# cases of failure

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## 1—10 of 277 matching pages

##### 1: 2.11 Remainder Terms; Stokes Phenomenon

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►Secondly, the asymptotic series represents an infinite class of functions, and the remainder depends on which member we have in mind.
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►In the transition through $\theta =\pi $, $\mathrm{erfc}\left(\sqrt{\frac{1}{2}\rho}c(\theta )\right)$ 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
$\rho =100$.
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►uniformly with respect to $\mathrm{ph}z$ in each case.
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##### 2: 23.23 Tables

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►2 in Abramowitz and Stegun (1964) gives values of $\mathrm{\wp}\left(z\right)$, ${\mathrm{\wp}}^{\prime}\left(z\right)$, and $\zeta \left(z\right)$ to 7 or 8D in the rectangular and rhombic cases, normalized so that ${\omega}_{1}=1$ and ${\omega}_{3}=\mathrm{i}a$ (rectangular case), or ${\omega}_{1}=1$ and ${\omega}_{3}=\frac{1}{2}+\mathrm{i}a$ (rhombic case), for $a$ = 1.
…05, and in the case of $\mathrm{\wp}\left(z\right)$ the user may deduce values for complex $z$ by application of the addition theorem (23.10.1).
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##### 3: 2.6 Distributional Methods

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2.6.6
$$S(x)\sim \frac{2\pi}{\sqrt{3}}\sum _{s=0}^{\mathrm{\infty}}{(-1)}^{s}\left(\genfrac{}{}{0.0pt}{}{-\frac{1}{3}}{s}\right){x}^{-s-(1/3)},$$
$x\to \mathrm{\infty}$.

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►Similarly, in the case
$\alpha =1$, we define
…In either case, we define the distribution associated with ${f}_{n}(t)$ by
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##### 4: 18.31 Bernstein–Szegő Polynomials

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►In consequence, ${p}_{n}(\mathrm{cos}\theta )$ can be given explicitly in terms of $\rho (\mathrm{cos}\theta )$ and sines and cosines, provided that $$ in the first case, $$ in the second case, and $$ in the third case.
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##### 5: 12.18 Methods of Computation

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►Because PCFs are special cases of confluent hypergeometric functions, the methods of computation described in §13.29 are applicable to PCFs.
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##### 6: 17.16 Mathematical Applications

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►Many special cases of $q$-series arise in the theory of partitions, a topic treated in §§27.14(i) and 26.9.
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##### 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

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►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values.
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►Whittaker’s notation ${D}_{\nu}\left(z\right)$ is useful when $\nu $ is a nonnegative integer (Hermite polynomial case).

##### 9: Possible Errors in DLMF

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►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 TeXfound under Encodings in the Info boxes (see About MathML).
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##### 10: 16.25 Methods of Computation

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►In these cases integration, or recurrence, in either a forward or a backward direction is unstable.
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