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31: 6.10 Other Series Expansions

… ►

$c_0=1,$

►

$c_1=-1,$

►

$c_2=\frac{1}{2},$

… ►

6.10.8 $$\mathrm{Ein}\left(z\right)=z{\mathrm{e}}^{-z/2}\left({𝗂}_0^{(1)}\left(\frac{1}{2}z\right)+\sum _{n=1}^{\mathrm{\infty }}\frac{2n+1}{n(n+1)}{𝗂}_n^{(1)}\left(\frac{1}{2}z\right)\right).$$

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Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\mathrm{e}$: base of natural logarithm, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§6.10(ii)

Permalink:

http://dlmf.nist.gov/6.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §6.10(ii), §6.10 and Ch.6

… ►An expansion for $E_1\left(z\right)$ can be obtained by combining (6.2.4) and (6.10.8).

32: 10.49 Explicit Formulas

… ►Define $a_k(\nu )$ as in (10.17.1). … ►Again, with $a_k(n+\frac{1}{2})$ as in (10.49.1), ►

10.49.8 $${𝗂}_n^{(1)}\left(z\right)=\frac{1}{2}{\mathrm{e}}^z\sum _{k=0}^n{(-1)}^k\frac{a_k(n+\frac{1}{2})}{z^{k+1}}+{(-1)}^{n+1}\frac{1}{2}{\mathrm{e}}^{-z}\sum _{k=0}^n\frac{a_k(n+\frac{1}{2})}{z^{k+1}}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer, $z$: complex variable and $a_k(\nu )$: polynomial coefficient

A&S Ref:

10.2.9 (modified)

Referenced by:

§10.52(ii)

Permalink:

http://dlmf.nist.gov/10.49.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.49(ii), §10.49 and Ch.10

►

${𝗂}_0^{(1)}\left(z\right)={\displaystyle \frac{\sinh z}{z}},$

… ► ${\sum }_{k=0}^n{a}_k(n+\frac{1}{2})z^{n-k}$ is sometimes called the Bessel polynomial of degree $n$. …

33: 1.12 Continued Fractions

… ► $b_0+{\displaystyle \frac{a_1}{b_1+}\frac{a_2}{b_2+}\mathrm{\cdots }}$ is equivalent to $b_0^{\prime }+{\displaystyle \frac{a_1^{\prime }}{b_1^{\prime }+}\frac{a_2^{\prime }}{b_2^{\prime }+}\mathrm{\cdots }}$ if there is a sequence ${\{d_n\}}_{n=0}^{\mathrm{\infty }}$, $d_0=1$,
$d_n\ne 0$, such that … ►when $p_k\ne 0$, $k=1,2,3,\mathrm{\dots }$. …when $c_k\ne 0$, $k=1,2,3,\mathrm{\dots }$. … ►The continued fraction ${\displaystyle \frac{a_1}{b_1+}\frac{a_2}{b_2+}\mathrm{\cdots }}$ converges when … ►Let the elements of the continued fraction ${\displaystyle \frac{1}{b_1+}\frac{1}{b_2+}\mathrm{\cdots }}$ satisfy …

34: 10.52 Limiting Forms

… ►

10.52.1 $${𝗃}_n\left(z\right),{𝗂}_n^{(1)}\left(z\right)\sim z^n/(2n+1)!!,$$

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Symbols:

$\sim$: asymptotic equality, $!!$: double factorial, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.4

Permalink:

http://dlmf.nist.gov/10.52.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.52(i), §10.52 and Ch.10

►

10.52.2 $$-{𝗒}_n\left(z\right),\mathrm{i}{𝗁}_n^{(1)}\left(z\right),-\mathrm{i}{𝗁}_n^{(2)}\left(z\right),{(-1)}^n{𝗂}_n^{(2)}\left(z\right),(2/\pi ){𝗄}_n\left(z\right)\sim (2n-1)!!/z^{n+1}.$$

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!!$: double factorial, $\mathrm{i}$: imaginary unit, ${𝗂}_n^{(2)}\left(z\right)$: modified spherical Bessel function, ${𝗄}_n\left(z\right)$: modified spherical Bessel function, ${𝗒}_n\left(z\right)$: spherical Bessel function of the second kind, ${𝗁}_n^{(1)}\left(z\right)$: spherical Bessel function of the third kind, ${𝗁}_n^{(2)}\left(z\right)$: spherical Bessel function of the third kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.5

Permalink:

http://dlmf.nist.gov/10.52.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.52(i), §10.52 and Ch.10

… ►

${𝗁}_n^{(1)}\left(z\right)\sim {\mathrm{i}}^{-n-1}{z}^{-1}{\mathrm{e}}^{\mathrm{i}z},$

►

${𝗁}_n^{(2)}\left(z\right)\sim {\mathrm{i}}^{n+1}{z}^{-1}{\mathrm{e}}^{-\mathrm{i}z},$

►

10.52.5 $${𝗂}_n^{(1)}\left(z\right)\sim {𝗂}_n^{(2)}\left(z\right)\sim \frac{1}{2}{z}^{-1}{\mathrm{e}}^z,$$ ,

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, ${𝗂}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $z$: complex variable and $\delta$: small positive constant

Permalink:

http://dlmf.nist.gov/10.52.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.52(ii), §10.52 and Ch.10

