relation to RC-function

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1: 16.7 Relations to Other Functions

§16.7 Relations to Other Functions

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2: 6.11 Relations to Other Functions

§6.11 Relations to Other Functions

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Incomplete Gamma Function

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Confluent Hypergeometric Function

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6.11.2 $$E_1\left(z\right)={\mathrm{e}}^{-z}U(1,1,z),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $z$: complex variable

Referenced by:

§6.11, 5th item, §6.6

Permalink:

http://dlmf.nist.gov/6.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

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6.11.3 $$\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right)=U(1,1,-\mathrm{i}z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{i}$: imaginary unit, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.11, 5th item

Permalink:

http://dlmf.nist.gov/6.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

3: 14 Legendre and Related Functions

Chapter 14 Legendre and Related Functions

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4: 19.10 Relations to Other Functions

§19.10 Relations to Other Functions

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§19.10(i) Theta and Elliptic Functions

►For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. … ►

§19.10(ii) Elementary Functions

… ►For relations to the Gudermannian function $\mathrm{gd}\left(x\right)$ and its inverse ${\mathrm{gd}}^{-1}\left(x\right)$ (§4.23(viii)), see (19.6.8) and …

5: 25.17 Physical Applications

§25.17 Physical Applications

… ►This relates to a suggestion of Hilbert and Pólya that the zeros are eigenvalues of some operator, and the Riemann hypothesis is true if that operator is Hermitian. … ►Quantum field theory often encounters formally divergent sums that need to be evaluated by a process of regularization: for example, the energy of the electromagnetic vacuum in a confined space (Casimir–Polder effect). It has been found possible to perform such regularizations by equating the divergent sums to zeta functions and associated functions (Elizalde (1995)).

6: 16.25 Methods of Computation

… ►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. They are similar to those described for confluent hypergeometric functions, and hypergeometric functions in §§13.29 and 15.19. There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). …Instead a boundary-value problem needs to be formulated and solved. …

7: 19.2 Definitions

… ►Bulirsch’s integrals are linear combinations of Legendre’s integrals that are chosen to facilitate computational application of Bartky’s transformation (Bartky (1938)). … ►

19.2.11 $$\mathrm{cel}(k_c,p,a,b)={\int }_0^{\pi /2}\frac{a{\cos }^2\theta +b{\sin }^2\theta }{{\cos }^2\theta +p{\sin }^2\theta }\frac{d\theta }{\sqrt{{\cos }^2\theta +k_c^2{\sin }^2\theta }},$$

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Defines:

$\mathrm{cel}(k_c,p,a,b)$: Bulirsch’s complete elliptic integral

Symbols:

$\pi$: 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 parameter, $b$: real parameter, $p$: real parameter not equal to zero and $k_c$: real or complex variable not equal to zero

Referenced by:

§19.2(iii)

Permalink:

http://dlmf.nist.gov/19.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(iii), §19.2 and Ch.19

… ►Lastly, corresponding to Legendre’s incomplete integral of the third kind we have ►

19.2.16 $$\mathrm{el3}(x,k_c,p)={\int }_0^{\arctan x}\frac{d\theta }{({\cos }^2\theta +p{\sin }^2\theta )\sqrt{{\cos }^2\theta +k_c^2{\sin }^2\theta }}=\mathrm{\Pi }(\arctan x,1-p,k),$$ $x^2\ne -1/p$.

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Defines:

$\mathrm{el3}(x,k_c,p)$: Bulirsch’s incomplete elliptic integral of the third kind

Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\int$: integral, $\arctan z$: arctangent function, $\sin z$: sine function, $k$: real or complex modulus, $p$: real parameter not equal to zero, $k_c$: real or complex variable not equal to zero and $x$: real or complex variable

Permalink:

http://dlmf.nist.gov/19.2.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(iii), §19.2 and Ch.19

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§19.2(iv) A Related Function: $R_C(x,y)$

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8: 9 Airy and Related Functions

Chapter 9 Airy and Related Functions

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9: 11 Struve and Related Functions

Chapter 11 Struve and Related Functions

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10: 16.24 Physical Applications

… ►For an extension to two-loop integrals see Moch et al. (2002). ►

§16.24(iii) $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ Symbols

… ►The coefficients of transformations between different coupling schemes of three angular momenta are related to the Wigner $\mathit{6}j$ symbols. …