generalized logarithms

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21: 14.23 Values on the Cut

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14.23.1 $$P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)={\mathrm{e}}^{\mp \mu \pi \mathrm{i}/2}{𝖯}_{\nu }^{\mu }\left(x\right),$$

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Symbols:

: Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.23

Permalink:

http://dlmf.nist.gov/14.23.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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14.23.2 $${𝑸}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)=\frac{{\mathrm{e}}^{\pm \mu \pi \mathrm{i}/2}}{\mathrm{\Gamma }\left(\nu +\mu +1\right)}\left({𝖰}_{\nu }^{\mu }\left(x\right)\mp \frac{1}{2}\pi \mathrm{i}{𝖯}_{\nu }^{\mu }\left(x\right)\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, : Ferrers function of the second kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.23

Permalink:

http://dlmf.nist.gov/14.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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14.23.4 $${𝖯}_{\nu }^{\mu }\left(x\right)={\mathrm{e}}^{\pm \mu \pi \mathrm{i}/2}{P}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right),$$

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Symbols:

: Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.3.2 (corrected)

Referenced by:

§14.23, §14.23, §14.3(iv)

Permalink:

http://dlmf.nist.gov/14.23.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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14.23.5 $${𝖰}_{\nu }^{\mu }\left(x\right)=\frac{1}{2}\mathrm{\Gamma }\left(\nu +\mu +1\right)\left({\mathrm{e}}^{-\mu \pi \mathrm{i}/2}{𝑸}_{\nu }^{\mu }\left(x+\mathrm{i}0\right)+{\mathrm{e}}^{\mu \pi \mathrm{i}/2}{𝑸}_{\nu }^{\mu }\left(x-\mathrm{i}0\right)\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.3.4 (modified)

Referenced by:

§14.23

Permalink:

http://dlmf.nist.gov/14.23.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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14.23.6 $${𝖰}_{\nu }^{\mu }\left(x\right)={\mathrm{e}}^{\mp \mu \pi \mathrm{i}/2}\mathrm{\Gamma }\left(\nu +\mu +1\right){𝑸}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)\pm \frac{1}{2}\pi \mathrm{i}{\mathrm{e}}^{\pm \mu \pi \mathrm{i}/2}{P}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.23, §14.23

Permalink:

http://dlmf.nist.gov/14.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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22: 2.11 Remainder Terms; Stokes Phenomenon

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2.11.5 $$E_p\left(z\right)=\frac{{\mathrm{e}}^{-z}{z}^{p-1}}{\mathrm{\Gamma }\left(p\right)}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}{t}^{p-1}}{1+t}dt$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral and $\int$: integral

Referenced by:

§2.11(iii)

Permalink:

http://dlmf.nist.gov/2.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(ii), §2.11 and Ch.2

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2.11.6 $$E_p\left(z\right)\sim \frac{{\mathrm{e}}^{-z}}{z}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(p\right)}_s}{z^s}$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm and $E_p\left(z\right)$: generalized exponential integral

Referenced by:

§2.11(ii), §2.11(iii), §2.11(iii), §2.11(iv), §2.11(iv)

Permalink:

http://dlmf.nist.gov/2.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(ii), §2.11 and Ch.2

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2.11.7 $$E_p\left(z\right)\sim \frac{2\pi \mathrm{i}{\mathrm{e}}^{-p\pi \mathrm{i}}}{\mathrm{\Gamma }\left(p\right)}{z}^{p-1}+\frac{{\mathrm{e}}^{-z}}{z}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(p\right)}_s}{z^s},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral and $\mathrm{i}$: imaginary unit

Referenced by:

§2.11(ii), §2.11(iv), §2.11(iv), §2.11(iv)

Permalink:

http://dlmf.nist.gov/2.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(ii), §2.11 and Ch.2

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2.11.10 $$E_p\left(z\right)=\frac{{\mathrm{e}}^{-z}}{z}\sum _{s=0}^{n-1}{(-1)}^s\frac{{\left(p\right)}_s}{z^s}+{(-1)}^n\frac{2\pi }{\mathrm{\Gamma }\left(p\right)}{z}^{p-1}{F}_{n+p}\left(z\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral and $F_p\left(z\right)$: terminant function

Referenced by:

§2.11(iii), §2.11(iv)

Permalink:

http://dlmf.nist.gov/2.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(iii), §2.11 and Ch.2

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2.11.11 $$F_{n+p}\left(z\right)=\frac{{\mathrm{e}}^{-z}}{2\pi }{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}{t}^{n+p-1}}{1+t}dt=\frac{\mathrm{\Gamma }\left(n+p\right)}{2\pi }\frac{E_{n+p}\left(z\right)}{z^{n+p-1}}.$$

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Defines:

$F_p\left(z\right)$: terminant function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral and $\int$: integral

Permalink:

http://dlmf.nist.gov/2.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(iii), §2.11 and Ch.2

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23: Bibliography S

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On the logarithmic solutions of the generalized hypergeometric equation when $p=q+1$ . Bull. Amer. Math. Soc. 45 (8), pp. 629–636.

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External Links:

Document, MathReview (H. Bateman), ZentralBlatt

Referenced by:

§16.8(ii).

Encodings:

BibTeX

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Relations among the fundamental solutions of the generalized hypergeometric equation when $p=q+1$. II. Logarithmic cases. Bull. Amer. Math. Soc. 45 (12), pp. 927–935.

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External Links:

Document, MathReview (C. S. Meijer), ZentralBlatt

Referenced by:

§16.8(ii).

Encodings:

BibTeX

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24: 6.2 Definitions and Interrelations

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§6.2(i) Exponential and Logarithmic Integrals

… ►As in the case of the logarithm (§4.2(i)) there is a cut along the interval $(-\mathrm{\infty },0]$ and the principal value is two-valued on $(-\mathrm{\infty },0)$. … ►The logarithmic integral is defined by ►

6.2.8 $$\mathrm{li}\left(x\right)={⨍}_0^x\frac{dt}{\ln t}=\mathrm{Ei}\left(\ln x\right),$$ $x>1$.

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Defines:

$\mathrm{li}\left(x\right)$: logarithmic integral

Symbols:

$dx$: differential of $x$ , $\mathrm{Ei}\left(x\right)$: exponential integral, $\ln z$: principal branch of logarithm function, ${⨍}_a^b$: Cauchy principal value and $x$: real variable

A&S Ref:

5.1.3

Referenced by:

§27.12, §6.12(i)

Permalink:

http://dlmf.nist.gov/6.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(i), §6.2 and Ch.6

►The generalized exponential integral $E_p\left(z\right)$, $p\in ℂ$, is treated in Chapter 8. …

25: 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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26: 2.3 Integrals of a Real Variable

… ►For extensions to oscillatory integrals with more general $t$-powers and logarithmic singularities see Wong and Lin (1978) and Sidi (2010). …

27: 31.3 Basic Solutions

… ►In general, one of them has a logarithmic singularity at $z=0$. …

28: 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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29: 6 Exponential, Logarithmic, Sine, and
Cosine Integrals

Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals

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30: 14.3 Definitions and Hypergeometric Representations

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14.3.7 $\mu +\nu \ne -1,-2,-3,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\mu$: general order, $\nu$: general degree and : real variable

A&S Ref:

8.1.3 (modified)

Permalink:

http://dlmf.nist.gov/14.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

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14.3.10

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mu$: general order, $\nu$: general degree and : real variable

Referenced by:

§14.20(i), §14.21(i), §14.21(iii), §14.6(i)

Permalink:

http://dlmf.nist.gov/14.3.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

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