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21: 6.10 Other Series Expansions

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6.10.6 $$\mathrm{Ei}\left(x\right)=\gamma +\ln \left|x\right|+\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n(x-a_n){\left({𝗂}_n^{(1)}\left(\frac{1}{2}x\right)\right)}^2,$$ $x\ne 0$,

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ei}\left(x\right)$: exponential integral, $\ln z$: principal branch of logarithm function, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $x$: real variable, $n$: nonnegative integer and $a_n$: coefficients

Referenced by:

§6.18(i)

Permalink:

http://dlmf.nist.gov/6.10.E6

Encodings:

pMML, png, TeX

See also:

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

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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).

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

§8.18 Asymptotic Expansions of $I_x(a,b)$

… ►If $b=1,2,3,\mathrm{\dots }$ and $n\ge b$, then the $O$-term can be omitted and the result is exact. ►If $b\to \mathrm{\infty }$ and $a$ and $x$ are fixed, with $a>0$ and , then (8.18.1), with $a$ and $b$ interchanged and $x$ replaced by $1-x$, can be combined with (8.17.4). … ►A recurrence relation for the $d_k$ can be found in Nemes and Olde Daalhuis (2016). … ►

8.18.18 $$I_x(a,b)=p,$$ $0\le p\le 1$,

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Symbols:

$I_x(a,b)$: incomplete beta function, $p$: parameter, $a$: parameter, $b$: parameter and $x$: variable

Permalink:

http://dlmf.nist.gov/8.18.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

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23: 2.3 Integrals of a Real Variable

… ►Other types of singular behavior in the integrand can be treated in an analogous manner. … ►

2.3.13 $$I(x)={\int }_a^b{\mathrm{e}}^{-xp(t)}q(t)dt$$

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Defines:

$I(x)$: integral (locally)

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $a$: left endpoint, $b$: right endpoint, $p(t)$: real function and $q(t)$: function

Referenced by:

item (c)

Permalink:

http://dlmf.nist.gov/2.3.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(iii), §2.3 and Ch.2

… ►Watson’s lemma can be regarded as a special case of this result. … ►oscillate rapidly and cancel themselves over most of the range. … ►A uniform approximation can be constructed by quadratic change of integration variable: …

24: 10.45 Functions of Imaginary Order

… ►and ${\tilde{I}}_{\nu }\left(x\right)$, ${\tilde{K}}_{\nu }\left(x\right)$ are real and linearly independent solutions of (10.45.1): … ►In consequence of (10.45.5)–(10.45.7), ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.45.1) when $x$ is large, and either ${\tilde{I}}_{\nu }\left(x\right)$ and $(1/\pi )\sinh \left(\pi \nu \right){\tilde{K}}_{\nu }\left(x\right)$, or ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$, comprise a numerically satisfactory pair when $x$ is small, depending whether $\nu \ne 0$ or $\nu =0$. … ►For graphs of ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$ see §10.26(iii). ►For properties of ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$, including uniform asymptotic expansions for large $\nu$ and zeros, see Dunster (1990a). In this reference ${\tilde{I}}_{\nu }\left(x\right)$ is denoted by $(1/\pi )\sinh \left(\pi \nu \right)L_{\mathrm{i}\nu }(x)$. …

25: 8.28 Software

… ►Citations in bulleted lists refer to papers for which research software has been made available and can be downloaded via the Web. …

26: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

… ►

10.46.2 $$I_{\nu }\left(z\right)={\left(\frac{1}{2}z\right)}^{\nu }\varphi (1,\nu +1;\frac{1}{4}{z}^2).$$

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Symbols:

$\varphi (\rho ,\beta ;z)$: generalized Bessel function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.46.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.46 and Ch.10

… ►The Laplace transform of $\varphi (\rho ,\beta ;z)$ can be expressed in terms of the Mittag-Leffler function: …

27: 1.2 Elementary Algebra

… ►If $𝐮$, $𝐯$, $\alpha$ and $\beta$ are real the complex conjugate bars can be omitted in (1.2.40)–(1.2.42). … ►A vector of $l^2$ norm unity is normalized and every non-zero vector $𝐯$ can be normalized via $𝐯\to 𝐯/\Vert 𝐯\Vert$. … ►The identity matrix $𝐈$, is defined as … ►We take ${𝐄}_n\subset {ℂ}^n$, but we can also restrict ourselves to vectors and matrices with only real elements. … ►Non-defective matrices are precisely the matrices which can be diagonalized via a similarity transformation of the form …

28: 36.12 Uniform Approximation of Integrals

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36.12.1 $$I(𝐲,k)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(\mathrm{i}kf(u;𝐲)\right)g(u,𝐲)du,$$

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Defines:

$I(𝐲,k)$: oscillatory integral (locally)

Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $k$: variable, $g(u,𝐲)$: function, $f(u,𝐲)$: function and $u(t;𝐲)$: mapping

Referenced by:

§36.13

Permalink:

http://dlmf.nist.gov/36.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §36.12(i), §36.12 and Ch.36

… ►As $𝐲$ varies as many as $K+1$ (real or complex) critical points of the smooth phase function $f$ can coalesce in clusters of two or more. …If , then we may evaluate the complex conjugate of $I$ for real values of $𝐲$ and $g$, and obtain $I$ by conjugation and analytic continuation. … ►This technique can be applied to generate a hierarchy of approximations for the diffraction catastrophes ${\mathrm{\Psi }}_K(𝐱;k)$ in (36.2.10) away from $𝐱=\mathrm{𝟎}$, in terms of canonical integrals ${\mathrm{\Psi }}_J\left(\xi (𝐱;k)\right)$ for . For example, the diffraction catastrophe ${\mathrm{\Psi }}_2(x,y;k)$ defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function ${\mathrm{\Psi }}_1\left(\xi (x,y;k)\right)$ when $k$ is large, provided that $x$ and $y$ are not small. …

29: 11.6 Asymptotic Expansions

… ►More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)). … ►These and higher coefficients $c_k(\lambda )$ can be computed via the representations in Nemes (2015b). … ►

11.6.9 $${𝐋}_{\nu }\left(\lambda \nu \right)\sim I_{\nu }\left(\lambda \nu \right),$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$,

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{ph}$: phase, $\nu$: real or complex order, $\delta$: arbitrary small positive constant and $\lambda$: parameter

Referenced by:

§11.6(iii)

Permalink:

http://dlmf.nist.gov/11.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(iii), §11.6 and Ch.11

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30: 23.20 Mathematical Applications

… ►An algebraic curve that can be put either into the form … ►Then $\mathrm{\varnothing }\subseteq T\subseteq I\subseteq K\subseteq C$. Both $T,K$ are subgroups of $C$, though $I$ may not be. …Both $T$ and $I$ are finite sets. …The order of a point (if finite and not already determined) can have only the values 3, 5, 6, 7, 9, 10, or 12, and so can be found from $2P=-P$, $4P=-P$, $4P=-2P$, $8P=P$, $8P=-P$, $8P=-2P$, or $8P=-4P$. …