# cases of failure

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##### 1: 2.11 Remainder Terms; Stokes Phenomenon
Secondly, the asymptotic series represents an infinite class of functions, and the remainder depends on which member we have in mind. … In the transition through $\theta=\pi$, $\operatorname{erfc}\left(\sqrt{\frac{1}{2}\rho}\;c(\theta)\right)$ changes very rapidly, but smoothly, from one form to the other; compare the graph of its modulus in Figure 2.11.1 in the case $\rho=100$. … uniformly with respect to $\operatorname{ph}z$ in each case. …
##### 2: 2.6 Distributional Methods
2.6.6 $S(x)\sim\frac{2\pi}{\sqrt{3}}\sum_{s=0}^{\infty}(-1)^{s}{\genfrac{(}{)}{0.0pt}% {}{-\frac{1}{3}}{s}}x^{-s-(1/3)},$ $x\to\infty$.
Similarly, in the case $\alpha=1$, we define …In either case, we define the distribution associated with $f_{n}(t)$ by …
##### 3: 23.23 Tables
2 in Abramowitz and Stegun (1964) gives values of $\wp\left(z\right)$, $\wp'\left(z\right)$, and $\zeta\left(z\right)$ to 7 or 8D in the rectangular and rhombic cases, normalized so that $\omega_{1}=1$ and $\omega_{3}=ia$ (rectangular case), or $\omega_{1}=1$ and $\omega_{3}=\tfrac{1}{2}+ia$ (rhombic case), for $a$ = 1. …05, and in the case of $\wp\left(z\right)$ the user may deduce values for complex $z$ by application of the addition theorem (23.10.1). …
##### 4: 18.31 Bernstein–Szegő Polynomials
In consequence, $p_{n}(\cos\theta)$ can be given explicitly in terms of $\rho(\cos\theta)$ and sines and cosines, provided that $\ell<2n$ in the first case, $\ell<2n+2$ in the second case, and $\ell<2n+1$ in the third case. …
##### 5: 12.18 Methods of Computation
Because PCFs are special cases of confluent hypergeometric functions, the methods of computation described in §13.29 are applicable to PCFs. …
##### 6: 17.16 Mathematical Applications
Many special cases of $q$-series arise in the theory of partitions, a topic treated in §§27.14(i) and 26.9. …
##### 7: 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. …As a corollary, special cases of the ${{}_{1}F_{1}}$ and ${{}_{2}F_{1}}$ functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer $G$-function. …
##### 8: 12.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. … Whittaker’s notation $D_{\nu}\left(z\right)$ is useful when $\nu$ is a nonnegative integer (Hermite polynomial case).
##### 9: Possible Errors in DLMF
There are also cases where browser bugs or poor fonts can be misleading; you can verify MathML display by comparing the to the images or found under Encodings in the Info boxes (see About MathML). …
##### 10: 16.25 Methods of Computation
In these cases integration, or recurrence, in either a forward or a backward direction is unstable. …