, and the $a_j$ must not take any of the values $q^{n+1}$. …

16: 5.13 Integrals

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17: 10.34 Analytic Continuation

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18: 10.69 Uniform Asymptotic Expansions for Large Order

… ►All fractional powers take their principal values. …

19: 14.1 Special Notation

… ►Multivalued functions take their principal values (§4.2(i)) unless indicated otherwise. …

20: 14.16 Zeros

… ► ${𝖰}_{\nu }^{\mu }\left(x\right)$ has $\max (\lceil \nu -|\mu |\rceil ,0)+k$ zeros in the interval $(-1,1)$, where $k$ can take one of the values $-1$, $0$, $1$, $2$, subject to $\max (\lceil \nu -|\mu |\rceil ,0)+k$ being even or odd according as $\cos \left(\nu \pi \right)$ and $\cos \left(\mu \pi \right)$ have opposite signs or the same sign. …