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11: 12.16 Mathematical Applications

… ►Sleeman (1968b) considers certain orthogonality properties of the PCFs and corresponding eigenvalues. …

12: 14.21 Definitions and Basic Properties

§14.21 Definitions and Basic Properties

… ► … ►

§14.21(iii) Properties

►Many of the properties stated in preceding sections extend immediately from the $x$-interval $(1,\mathrm{\infty })$ to the cut $z$-plane $ℂ\backslash (-\mathrm{\infty },1]$. …

13: 4.16 Elementary Properties

§4.16 Elementary Properties

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Table 4.16.3: Trigonometric functions: interrelations. …

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	$\sin \theta =a$	$\cos \theta =a$	$\tan \theta =a$	$\csc \theta =a$	$\sec \theta =a$	$\cot \theta =a$
…

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14: 5.2 Definitions

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5.2.1 $$\mathrm{\Gamma }\left(z\right)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{z-1}dt,$$ $\mathrm{\Re }z>0$.

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Defines:

$\mathrm{\Gamma }\left(z\right)$: gamma function

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

6.1.1

Referenced by:

§10.43(iii), (25.11.27), (25.11.28), (25.5.6), (25.5.7), §5.9(i), §5.9(ii), §8.21(ii), (9.12.17)

Permalink:

http://dlmf.nist.gov/5.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.2(i), §5.2(i), §5.2 and Ch.5

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5.2.2 $$\psi \left(z\right)={\mathrm{\Gamma }}^{\prime }\left(z\right)/\mathrm{\Gamma }\left(z\right),$$ $z\ne 0,-1,-2,\mathrm{\dots }$.

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Defines:

$\psi \left(z\right)$: psi (or digamma) function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $z$: complex variable

A&S Ref:

6.3.1

Permalink:

http://dlmf.nist.gov/5.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.2(i), §5.2(i), §5.2 and Ch.5

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15: 24.10 Arithmetic Properties

§24.10 Arithmetic Properties

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16: 28.7 Analytic Continuation of Eigenvalues

§28.7 Analytic Continuation of Eigenvalues

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28.7.4 $$\sum _{n=0}^{\mathrm{\infty }}\left(b_{2n+2}\left(q\right)-{(2n+2)}^2\right)=0.$$

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Symbols:

$b_n\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $n$: integer

Permalink:

http://dlmf.nist.gov/28.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.7 and Ch.28

17: 4.1 Special Notation

… ►It is assumed the user is familiar with the definitions and properties of elementary functions of real arguments $x$. The main purpose of the present chapter is to extend these definitions and properties to complex arguments $z$. …

18: 29.17 Other Solutions

… ►For properties of these solutions see Arscott (1964b, §9.7), Erdélyi et al. (1955, §15.5.1), Shail (1980), and Sleeman (1966b). … ►Lamé–Wangerin functions are solutions of (29.2.1) with the property that ${(\mathrm{sn}(z,k))}^{1/2}w(z)$ is bounded on the line segment from $\mathrm{i}{K}^{\prime }$ to $2K+\mathrm{i}{K}^{\prime }$. …

19: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

… ►Some properties are included as special cases of properties given in §31.15 below.

20: 4.28 Definitions and Periodicity

… ►As a consequence, many properties of the hyperbolic functions follow immediately from the corresponding properties of the trigonometric functions. ►

Periodicity and Zeros

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