generalized exponentials and logarithms

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11: 13.6 Relations to Other Functions

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13.6.6 $$U(a,a,z)=z^{1-a}U(1,2-a,z)=z^{1-a}{\mathrm{e}}^z{E}_a\left(z\right)={\mathrm{e}}^z\mathrm{\Gamma }(1-a,z).$$

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Symbols:

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

A&S Ref:

13.6.28 (in different form)

Permalink:

http://dlmf.nist.gov/13.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(ii), §13.6 and Ch.13

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12: 6.2 Definitions and Interrelations

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§6.2(i) Exponential and Logarithmic Integrals

… ► $\mathrm{Ein}\left(z\right)$ is sometimes called the complementary exponential integral. … ► … ►The logarithmic integral is defined by … ►The generalized exponential integral $E_p\left(z\right)$, $p\in ℂ$, is treated in Chapter 8. …

13: 13.18 Relations to Other Functions

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13.18.17 $$W_{\frac{1}{2}\alpha +\frac{1}{2}+n,\frac{1}{2}\alpha }\left(z\right)={(-1)}^n{\left(\alpha +1\right)}_n{M}_{\frac{1}{2}\alpha +\frac{1}{2}+n,\frac{1}{2}\alpha }\left(z\right)={(-1)}^{n}n!{\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}\alpha +\frac{1}{2}}{L}_n^{(\alpha )}\left(z\right).$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

Referenced by:

(33.14.14)

Permalink:

http://dlmf.nist.gov/13.18.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(v), §13.18(v), §13.18 and Ch.13

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14: 6.4 Analytic Continuation

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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:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

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15: Bibliography

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A general mode theory for the elliptic disk microstrip antenna. IEEE Trans. Antennas and Propagation 43 (6), pp. 560–568.

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External Links:

Document

Referenced by:

5th item.

Encodings:

BibTeX

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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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Computation of exponential integrals. ACM Trans. Math. Software 6 (3), pp. 365–377.

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External Links:

ISSN 0098-3500, Document, MathReview (M. M. Chawla), ZentralBlatt

Referenced by:

§8.25(v).

Encodings:

BibTeX

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Generalized elliptic integrals and modular equations. Pacific J. Math. 192 (1), pp. 1–37.

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External Links:

ISSN 0030-8730, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.35(i).

Encodings:

BibTeX

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Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.

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External Links:

MathReview (N. G. de Bruijn), ZentralBlatt

Referenced by:

(25.11.41), (25.11.42), §25.11(xii).

Encodings:

BibTeX

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16: Bibliography M

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Catastrophe optics of spheroidal drops and generalized rainbows. J. Quantit. Spec. and Rad. Trans. 63, pp. 341–351.

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External Links:

Document

Referenced by:

§36.14(ii).

Encodings:

BibTeX

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Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.

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External Links:

ISSN 0036-1429, Document, MathReview (M. Brannigan), ZentralBlatt

Referenced by:

§4.45(ii).

Encodings:

BibTeX

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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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A class of generalized hypergeometric summations. J. Comput. Appl. Math. 87 (1), pp. 79–85.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§16.10.

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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17: 4.2 Definitions

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4.2.14 $${\log }_{\mathrm{e}}z=\ln z,$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, ${\log }_{a}z$: logarithm to general base, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.21

Permalink:

http://dlmf.nist.gov/4.2.E14

Encodings:

pMML, png, TeX

See also:

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

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4.2.15 $${\log }_{10}z=(\ln z)/(\ln 10)=({\log }_{10}\mathrm{e})\ln z,$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, ${\log }_{a}z$: logarithm to general base, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.22

Permalink:

http://dlmf.nist.gov/4.2.E15

Encodings:

pMML, png, TeX

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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

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

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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:

pMML, png, TeX

See also:

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

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18: 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:

pMML, png, TeX

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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19: 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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

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

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20: 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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