generalized exponentials

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21: Bibliography

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Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions. J. Heat Transfer 118 (3), pp. 789–792.

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External Links:

Document

Referenced by:

§10.73(iv), §8.24(iii).

Encodings:

BibTeX

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22: Bibliography D

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Asymptotics of the generalized exponential integral, and error bounds in the uniform asymptotic smoothing of its Stokes discontinuities. Proc. Roy. Soc. London Ser. A 452, pp. 1351–1367.

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External Links:

ISSN 0962-8444, FullText, MathReview (Alexander Tovbis), ZentralBlatt

Referenced by:

§2.11(iii), §8.20(ii).

Encodings:

BibTeX

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23: Bibliography M

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The generalized integro-exponential function. Math. Comp. 44 (170), pp. 443–458.

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External Links:

ISSN 0025-5718, FullText, MathReview (S. Conde), ZentralBlatt

Referenced by:

§8.19(v), §8.19(xi).

Encodings:

BibTeX

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Tables of Generalized Exponential Integrals. NPL Mathematical Tables, Vol. III, Her Majesty’s Stationery Office, London.

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External Links:

MathReview (L. Fox)

Referenced by:

§8.25(v).

Encodings:

BibTeX

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24: 6.4 Analytic Continuation

… ►The general value of $E_1\left(z\right)$ is given by ►

6.4.1 $$E_1\left(z\right)=\mathrm{Ein}\left(z\right)-\mathrm{Ln}z-\gamma ;$$

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $E_1\left(z\right)$: exponential integral, $\mathrm{Ln}z$: general logarithm function and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E1

Encodings:

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See also:

Annotations for §6.4 and Ch.6

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25: 4.2 Definitions

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4.2.17 $${\log }_{10}\mathrm{e}=\mathrm{0.43429\; 44819\; 03251\; 82765}\mathrm{\dots },$$

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Symbols:

$\mathrm{e}$: base of natural logarithm and ${\log }_{a}z$: logarithm to general base

A&S Ref:

4.1.22 (with 10D value)

Notes:

For more digits see OEIS Sequence A002285; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/4.2.E17

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See also:

Annotations for §4.2(ii), §4.2 and Ch.4

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4.2.25 $$\exp z=\zeta ⟺z=\mathrm{Ln}\zeta .$$

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Symbols:

$\exp z$: exponential function, $\mathrm{Ln}z$: general logarithm function and $z$: complex variable

A&S Ref:

4.2.4

Permalink:

http://dlmf.nist.gov/4.2.E25

Encodings:

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See also:

Annotations for §4.2(iii), §4.2 and Ch.4

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4.2.26 $$z^a=\exp \left(a\mathrm{Ln}z\right),$$ $z\ne 0$.

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Symbols:

$\exp z$: exponential function, $\mathrm{Ln}z$: general logarithm function, $a$: real or complex constant and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.2.E26

Encodings:

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See also:

Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

… ►but the general value of ${\mathrm{e}}^z$ is … ►

4.2.35 $$z^a=w⟺z=\exp \left(\frac{1}{a}\mathrm{Ln}w\right).$$

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Defines:

$a\ne 0$: complex constant (locally)

Symbols:

$\exp z$: exponential function, $\mathrm{Ln}z$: general logarithm function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.2.E35

Encodings:

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See also:

Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

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26: 8.7 Series Expansions

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8.7.6 $$\mathrm{\Gamma }(a,x)=x^a{\mathrm{e}}^{-x}\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(a)}\left(x\right)}{n+1},$$ $x>0$, .

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{\Re }$: real part, $x$: real variable, $z$: complex variable, $a$: parameter and $n$: nonnegative integer

Proof sketch:

Integrate (18.12.13) from $z=0$ to $z=1$ and for the integral on the left-hand side use the substitution $z=\frac{t}{1+t}$. The resulting integral is (8.6.5). The constraint follows from (18.15.14).

Referenced by:

Erratum (V1.1.8) for Equation (8.7.6)

Permalink:

http://dlmf.nist.gov/8.7.E6

Encodings:

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Addition (effective with 1.1.8):

The constraint was updated to include .

