relation to infinite partial fractions

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31: 13.18 Relations to Other Functions

§13.18 Relations to Other Functions

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§13.18(i) Elementary Functions

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§13.18(iv) Parabolic Cylinder Functions

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§13.18(v) Orthogonal Polynomials

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Laguerre Polynomials

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32: 7.18 Repeated Integrals of the Complementary Error Function

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§7.18(iv) Relations to Other Functions

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Hermite Polynomials

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Confluent Hypergeometric Functions

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Parabolic Cylinder Functions

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Probability Functions

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33: 12.7 Relations to Other Functions

§12.7 Relations to Other Functions

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§12.7(i) Hermite Polynomials

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§12.7(ii) Error Functions, Dawson’s Integral, and Probability Function

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§12.7(iii) Modified Bessel Functions

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§12.7(iv) Confluent Hypergeometric Functions

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34: 7.11 Relations to Other Functions

§7.11 Relations to Other Functions

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Incomplete Gamma Functions and Generalized Exponential Integral

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7.11.1 $$\mathrm{erf}z=\frac{1}{\sqrt{\pi }}\gamma (\frac{1}{2},z^2),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\gamma (a,z)$: incomplete gamma function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/7.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

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Confluent Hypergeometric Functions

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Generalized Hypergeometric Functions

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35: 13.6 Relations to Other Functions

§13.6 Relations to Other Functions

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§13.6(iv) Parabolic Cylinder Functions

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§13.6(v) Orthogonal Polynomials

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Laguerre Polynomials

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§13.6(vi) Generalized Hypergeometric Functions

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36: 8.19 Generalized Exponential Integral

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§8.19(i) Definition and Integral Representations

… ►In Figures 8.19.2–8.19.5, height corresponds to the absolute value of the function and color to the phase. … ►

§8.19(v) Recurrence Relation and Derivatives

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§8.19(vi) Relation to Confluent Hypergeometric Function

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§8.19(vii) Continued Fraction

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37: 19.5 Maclaurin and Related Expansions

§19.5 Maclaurin and Related Expansions

… ► … ►Coefficients of terms up to ${\lambda }^{49}$ are given in Lee (1990), along with tables of fractional errors in $K\left(k\right)$ and $E\left(k\right)$, $0.1\le k^2\le 0.9999$, obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9). ►

19.5.8 $$K\left(k\right)=\frac{\pi }{2}{\left(1+2\sum _{n=1}^{\mathrm{\infty }}{q}^{n^2}\right)}^2,$$ ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $q$: nome, $n$: nonnegative integer and $k$: real or complex modulus

Referenced by:

§19.5, §19.5

Permalink:

http://dlmf.nist.gov/19.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

… ►An infinite series for $\ln K\left(k\right)$ is equivalent to the infinite product …

38: 14.32 Methods of Computation

… ►Essentially the same comments that are made in §15.19 concerning the computation of hypergeometric functions apply to the functions described in the present chapter. …In other cases recurrence relations (§14.10) provide a powerful method when applied in a stable direction (§3.6); see Olver and Smith (1983) and Gautschi (1967). … ►

Evaluation (§3.10) of the continued fractions given in §14.14. See Gil and Segura (2000).

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39: 12.17 Physical Applications

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12.17.2 $${\nabla }^2=\frac{{\partial }^2}{{\partial x}^2}+\frac{{\partial }^2}{{\partial y}^2}+\frac{{\partial }^2}{{\partial z}^2}$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable, $y$: real variable and $z$: complex variable

Permalink:

http://dlmf.nist.gov/12.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.17 and Ch.12

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12.17.4 $$\frac{1}{{\xi }^2+{\eta }^2}\left(\frac{{\partial }^{2}w}{{\partial \xi }^2}+\frac{{\partial }^{2}w}{{\partial \eta }^2}\right)+\frac{{\partial }^{2}w}{{\partial \zeta }^2}+k^{2}w=0.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: constant, $\xi$: coordinate, $\eta$: coordinate and $\zeta$: coordinate

Permalink:

http://dlmf.nist.gov/12.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.17 and Ch.12

… ►In a similar manner coordinates of the paraboloid of revolution transform the Helmholtz equation into equations related to the differential equations considered in this chapter. … ►Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator. ►Problems on high-frequency scattering in homogeneous media by parabolic cylinders lead to asymptotic methods for integrals involving PCFs. …

40: Qiming Wang

… ►She started to work for NIST in 1990 and was on the staff of the Visualization and Usability Group in the Information Access Division of the Information Technology Laboratory in the National Institute of Standards and Technology when she retired in March, 2008. … ►She has applied VRML and X3D techniques to several different fields including interactive mathematical function visualization, 3D human body modeling, and manufacturing-related modeling. …