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31: Guide to Searching the DLMF

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Table 1: Query Examples

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Query	Matching records contain
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J_n@(z)=	the math fragment $J_n(z)=$, emphasizing more that $J_n$ is a function.
…

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… ►Wildcards allow matching patterns and marking parts of an expression that don’t matter (as for example, which variable name the author uses for a function): … ►You can use in math queries all the symbols and commands defined in LaTeX (you can omit the $\backslash$), and some additional convenient ones, as well as the special functions’ names: … ►The syntax of the special functions can be LaTeX-like or as employed in widely used computer algebra systems. … ►DLMF search is generally case-insensitive except when it is important to be case-sensitive, as when two different special functions have the same standard names but one name has a lower-case initial and the other is has an upper-case initial, such as si and Si, gamma and Gamma. …

32: 29.20 Methods of Computation

… ►The eigenvalues $a_{\nu }^m\left(k^2\right)$, $b_{\nu }^m\left(k^2\right)$, and the Lamé functions ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$, can be calculated by direct numerical methods applied to the differential equation (29.2.1); see §3.7. … ►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). …

33: Bibliography E

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Interlacing properties of the zeros of Bessel functions. Atti Sem. Mat. Fis. Univ. Modena XLII (2), pp. 525–529.

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External Links:

ISSN 0041-8986, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§10.21(i).

Encodings:

BibTeX

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The zeros of the complementary error function. Numer. Algorithms 49 (1-4), pp. 153–157.

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External Links:

ISSN 1017-1398, Document, FullText, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§7.13(ii).

Encodings:

BibTeX

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On Lamé functions. Philos. Mag. (7) 31, pp. 123–130.

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External Links:

MathReview (G. Szegő), ZentralBlatt

Referenced by:

§29.3(ii).

Encodings:

BibTeX

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On algebraic Lamé functions. Philos. Mag. (7) 32, pp. 348–350.

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External Links:

MathReview (G. Szegő), ZentralBlatt

Referenced by:

§29.17(ii).

Encodings:

BibTeX

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Integral equations for Heun functions. Quart. J. Math., Oxford Ser. 13, pp. 107–112.

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External Links:

Document, MathReview (E. Rothe), ZentralBlatt

Referenced by:

§31.10(i).

Encodings:

BibTeX

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34: 28.19 Expansions in Series of ${\mathrm{me}}_{\nu +2n}$ Functions

… ►Let $q$ be a normal value (§28.12(i)) with respect to $\nu$, and $f(z)$ be a function that is analytic on a doubly-infinite open strip $S$ that contains the real axis. … ►

28.19.1 $$f(z+\pi )={\mathrm{e}}^{\mathrm{i}\nu \pi }f(z).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\nu$: complex parameter and $f(z)$: function

Permalink:

http://dlmf.nist.gov/28.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.19 and Ch.28

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28.19.3 $$f_n=\frac{1}{\pi }{\int }_0^{\pi }f(z){\mathrm{me}}_{\nu +2n}(-z,q)dz.$$

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Defines:

$f_n$: coefficients (locally)

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $\nu$: complex parameter and $f(z)$: function

Permalink:

http://dlmf.nist.gov/28.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.19 and Ch.28

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35: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

… ►Such a solution is given in terms of a Riemann theta function with two phases. …

36: 4.40 Integrals

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4.40.7 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-x}\frac{\sin \left(ax\right)}{\sinh x}dx=\frac{1}{2}\pi \coth \left(\frac{1}{2}\pi a\right)-\frac{1}{a},$$ $a\ne 0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $a$: real or complex constant and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.40.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iii), §4.40 and Ch.4

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4.40.8 $${\int }_0^{\mathrm{\infty }}\frac{\sinh \left(ax\right)}{\sinh \left(\pi x\right)}dx=\frac{1}{2}\tan \left(\frac{1}{2}a\right),$$ ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral, $\tan z$: tangent function, $a$: real or complex constant and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.40.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iii), §4.40 and Ch.4

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4.40.9 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{ax}}{{\left(\cosh \left(\frac{1}{2}x\right)\right)}^2}dx=\frac{4\pi a}{\sin \left(\pi a\right)},$$ ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $a$: real or complex constant and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.40.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iii), §4.40 and Ch.4

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4.40.10 $${\int }_0^{\mathrm{\infty }}\frac{\tanh \left(ax\right)-\tanh \left(bx\right)}{x}dx=\ln \left(\frac{a}{b}\right),$$ $a>0$, $b>0$.

