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31: 1.17 Integral and Series Representations of the Dirac Delta

… ►for a suitably chosen sequence of functions ${\delta }_n\left(x\right)$, $n=1,2,\mathrm{\dots }$. … ►In this case … ►However, for $n=1,2,\mathrm{\dots }$, …provided that $\varphi (x)$ is continuous when $x\in (-\mathrm{\infty },\mathrm{\infty })$, and for each $a$, ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$ (as in the case of (1.17.6)). … ►(1.17.22)–(1.17.24) are special cases of Morse and Feshbach (1953a, Eq. (6.3.11)). …

32: 8.2 Definitions and Basic Properties

… ►However, when the integration paths do not cross the negative real axis, and in the case of (8.2.2) exclude the origin, $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ take their principal values; compare §4.2(i). … ►When $z\ne 0$, $\mathrm{\Gamma }(a,z)$ is an entire function of $a$, and $\gamma (a,z)$ is meromorphic with simple poles at $a=-n$, $n=0,1,2,\mathrm{\dots }$, with residue ${(-1)}^n/n!$. ►For $m\in \mathbb{Z}$, …For example, in the case $m=-1$ we have … ►If $w=\gamma (a,z)$ or $\mathrm{\Gamma }(a,z)$, then …

33: 28.11 Expansions in Series of Mathieu Functions

… ►Let $f(z)$ be a $2\pi$-periodic function that is analytic in an open doubly-infinite strip $S$ that contains the real axis, and $q$ be a normal value (§28.7). …See Meixner and Schäfke (1954, §2.28), and for expansions in the case of the exceptional values of $q$ see Meixner et al. (1980, p. 33). … ►

28.11.4 $$\cos 2mz=\sum _{n=0}^{\mathrm{\infty }}{A}_{2m}^{2n}(q){\mathrm{ce}}_{2n}(z,q),$$ $m\ne 0$,

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E4

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See also:

Annotations for §28.11, §28.11 and Ch.28

►

28.11.5 $$\cos (2m+1)z=\sum _{n=0}^{\mathrm{\infty }}{A}_{2m+1}^{2n+1}(q){\mathrm{ce}}_{2n+1}(z,q),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E5

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See also:

Annotations for §28.11, §28.11 and Ch.28

… ►

28.11.7 $$\sin (2m+2)z=\sum _{n=0}^{\mathrm{\infty }}{B}_{2m+2}^{2n+2}(q){\mathrm{se}}_{2n+2}(z,q).$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E7

Encodings:

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See also:

Annotations for §28.11, §28.11 and Ch.28

34: 28.31 Equations of Whittaker–Hill and Ince

… ►and $m=0,1,\mathrm{\dots },n$ in all cases. … ►with $\eta =a_p^m(\xi )$, $\eta =b_p^m(\xi )$, respectively. … ►For $m_1\ne m_2$, …More important are the double orthogonality relations for $p_1\ne p_2$ or $m_1\ne m_2$ or both, given by …and also for all $p_1,p_2,m_1,m_2$, given by …

35: 13.11 Series

… ►

13.11.1 $$M(a,b,z)=\mathrm{\Gamma }\left(a-\frac{1}{2}\right){\mathrm{e}}^{\frac{1}{2}z}{\left(\frac{1}{4}z\right)}^{\frac{1}{2}-a}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(2a-1\right)}_s{\left(2a-b\right)}_s}{{\left(b\right)}_{s}s!}\left(a-\frac{1}{2}+s\right)I_{a-\frac{1}{2}+s}\left(\frac{1}{2}z\right),$$ $a+\frac{1}{2},b\ne 0,-1,-2,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.6(iii), §13.6(iii), §8.7

Permalink:

http://dlmf.nist.gov/13.11.E1

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Annotations for §13.11 and Ch.13

►

13.11.2 $$M(a,b,z)=\mathrm{\Gamma }\left(b-a-\frac{1}{2}\right){\mathrm{e}}^{\frac{1}{2}z}{\left(\frac{1}{4}z\right)}^{a-b+\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(2b-2a-1\right)}_s{\left(b-2a\right)}_s(b-a-\frac{1}{2}+s)}{{\left(b\right)}_{s}s!}{I}_{b-a-\frac{1}{2}+s}\left(\frac{1}{2}z\right),$$ $b-a+\frac{1}{2},b\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $s$: nonnegative integer and $z$: complex variable

Proof sketch:

Combine (13.24.1) with (13.14.4).

