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31: 6.4 Analytic Continuation

… ►Analytic continuation of the principal value of $E_1\left(z\right)$ yields a multi-valued function with branch points at $z=0$ and $z=\mathrm{\infty }$. … ►

6.4.3 $$E_1\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\mathrm{Ein}\left(-z\right)-\ln z-\gamma \mp \pi \mathrm{i},$$ $|\mathrm{ph}z|\le \pi$.

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

… ►Unless indicated otherwise, in the rest of this chapter and elsewhere in the DLMF the functions $E_1\left(z\right)$, $\mathrm{Ci}\left(z\right)$, $\mathrm{Chi}\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ assume their principal values, that is, the branches that are real on the positive real axis and two-valued on the negative real axis. …

32: 4.7 Derivatives and Differential Equations

… ►

4.7.1 $$\frac{d}{dz}\ln z=\frac{1}{z},$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.46

Permalink:

http://dlmf.nist.gov/4.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(i), §4.7 and Ch.4

… ►

4.7.3 $$\frac{d^n}{{dz}^n}\ln z={(-1)}^{n-1}(n-1)!z^{-n},$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $n$: integer and $z$: complex variable

A&S Ref:

4.1.47

Permalink:

http://dlmf.nist.gov/4.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(i), §4.7 and Ch.4

… ►

4.7.9 $$\frac{d}{dz}{a}^z=a^z\ln a,$$ $a\ne 0$.

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\ln z$: principal branch of logarithm function, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.2.51

Permalink:

http://dlmf.nist.gov/4.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(ii), §4.7 and Ch.4

►When $a^z$ is a general power, $\ln a$ is replaced by the branch of $\mathrm{Ln}a$ used in constructing $a^z$. …

33: 6.8 Inequalities

… ►

6.8.1

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

5.1.20

Referenced by:

§6.8

Permalink:

http://dlmf.nist.gov/6.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.8 and Ch.6

…

34: 25.8 Sums

… ►

25.8.4 $$\sum _{k=1}^{\mathrm{\infty }}\frac{{(-1)}^k}{k}(\zeta \left(nk\right)-1)=\ln \left(\prod _{j=0}^{n-1}\mathrm{\Gamma }\left(2-{\mathrm{e}}^{(2j+1)\pi \mathrm{i}/n}\right)\right),$$ $n=2,3,4,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $n$: nonnegative integer

Keywords:

infinite series

Source:

Adamchik and Srivastava (1998, (2.7), p. 135)

Permalink:

http://dlmf.nist.gov/25.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

… ►

25.8.7 $$\sum _{k=2}^{\mathrm{\infty }}\frac{\zeta \left(k\right)}{k}{z}^k=-\gamma z+\ln \mathrm{\Gamma }\left(1-z\right),$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

Keywords:

infinite series

Proof sketch:

Derivable from (25.8.5) by dividing by $z$ and integrating.

Permalink:

http://dlmf.nist.gov/25.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

►

25.8.8 $$\sum _{k=1}^{\mathrm{\infty }}\frac{\zeta \left(2k\right)}{k}{z}^{2k}=\ln \left(\frac{\pi z}{\sin \left(\pi z\right)}\right),$$ .

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $k$: nonnegative integer and $z$: complex variable

Keywords:

infinite series

Proof sketch:

Derivable from (25.8.6) by dividing by $z$ and integrating.

Permalink:

http://dlmf.nist.gov/25.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

►

25.8.9 $$\sum _{k=1}^{\mathrm{\infty }}\frac{\zeta \left(2k\right)}{(2k+1)2^{2k}}=\frac{1}{2}-\frac{1}{2}\ln 2.$$

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function and $k$: nonnegative integer

Keywords:

infinite series

Source:

Srivastava and Choi (2001, (467), p. 212)

Permalink:

http://dlmf.nist.gov/25.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

…

35: 25.12 Polylogarithms

… ►Other notations and names for ${\mathrm{Li}}_2\left(z\right)$ include $S_2(z)$ (Kölbig et al. (1970)), Spence function $\mathrm{Sp}(z)$ (’t Hooft and Veltman (1979)), and ${\mathrm{L}}_2(z)$ (Maximon (2003)). ►In the complex plane ${\mathrm{Li}}_2\left(z\right)$ has a branch point at $z=1$. The principal branch has a cut along the interval $[1,\mathrm{\infty })$ and agrees with (25.12.1) when $|z|\le 1$; see also §4.2(i). The remainder of the equations in this subsection apply to principal branches. … ►

25.12.6 $${\mathrm{Li}}_2\left(x\right)+{\mathrm{Li}}_2\left(1-x\right)=\frac{1}{6}{\pi }^2-(\ln x)\ln \left(1-x\right),$$ .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{Li}}_2\left(z\right)$: dilogarithm, $\ln z$: principal branch of logarithm function and $x$: real variable

Keywords:

connection formula

Source:

Maximon (2003, (3.3), p. 2808)

Permalink:

http://dlmf.nist.gov/25.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

…

36: 4.26 Integrals

… ►The results in §§4.26(ii) and 4.26(iv) can be extended to the complex plane by using continuous branches and avoiding singularities. … ►

4.26.3 $$\int \tan xdx=-\ln \left(\cos x\right),$$ .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $\tan z$: tangent function and $x$: real variable

A&S Ref:

4.3.115

Permalink:

http://dlmf.nist.gov/4.26.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(ii), §4.26 and Ch.4