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Symbols:

$dx$: differential of $x$ , $\tanh z$: hyperbolic tangent function, $\int$: integral, $\ln z$: principal branch of logarithm function, $a$: real or complex constant and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.40.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iii), §4.40 and Ch.4

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37: 11.1 Special Notation

… ►The functions treated in this chapter are the Struve functions ${𝐇}_{\nu }\left(z\right)$ and ${𝐊}_{\nu }\left(z\right)$, the modified Struve functions ${𝐋}_{\nu }\left(z\right)$ and ${𝐌}_{\nu }\left(z\right)$, the Lommel functions $s_{\mu ,\nu }\left(z\right)$ and $S_{\mu ,\nu }\left(z\right)$, the Anger function ${𝐉}_{\nu }\left(z\right)$, the Weber function ${𝐄}_{\nu }\left(z\right)$, and the associated Anger–Weber function ${𝐀}_{\nu }\left(z\right)$.

38: 8.2 Definitions and Basic Properties

… ►The general values of the incomplete gamma functions $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ are defined by …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). … ►

8.2.5 $$P(a,z)+Q(a,z)=1.$$

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Symbols:

$P(a,z)$: normalized incomplete gamma function, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.2(i), §8.2 and Ch.8

… ►The function ${\gamma }^{\ast }(a,z)$ is entire in $z$ and $a$. 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 ${(-1)}^n/n!$. …

39: 5.12 Beta Function

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5.12.2 $${\int }_0^{\pi /2}{\sin }^{2a-1}\theta {\cos }^{2b-1}\theta d\theta =\frac{1}{2}\mathrm{B}(a,b).$$

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Symbols:

$\mathrm{B}(a,b)$: beta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $a$: real or complex variable and $b$: real or complex variable

A&S Ref:

6.2.1 and 6.2.2

Referenced by:

§10.22(ii)

Permalink:

http://dlmf.nist.gov/5.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.12, §5.12 and Ch.5

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5.12.3 $${\int }_0^{\mathrm{\infty }}\frac{t^{a-1}dt}{{(1+t)}^{a+b}}=\mathrm{B}(a,b).$$

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Symbols:

$\mathrm{B}(a,b)$: beta function, $dx$: differential of $x$ , $\int$: integral, $a$: real or complex variable and $b$: real or complex variable

A&S Ref:

6.2.1 and 6.2.2

Referenced by:

§5.12

Permalink:

http://dlmf.nist.gov/5.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.12, §5.12 and Ch.5

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5.12.6 $${\int }_0^{\pi }{(\sin t)}^{a-1}{\mathrm{e}}^{\mathrm{i}bt}dt=\frac{\pi }{2^{a-1}}\frac{{\mathrm{e}}^{\mathrm{i}\pi b/2}}{a\mathrm{B}(\frac{1}{2}(a+b+1),\frac{1}{2}(a-b+1))},$$ $\mathrm{\Re }a>0$.

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Symbols:

$\mathrm{B}(a,b)$: beta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§10.22(ii), §5.12

Permalink:

http://dlmf.nist.gov/5.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §5.12, §5.12 and Ch.5

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5.12.10 $$\frac{1}{2\pi \mathrm{i}}{\int }_0^{(1+)}{t}^{a-1}{(t-1)}^{b-1}dt=\frac{\sin \left(\pi b\right)}{\pi }\mathrm{B}(a,b),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{B}(a,b)$: beta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.12

Permalink:

http://dlmf.nist.gov/5.12.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §5.12, §5.12 and Ch.5

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5.12.12 $${\int }_P^{(1+,0+,1-,0-)}{t}^{a-1}{(1-t)}^{b-1}dt=-4{\mathrm{e}}^{\pi \mathrm{i}(a+b)}\sin \left(\pi a\right)\sin \left(\pi b\right)\mathrm{B}(a,b),$$

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Symbols:

$\mathrm{B}(a,b)$: beta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $a$: real or complex variable, $b$: real or complex variable and $P$: point

Referenced by:

§5.12, §5.12

Permalink:

http://dlmf.nist.gov/5.12.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §5.12, §5.12 and Ch.5

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40: 8.23 Statistical Applications

… ►The functions $P(a,x)$ and $Q(a,x)$ are used extensively in statistics as the probability integrals of the gamma distribution; see Johnson et al. (1994, pp. 337–414). …The function ${\mathrm{B}}_x(a,b)$ and its normalization $I_x(a,b)$ play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of $Q(a,x)$; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319). …