case m=2

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11: 13.7 Asymptotic Expansions for Large Argument

… ►unless $a=0,-1,\mathrm{\dots }$ and $b-a=0,-1,\mathrm{\dots }$. … ►Also, when $z\in {\text{R}}_1\cup {\text{R}}_2\cup {\overline{\text{R}}}_2$ … ►where $m$ is an arbitrary nonnegative integer, and …Then as $z\to \mathrm{\infty }$ with $\left|\left|z\right|-n\right|$ bounded and $a,b,m$ fixed … ►For the special case $\mathrm{ph}z=\pm \pi$ see Paris (2013). …

12: 36.2 Catastrophes and Canonical Integrals

… ►Special cases: $K=1$, fold catastrophe; $K=2$, cusp catastrophe; $K=3$, swallowtail catastrophe. … ►with the contour passing to the lower right of $u=0$. … ►

§36.2(ii) Special Cases

… ►Addendum: For further special cases see §36.2(iv) … ►

§36.2(iv) Addendum to 36.2(ii) Special Cases

…

13: 10.16 Relations to Other Functions

… ►

10.16.5 $$J_{\nu }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu }{\mathrm{e}}^{\mp \mathrm{i}z}}{\mathrm{\Gamma }\left(\nu +1\right)}M(\nu +\frac{1}{2},2\nu +1,\pm 2\mathrm{i}z),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.69 (modified)

Referenced by:

§10.16, §10.39

Permalink:

http://dlmf.nist.gov/10.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.16, §10.16 and Ch.10

… ►For the functions $M$ and $U$ see §13.2(i). ►

10.16.7 $$J_{\nu }\left(z\right)=\frac{{\mathrm{e}}^{\mp (2\nu +1)\pi \mathrm{i}/4}}{2^{2\nu }\mathrm{\Gamma }\left(\nu +1\right)}{(2z)}^{-\frac{1}{2}}{M}_{0,\nu }\left(\pm 2\mathrm{i}z\right),$$ $2\nu \ne -1,-2,-3,\mathrm{\dots }$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.16, §10.16

Permalink:

http://dlmf.nist.gov/10.16.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.16, §10.16 and Ch.10

… ►For the functions $M_{0,\nu }$ and $W_{0,\nu }$ see §13.14(i). ►In all cases principal branches correspond at least when $|\mathrm{ph}z|\le \frac{1}{2}\pi$. …

14: 13.2 Definitions and Basic Properties

… ►In other cases … ► … ►When $b=n+1$, $n=0,1,2,\mathrm{\dots }$, and $a=-m$, $m=0,1,2,\mathrm{\dots }$, … ►In all other cases … ►Except when $a=0,-1,\mathrm{\dots }$ (polynomial cases), …

15: 10.70 Zeros

… ►Let $\mu =4{\nu }^2$ and $f(t)$ denote the formal series …If $m$ is a large positive integer, then ►

$$\text{zeros of }{\mathrm{ber}}_{\nu }x\sim \sqrt{2}(t-f(t)),$$ $t=(m-\frac{1}{2}\nu -\frac{3}{8})\pi$,

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$$\text{zeros of }{\mathrm{bei}}_{\nu }x\sim \sqrt{2}(t-f(t)),$$ $t=(m-\frac{1}{2}\nu +\frac{1}{8})\pi$,

… ►In the case $\nu =0$, numerical tabulations (Abramowitz and Stegun (1964, Table 9.12)) indicate that each of (10.70.2) corresponds to the $m$th zero of the function on the left-hand side. …

16: 2.10 Sums and Sequences

… ►In both expansions the remainder term is bounded in absolute value by the first neglected term in the sum, and has the same sign, provided that in the case of (2.10.7), truncation takes place at $s=2m-1$, where $m$ is any positive integer satisfying $m\ge \frac{1}{2}(\alpha +1)$. …

17: 15.2 Definitions and Analytical Properties

… ►For example, when $a=-m$, $m=0,1,2,\mathrm{\dots }$, and $c\ne 0,-1,-2,\mathrm{\dots }$, $F(a,b;c;z)$ is a polynomial: …

18: 3.9 Acceleration of Convergence

… ►where $H_m$ is the Hankel determinant … ►Then $t_{n,2k}={\epsilon }_{2k}^{(n)}$. Aitken’s ${\mathrm{\Delta }}^2$-process is the case $k=1$. ►If $s_n$ is the $n$th partial sum of a power series $f$, then $t_{n,2k}={\epsilon }_{2k}^{(n)}$ is the Padé approximant ${[(n+k)/k]}_f$ (§3.11(iv)). … ►with $s_{\mathrm{\infty }}=\frac{1}{12}{\pi }^2=\mathrm{0.82246\hspace{0.33em}70334\hspace{0.33em}24}\mathrm{\dots }$. …

19: 36.10 Differential Equations

… ►

Special Cases

… ► $K=2$, cusp: … ► $K=3$, swallowtail: … ►

Special Cases

… ► $K=2$, cusp: …

20: 15.12 Asymptotic Approximations

… ►

and $|c+n|\ge \delta$ for all $n\in \{0,1,2,\mathrm{\dots }\}$.

►

$\mathrm{\Re }z=\frac{1}{2}$ and $\left|\mathrm{ph}c\right|\le \pi -\delta$.

… ►Then for fixed $m\in \{0,1,2,\mathrm{\dots }\}$, … ►For the more general case in which $a^2=o\left(c\right)$ and $b^2=o\left(c\right)$ see Wagner (1990). … ►where $q_0(z)=1$ and $q_s(z)$, $s=1,2,\mathrm{\dots }$, are defined by the generating function …