A&S Ref:

13.3.6

Referenced by:

§13.11, §13.6(iii), §13.6(iii), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/13.11.E2

Encodings:

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Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §13.11 and Ch.13

►(13.6.9), (13.6.11_1) and (13.6.11_2) are special cases. ►

13.11.3 $$𝐌(a,b,z)={\mathrm{e}}^{\frac{1}{2}z}\sum _{s=0}^{\mathrm{\infty }}{A}_s{\left(b-2a\right)}^{\frac{1}{2}(1-b-s)}{\left(\frac{1}{2}z\right)}^{\frac{1}{2}(1-b+s)}{J}_{b-1+s}\left(\sqrt{2z(b-2a)}\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $s$: nonnegative integer, $z$: complex variable and $A_n$: coefficients

Source:

Slater (1960, §3.8)

A&S Ref:

13.3.7

Referenced by:

§13.11, Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/13.11.E3

Encodings:

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Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §13.11 and Ch.13

… ►

$A_2={\displaystyle \frac{1}{2}}b,$

…

36: 18.21 Hahn Class: Interrelations

… ►

$M_n(x;\beta ,c)=M_x(n;\beta ,c),$ $n,x=0,1,2,\mathrm{\dots }$.

… ►

§18.21(ii) Limit Relations and Special Cases

… ►

18.21.4 $$\underset{N\to \mathrm{\infty }}{lim}{Q}_n(x;\beta -1,N(c^{-1}-1),N)=M_n(x;\beta ,c).$$

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Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $M_n(x;\beta ,c)$: Meixner polynomial, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii)

Permalink:

http://dlmf.nist.gov/18.21.E4

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Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

… ►

18.21.7 $$\underset{\beta \to \mathrm{\infty }}{lim}{M}_n(x;\beta ,a{(a+\beta )}^{-1})=C_n(x;a).$$

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Symbols:

$C_n(x;a)$: Charlier polynomial, $M_n(x;\beta ,c)$: Meixner polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii), §18.22(iii)

Permalink:

http://dlmf.nist.gov/18.21.E7

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Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

… ►

►►

Figure 18.21.1: Askey scheme. …(This is with the convention that the real and imaginary parts of the parameters are counted separately in the case of the continuous Hahn polynomials.) Magnify

37: 18.39 Applications in the Physical Sciences

… ►In the case of a single electron, charge $-e$ and mass $m_e$, interacting with a fixed (infinite mass) nucleus of charge $+Ze$ at the co-ordinate origin, with the choice of SI units, $V(r)=-Z{e}^2/(4\pi {ϵ}_{0}r)$. …

38: 18.14 Inequalities

… ►When $(\alpha +\frac{1}{2})(\beta +\frac{1}{2})>0$ choose $m$ so that … ►except that when $\alpha =\beta =-\frac{1}{2}$ (Chebyshev case) $|P_n^{(\alpha ,\beta )}\left(x_{n,m}\right)|$ is constant. … ►When $\alpha >-\frac{1}{2}$ choose $m$ so that … ►Also, when $\alpha \le -\frac{1}{2}$ … ►The case $\beta =0$ of (18.14.26) is the Askey–Gasper inequality (18.38.3). …

39: 13.6 Relations to Other Functions

… ►Special cases are the error functions … ►When $b=2a$ the Kummer functions can be expressed as modified Bessel functions. …and in the case that $b-2a$ is an integer we have …Note that (13.6.11_1) and (13.6.11_2) are special cases of (13.11.1) and (13.11.2), respectively … ►Special cases of §13.6(iv) are as follows. …

40: 10.22 Integrals

… ►When $\alpha =m=1,2,3,\mathrm{\dots }$ the left-hand side of (10.22.36) is the $m$th repeated integral of $J_{\nu }\left(x\right)$ (§§1.4(v) and 1.15(vi)). … ►For the confluent hypergeometric function $𝐌$ see §13.2(i). … ►

Weber–Schafheitlin Discontinuous Integrals, including Special Cases

… ►When $n=0,1,2,\mathrm{\dots }$ and $\mathrm{\Re }\mu >-n-1$, … ►Equation (10.22.70) also remains valid if the order $\nu +1$ of the $J$ functions on both sides is replaced by $\nu +2n-3$, $n=1,2,\mathrm{\dots }$, and the constraint $\mathrm{\Re }\nu >-\frac{3}{2}$ is replaced by $\mathrm{\Re }\nu >-n+\frac{1}{2}$. …