ortho-normality relation

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1: 16.7 Relations to Other Functions

§16.7 Relations to Other Functions

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2: Gergő Nemes

… ►As of September 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. … ►

8 Incomplete Gamma and Related Functions

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9 Airy and Related Functions

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11 Struve and Related Functions

3: 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

4: 25 Zeta and Related Functions

Chapter 25 Zeta and Related Functions

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5: 17 q-Hypergeometric and Related Functions

Chapter 17 $q$-Hypergeometric and Related Functions

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6: 14 Legendre and Related Functions

Chapter 14 Legendre and Related Functions

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7: 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 …

8: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

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9: 9 Airy and Related Functions

Chapter 9 Airy and Related Functions

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10: 11 Struve and Related Functions

Chapter 11 Struve and Related Functions

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