Suggested 2022-10-14 by Walter Gautschi

See also:

Annotations for §8.7 and Ch.8

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27: Bibliography W

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Infinitely differentiable generalized logarithmic and exponential functions. Math. Comp. 57 (196), pp. 723–733.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§4.12.

Encodings:

BibTeX

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28: 4.8 Identities

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4.8.8 $$\mathrm{Ln}\left(\exp z\right)=z+2k\pi \mathrm{i},$$ $k\in \mathbb{Z}$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $\mathrm{Ln}z$: general logarithm function, $k$: integer and $z$: complex variable

A&S Ref:

4.2.2

Permalink:

http://dlmf.nist.gov/4.8.E8

Encodings:

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See also:

Annotations for §4.8(i), §4.8 and Ch.4

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4.8.10 $$\exp \left(\ln z\right)=\exp \left(\mathrm{Ln}z\right)=z.$$

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Symbols:

$\exp z$: exponential function, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.2.4

Referenced by:

(25.10.2), §4.8(i)

Permalink:

http://dlmf.nist.gov/4.8.E10

Encodings:

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See also:

Annotations for §4.8(i), §4.8 and Ch.4

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29: 18.39 Applications in the Physical Sciences

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18.39.29 $${\psi }_{p,l}(r)=\sqrt{\frac{Zp!}{(p+2l+1)!}}\frac{{\mathrm{e}}^{-{\rho }_p/2}{\rho }_p^{l+1}}{p+l+1}{L}_p^{(2l+1)}\left({\rho }_p\right),$$ $p=0,1,2,\mathrm{\dots }$; $l=0,1,2,\mathrm{\dots }$,

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$) and $l$: nonnegative integer

Referenced by:

§18.39(ii), §18.39(ii), §18.39(ii)

Permalink:

http://dlmf.nist.gov/18.39.E29

Encodings:

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See also:

Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

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18.39.34 $${\psi }_{n,l}(r)=\frac{1}{n}\sqrt{\frac{Z(n-l-1)!}{(n+l)!}}{\mathrm{e}}^{-{\rho }_n/2}{\rho }_n^{l+1}{L}_{n-l-1}^{(2l+1)}\left({\rho }_n\right),$$ $n=1,2,\mathrm{\dots },l=0,1,\mathrm{\dots }n-1$,

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $l$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.39.E34

Encodings:

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See also:

Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

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18.39.37 $$R_{n,l}(r)=\frac{2}{n^2}\sqrt{\frac{Z^3(n-l-1)!}{(n+l)!}}{\mathrm{e}}^{-{\rho }_n/2}{\rho }_n^l{L}_{n-l-1}^{(2l+1)}\left({\rho }_n\right),$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $l$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§18.39(ii)

Permalink:

http://dlmf.nist.gov/18.39.E37

Encodings:

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See also:

Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

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18.39.44 $${\varphi }_{n,l}(sr)={(sr)}^{l+1}{\mathrm{e}}^{-sr/2}{L}_n^{(2l+1)}\left(sr\right),$$ $n=0,1,2,\mathrm{\dots }$,

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $l$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§18.39(iv), §18.39(iv), §18.39(iv)

Permalink:

http://dlmf.nist.gov/18.39.E44

Encodings:

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See also:

Annotations for §18.39(iv), §18.39 and Ch.18

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30: 18.12 Generating Functions

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18.12.13 $${(1-z)}^{-\alpha -1}\exp \left(\frac{xz}{z-1}\right)=\sum _{n=0}^{\mathrm{\infty }}{L}_n^{(\alpha )}\left(x\right)z^n,$$ .

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\exp z$: exponential function, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.15

Referenced by:

§18.12, §18.18(ii), §18.18(iv), §18.18(viii), §18.2(xii), (8.7.6)

Permalink:

http://dlmf.nist.gov/18.12.E13

Encodings:

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See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.14 $$\mathrm{\Gamma }\left(\alpha +1\right){(xz)}^{-\frac{1}{2}\alpha }{\mathrm{e}}^z{J}_{\alpha }\left(2\sqrt{xz}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(\alpha )}\left(x\right)}{{\left(\alpha +1\right)}_n}{z}^n.$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.16

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E14

Encodings:

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See also:

Annotations for §18.12, §18.12 and Ch.18

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