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21: 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. …

22: 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. …

23: 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.
…

…

24: 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 …

25: 10.22 Integrals

… ►

10.22.2 $$\int z^{\nu }{𝒞}_{\nu }\left(z\right)dz={\pi }^{\frac{1}{2}}{2}^{\nu -1}\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)z\left({𝒞}_{\nu }\left(z\right){𝐇}_{\nu -1}\left(z\right)-{𝒞}_{\nu -1}\left(z\right){𝐇}_{\nu }\left(z\right)\right),$$ $\nu \ne -\frac{1}{2}$.

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Symbols:

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

A&S Ref:

11.17 (Case $\nu =0$ only)

Permalink:

http://dlmf.nist.gov/10.22.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(i), §10.22 and Ch.10

… ►

10.22.13 $${\int }_0^{\frac{1}{2}\pi }{J}_{2\nu }\left(2z\cos \theta \right)\cos \left(2\mu \theta \right)d\theta =\frac{1}{2}\pi J_{\nu +\mu }\left(z\right)J_{\nu -\mu }\left(z\right),$$ $\mathrm{\Re }\nu >-\frac{1}{2}$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $n$: integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

11.4.7 (Case $\nu =n$, $\mu =0$.)

Referenced by:

§10.22(ii)

Permalink:

http://dlmf.nist.gov/10.22.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

►

10.22.14 $${\int }_0^{\pi }{J}_{2\nu }\left(2z\sin \theta \right)\cos \left(2\mu \theta \right)d\theta =\pi \cos \left(\mu \pi \right)J_{\nu +\mu }\left(z\right)J_{\nu -\mu }\left(z\right),$$ $\mathrm{\Re }\nu >-\frac{1}{2}$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $n$: integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

11.4.8 (Case $\nu =0$, $\mu =n$.)

Referenced by:

§10.22(ii)

Permalink:

http://dlmf.nist.gov/10.22.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

… ►

10.22.47 $${\int }_0^{\mathrm{\infty }}\frac{t^{\nu }{Y}_{\nu }\left(at\right)}{t^2+b^2}dt=-b^{\nu -1}{K}_{\nu }\left(ab\right),$$ .

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $\nu$: complex parameter

Keywords:

Mellin transform

A&S Ref:

11.4.46 (is the special case $\nu =0$.)

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.22.E47

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22 and Ch.10

… ►

10.22.65

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $dx$: differential of $x$ , $\int$: integral and $\ln z$: principal branch of logarithm function

A&S Ref:

11.4.43 (Case $b=0$.)

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.22.E65

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

…

26: 10.19 Asymptotic Expansions for Large Order

… ►For higher coefficients in (10.19.8) in the case $a=0$ (that is, in the expansions of $J_{\nu }\left(\nu \right)$ and $Y_{\nu }\left(\nu \right)$), see Watson (1944, §8.21), Temme (1997), and Jentschura and Lötstedt (2012). …

27: 18.7 Interrelations and Limit Relations

… ►

18.7.25

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

Source:

Andrews et al. (1999, (6.4.13))

Referenced by:

§18.7(i), §18.7(iii), Erratum (V1.2.0) for Equation (18.7.25)

Permalink:

http://dlmf.nist.gov/18.7.E25

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

We included the case $n=0$.

See also:

Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

…

28: 10.2 Definitions

… ►Table 10.2.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.2.1) for the stated intervals or regions in the case $\mathrm{\Re }\nu \ge 0$. …

29: 19.20 Special Cases

… ►In the complete case ($x=0$) (19.20.14) reduces to …

30: 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) $x,y\ge 0$. …