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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: 19.10 Relations to Other Functions

§19.10 Relations to Other Functions

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§19.10(i) Theta and Elliptic Functions

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§19.10(ii) Elementary Functions

… ►In each case when $y=1$, the quantity multiplying $R_C$ supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0. ►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 …

4: 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. …

5: 16.24 Physical Applications

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§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. …

6: 13.18 Relations to Other Functions

§13.18 Relations to Other Functions

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§13.18(ii) Incomplete Gamma 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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7: 25.13 Periodic Zeta Function

§25.13 Periodic Zeta Function

►The notation $F(x,s)$ is used for the polylogarithm ${\mathrm{Li}}_s\left({\mathrm{e}}^{2\pi \mathrm{i}x}\right)$ with $x$ real: … ►Also, …

8: 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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Confluent Hypergeometric Functions

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Generalized Hypergeometric Functions

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9: 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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10: 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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