►

4.26.4 $$\int \csc xdx=\ln \left(\tan \frac{1}{2}x\right),$$ .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $\tan z$: tangent function and $x$: real variable

A&S Ref:

4.3.116

Permalink:

http://dlmf.nist.gov/4.26.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(ii), §4.26 and Ch.4

… ►

4.26.6 $$\int \cot xdx=\ln \left(\sin x\right),$$ .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $\sin z$: sine function and $x$: real variable

A&S Ref:

4.3.118

Permalink:

http://dlmf.nist.gov/4.26.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(ii), §4.26 and Ch.4

… ►

4.26.16 $$\int \arctan xdx=x\arctan x-\frac{1}{2}\ln \left(1+x^2\right),$$ ,

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.4.60 (modified)

Permalink:

http://dlmf.nist.gov/4.26.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(iv), §4.26 and Ch.4

…

37: 9.16 Physical Applications

… ►Airy functions are applied in many branches of both classical and quantum physics. … ►The KdV equation and solitons have applications in many branches of physics, including plasma physics lattice dynamics, and quantum mechanics. …

38: 10.31 Power Series

… ►

10.31.1 $$K_n\left(z\right)=\frac{1}{2}{(\frac{1}{2}z)}^{-n}\sum _{k=0}^{n-1}\frac{(n-k-1)!}{k!}{(-\frac{1}{4}{z}^2)}^k+{(-1)}^{n+1}\ln \left(\frac{1}{2}z\right)I_n\left(z\right)+{(-1)}^n\frac{1}{2}{(\frac{1}{2}z)}^n\sum _{k=0}^{\mathrm{\infty }}\left(\psi \left(k+1\right)+\psi \left(n+k+1\right)\right)\frac{{(\frac{1}{4}{z}^2)}^k}{k!(n+k)!},$$

ⓘ

Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.6.11

Referenced by:

§10.30(i), §10.65(ii)

Permalink:

http://dlmf.nist.gov/10.31.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.31 and Ch.10

… ►

10.31.2 $$K_0\left(z\right)=-\left(\ln \left(\frac{1}{2}z\right)+\gamma \right)I_0\left(z\right)+\frac{\frac{1}{4}{z}^2}{{(1!)}^2}+(1+\frac{1}{2})\frac{{(\frac{1}{4}{z}^2)}^2}{{(2!)}^2}+(1+\frac{1}{2}+\frac{1}{3})\frac{{(\frac{1}{4}{z}^2)}^3}{{(3!)}^2}+\mathrm{\cdots }.$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

9.6.13

Referenced by:

§10.45

Permalink:

http://dlmf.nist.gov/10.31.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.31 and Ch.10

…

39: 16.2 Definition and Analytic Properties

… ►The branch obtained by introducing a cut from $1$ to $+\mathrm{\infty }$ on the real axis, that is, the branch in the sector $|\mathrm{ph}\left(1-z\right)|\le \pi$, is the principal branch (or principal value) of ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;z)$; compare §4.2(i). Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at $z=0,1$, and $\mathrm{\infty }$. Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values. … ►When $p\le q+1$ and $z$ is fixed and not a branch point, any branch of ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ is an entire function of each of the parameters $a_1,\mathrm{\dots },a_p,b_1,\mathrm{\dots },b_q$.

40: 19.27 Asymptotic Approximations and Expansions

… ►

19.27.2 $$R_F(x,y,z)=\frac{1}{2\sqrt{z}}\left(\ln \frac{8z}{a+g}\right)\left(1+O\left(\frac{a}{z}\right)\right),$$ $a/z\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\ln z$: principal branch of logarithm function, $a$ and $g$

Referenced by:

§19.20(iii), §19.27(ii)

Permalink:

http://dlmf.nist.gov/19.27.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(ii), §19.27 and Ch.19

… ►

19.27.4 $$R_G(x,y,z)=\frac{\sqrt{z}}{2}\left(1+O\left(\frac{a}{z}\ln \frac{z}{a}\right)\right),$$ $a/z\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\ln z$: principal branch of logarithm function and $a$

Permalink:

http://dlmf.nist.gov/19.27.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iii), §19.27 and Ch.19

… ►

19.27.6 $$R_G(0,y,z)=\frac{\sqrt{z}}{2}+\frac{y}{8\sqrt{z}}\left(\ln \left(\frac{16z}{y}\right)-1\right)\left(1+O\left(\frac{y}{z}\right)\right),$$ $y/z\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/19.27.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iii), §19.27 and Ch.19

… ►

19.27.7 $$R_D(x,y,z)=\frac{3}{2z^{3/2}}\left(\ln \left(\frac{8z}{a+g}\right)-2\right)\left(1+O\left(\frac{a}{z}\right)\right),$$ $a/z\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\ln z$: principal branch of logarithm function, $a$ and $g$

Referenced by:

§19.20(iv)

Permalink:

http://dlmf.nist.gov/19.27.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iv), §19.27 and Ch.19

… ►

19.27.9 $$R_D(x,y,z)=\frac{3}{\sqrt{xz}(\sqrt{y}+\sqrt{z})}\left(1+O\left(\frac{b}{x}\ln \frac{x}{b}\right)\right),$$ $b/x\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\ln z$: principal branch of logarithm function and $b$

Permalink:

http://dlmf.nist.gov/19.27.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iv), §19.27 and Ch.19

…