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21: 33.19 Power-Series Expansions in $r$

… ►The expansions (33.19.1) and (33.19.3) converge for all finite values of $r$, except $r=0$ in the case of (33.19.3).

22: 8.18 Asymptotic Expansions of $I_x(a,b)$

… ►for each $n=0,1,2,\mathrm{\dots }$. … ►

Symmetric Case

… ►All of the $c_k(\eta )$ are analytic at $\eta =0$. ►

General Case

… ►Let $\mu =b/a$, and $x_0$ again be as in (8.18.8). …

23: 18.18 Sums

… ►This is the case $\alpha =\beta =0$ of Jacobi. … ►These Poisson kernels are positive, provided that $x,y$ are real, , and in the case of (18.18.27) $x,y\ge 0$. …

24: 18.38 Mathematical Applications

… ►also the case $\beta =0$ of (18.14.26), was used in de Branges’ proof of the long-standing Bieberbach conjecture concerning univalent functions on the unit disk in the complex plane. …

25: 10.43 Integrals

… ►

10.43.2 $$\int z^{\nu }{𝒵}_{\nu }\left(z\right)dz={\pi }^{\frac{1}{2}}{2}^{\nu -1}\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)z\left({𝒵}_{\nu }\left(z\right){𝐋}_{\nu -1}\left(z\right)-{𝒵}_{\nu -1}\left(z\right){𝐋}_{\nu }\left(z\right)\right),$$ $\nu \ne -\frac{1}{2}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

11.1.8 (Case $\nu =0$ only)

Referenced by:

§10.43(i)

Permalink:

http://dlmf.nist.gov/10.43.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(i), §10.43 and Ch.10

… ►

10.43.25 $${\int }_0^{\mathrm{\infty }}{K}_{\nu }\left(bt\right)\exp \left(-p^2{t}^2\right)dt=\frac{\sqrt{\pi }}{4p}\sec \left(\frac{1}{2}\pi \nu \right)\exp \left(\frac{b^2}{8p^2}\right)K_{\frac{1}{2}\nu }\left(\frac{b^2}{8p^2}\right),$$ , $\mathrm{\Re }\left(p^2\right)>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $\sec z$: secant function and $\nu$: complex parameter

A&S Ref:

11.4.32 (Case $\nu =0$)

Referenced by:

§10.43(iv)

Permalink:

http://dlmf.nist.gov/10.43.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iv), §10.43 and Ch.10

…

26: 3.6 Linear Difference Equations

… ►Within this framework forward and backward recursion may be regarded as the special cases $\mathrm{\ell }=0$ and $\mathrm{\ell }=k$, respectively. …

27: 20.1 Special Notation

… ► ►►►

$m$, $n$	integers.
…
$q$ $(\in ℂ)$	the nome, $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$, . Since $\tau$ is not a single-valued function of $q$, it is assumed that $\tau$ is known, even when $q$ is specified. Most applications concern the rectangular case $\mathrm{\Re }\tau =0$, $\mathrm{\Im }\tau >0$, so that and $\tau$ and $q$ are uniquely related.
…

…

28: 26.9 Integer Partitions: Restricted Number and Part Size

… ►In the present chapter $m\ge n\ge 0$ in all cases. …

29: 36.4 Bifurcation Sets

… ►

Special Cases

► $K=1$, fold bifurcation set: … ► $K=2$, cusp bifurcation set: … ► $K=3$, swallowtail bifurcation set: … ►The $+$ sign labels the cusped sheet; the $-$ sign labels the sheet that is smooth for $z\ne 0$ (see Figure 36.4.4). …

30: 32.10 Special Function Solutions

… ►In the case when $n=0$ in (32.10.15), the Riccati equation is … ►In the case when $n=0$ in (32.10.23), the Riccati equation is …