…

35: 15.11 Riemann’s Differential Equation

… ►

15.11.2 $$a_1+a_2+b_1+b_2+c_1+c_2=1.$$

ⓘ

Symbols:

$a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(i), §15.11 and Ch.15

►Here $\{a_1,a_2\}$, $\{b_1,b_2\}$, $\{c_1,c_2\}$ are the exponent pairs at the points $\alpha$, $\beta$, $\gamma$, respectively. Cases in which there are fewer than three singularities are included automatically by allowing the choice $\{0,1\}$ for exponent pairs. … ►where $\kappa$, $\lambda$, $\mu$, $\nu$ are real or complex constants such that $\kappa \nu -\lambda \mu =1$. … ►

15.11.8

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Symbols:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\lambda$: constant and $\mu$: constant

Permalink:

http://dlmf.nist.gov/15.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(ii), §15.11 and Ch.15

…

36: 16.4 Argument Unity

… ►The function ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;z)$ is well-poised if …It is very well-poised if it is well-poised and $a_1=b_1+1$. … ►The special case ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;1)$ is $k$-balanced if $a_{q+1}$ is a nonpositive integer and … ►with limiting form $a\left(\psi \left(a+n+1\right)-\psi \left(a\right)\right)=\frac{a\frac{d}{da}{\left(a\right)}_{n+1}}{{\left(a\right)}_{n+1}}$ in the case that $c=a+1$. … ►Transformations for both balanced ${}_4{}^{}F_3^{}\left(1\right)$ and very well-poised ${}_7{}^{}F_6^{}\left(1\right)$ are included in Bailey (1964, pp. 56–63). …

37: 24.12 Zeros

… ►In the interval $0\le x\le 1$ the only zeros of $B_{2n+1}\left(x\right)$, $n=1,2,\mathrm{\dots }$, are $0,\frac{1}{2},1$, and the only zeros of $B_{2n}\left(x\right)-B_{2n}$, $n=1,2,\mathrm{\dots }$, are $0,1$. … ►Then the zeros in the interval are $1-x_j^{(n)}$. … ►When $n$ is odd $x_1^{(n)}=\frac{1}{2}$, $x_2^{(n)}=1$ $(n\ge 3)$, and as $n\to \mathrm{\infty }$ with $m(\ge 1)$ fixed, … ►Then the zeros in the interval are $1-y_j^{(n)}$. … ►When $n$ is odd $y_1^{(n)}=\frac{1}{2}$, …

38: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►Let $p$ and $q$ be nonnegative integers; $a_1,\mathrm{\dots },a_p\in ℂ$; $b_1,\mathrm{\dots },b_q\in ℂ$; $-b_j+\frac{1}{2}(k+1)\notin \mathbb{N}$, $1\le j\le q$, $1\le k\le m$. … ►If $-a_j+\frac{1}{2}(k+1)\in \mathbb{N}$ for some $j,k$ satisfying $1\le j\le p$, $1\le k\le m$, then the series expansion (35.8.1) terminates. … ►Let $c=b_1+b_2-a_1-a_2-a_3$. … ►Let $a_1+a_2+a_3+\frac{1}{2}(m+1)=b_1+b_2$; one of the $a_j$ be a negative integer; $\mathrm{\Re }\left(b_1-a_1\right)$, $\mathrm{\Re }\left(b_1-a_2\right)$, $\mathrm{\Re }\left(b_1-a_3\right)$, $\mathrm{\Re }\left(b_1-a_1-a_2-a_3\right)>\frac{1}{2}(m-1)$. … ►Again, let $c=b_1+b_2-a_1-a_2-a_3$. …

39: 6.9 Continued Fraction

… ►

6.9.1 $$E_1\left(z\right)=\frac{{\mathrm{e}}^{-z}}{z+}\frac{1}{1+}\frac{1}{z+}\frac{2}{1+}\frac{2}{z+}\frac{3}{1+}\frac{3}{z+}\mathrm{\cdots },$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

5.1.22 (special case of)

Referenced by:

§6.18(i)

Permalink:

http://dlmf.nist.gov/6.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.9 and Ch.6

…

40: 8.5 Confluent Hypergeometric Representations

… ►For the confluent hypergeometric functions $M$, $𝐌$, $U$, and the Whittaker functions $M_{\kappa ,\mu }$ and $W_{\kappa ,\mu }$, see §§13.2(i) and 13.14(i). ►

8.5.1 $$\gamma (a,z)=a^{-1}{z}^a{\mathrm{e}}^{-z}M(1,1+a,z)=a^{-1}{z}^{a}M(a,1+a,-z),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.12

Referenced by:

§8.11(ii), §8.14

Permalink:

http://dlmf.nist.gov/8.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8

… ►

8.5.3 $$\mathrm{\Gamma }(a,z)={\mathrm{e}}^{-z}U(1-a,1-a,z)=z^a{\mathrm{e}}^{-z}U(1,1+a,z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.14, §8.19(vi)

Permalink:

http://dlmf.nist.gov/8.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8

►

8.5.4 $$\gamma (a,z)=a^{-1}{z}^{\frac{1}{2}a-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{2}z}{M}_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.5

Permalink:

http://dlmf.nist.gov/8.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8

►

8.5.5 $$\mathrm{\Gamma }(a,z)={\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}a-\frac{1}{2}}{W}_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8