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1: 4.37 Inverse Hyperbolic Functions

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§4.37(iv) Logarithmic Forms

… ►For the corresponding results for $\mathrm{arccsch}z$, $\mathrm{arcsech}z$, and $\mathrm{arccoth}z$, use (4.37.7)–(4.37.9); compare §4.23(iv). …

2: 4.45 Methods of Computation

… ►The inverses $\mathrm{arcsinh}$, $\mathrm{arccosh}$, and $\mathrm{arctanh}$ can be computed from the logarithmic forms given in §4.37(iv), with real arguments. … ►The trigonometric functions may be computed from the definitions (4.14.1)–(4.14.7), and their inverses from the logarithmic forms in §4.23(iv), followed by (4.23.7)–(4.23.9). …

3: 4.23 Inverse Trigonometric Functions

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§4.23(iv) Logarithmic Forms

… ►Care needs to be taken on the cuts, for example, if then $1/(x+\mathrm{i}0)=(1/x)-\mathrm{i}0$. …

4: 4.38 Inverse Hyperbolic Functions: Further Properties

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4.38.2 $$\mathrm{arcsinh}z=\ln \left(2z\right)+\frac{1}{2}\frac{1}{2z^2}-\frac{1\cdot 3}{2\cdot 4}\frac{1}{4z^4}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{1}{6z^6}-\mathrm{\cdots },$$ $\mathrm{\Re }z>0$, $|z|>1$.

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Symbols:

$\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.6.31 (misses a condition on $z$.)

Referenced by:

§4.38(i)

Permalink:

http://dlmf.nist.gov/4.38.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(i), §4.38 and Ch.4

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4.38.3 $$\mathrm{arccosh}z=\ln \left(2z\right)-\frac{1}{2}\frac{1}{2z^2}-\frac{1\cdot 3}{2\cdot 4}\frac{1}{4z^4}-\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{1}{6z^6}-\mathrm{\cdots },$$ $|z|>1$.

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.6.32

Referenced by:

§4.38(i)

Permalink:

http://dlmf.nist.gov/4.38.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(i), §4.38 and Ch.4

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5: 27.11 Asymptotic Formulas: Partial Sums

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27.11.4 $$\sum _{n\le x}{\sigma }_1\left(n\right)=\frac{{\pi }^2}{12}{x}^2+O\left(x\ln x\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\ln z$: principal branch of logarithm function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.3 III (in slightly different form)

Permalink:

http://dlmf.nist.gov/27.11.E4

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.11 and Ch.27

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27.11.6 $$\sum _{n\le x}\varphi \left(n\right)=\frac{3}{{\pi }^2}{x}^2+O\left(x\ln x\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\varphi \left(n\right)$: Euler’s totient, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.2 III (in slightly different form)

Permalink:

http://dlmf.nist.gov/27.11.E6

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.11 and Ch.27

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6: 4.3 Graphics

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Figure 4.3.1: $\ln x$ and ${\mathrm{e}}^x$. … Magnify

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§4.3(ii) Complex Arguments: Conformal Maps

►Figure 4.3.2 illustrates the conformal mapping of the strip onto the whole $w$-plane cut along the negative real axis, where $w={\mathrm{e}}^z$ and $z=\ln w$ (principal value). … ►

► … ►

Figure 4.3.2: Conformal mapping of exponential and logarithm. … Magnify

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Figure 4.3.3: $\ln \left(x+\mathrm{i}y\right)$ (principal value). … Magnify 3D Help

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7: 8.9 Continued Fractions

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8.9.1 $$\mathrm{\Gamma }\left(a+1\right){\mathrm{e}}^z{\gamma }^{\ast }(a,z)=\frac{1}{1-}\frac{z}{a+1+}\frac{z}{a+2-}\frac{(a+1)z}{a+3+}\frac{2z}{a+4-}\frac{(a+2)z}{a+5+}\frac{3z}{a+6-}\mathrm{\cdots },$$ $a\ne -1,-2,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.31 (in modified form.)

Permalink:

http://dlmf.nist.gov/8.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.9 and Ch.8

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8: 6.7 Integral Representations

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6.7.1 $${\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-at}}{t+b}dt={\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{\mathrm{i}at}}{t+\mathrm{i}b}dt={\mathrm{e}}^{ab}{E}_1\left(ab\right),$$ $a>0$, $b>0$,

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit and $\int$: integral

A&S Ref:

5.1.28 (modified form of) 5.1.29 (modified form of)

Referenced by:

§6.7(i)

Permalink:

http://dlmf.nist.gov/6.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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6.7.3 $${\int }_x^{\mathrm{\infty }}\frac{{\mathrm{e}}^{\mathrm{i}t}}{a^2+t^2}dt=\frac{\mathrm{i}}{2a}\left({\mathrm{e}}^a{E}_1\left(a-\mathrm{i}x\right)-{\mathrm{e}}^{-a}{E}_1\left(-a-\mathrm{i}x\right)\right),$$ $a>0$, $x>0$,

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\int$: integral and $x$: real variable

A&S Ref:

5.1.41 (modified form of)

Referenced by:

§6.7(i)

Permalink:

http://dlmf.nist.gov/6.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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6.7.4 $${\int }_x^{\mathrm{\infty }}\frac{t{\mathrm{e}}^{\mathrm{i}t}}{a^2+t^2}dt=\frac{1}{2}\left({\mathrm{e}}^a{E}_1\left(a-\mathrm{i}x\right)+{\mathrm{e}}^{-a}{E}_1\left(-a-\mathrm{i}x\right)\right),$$ $a>0$, $x>0$.

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\int$: integral and $x$: real variable

A&S Ref:

5.1.42 (modified form of)

Permalink:

http://dlmf.nist.gov/6.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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6.7.5 $${\int }_x^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t}}{a^2+t^2}dt=-\frac{1}{2a\mathrm{i}}\left({\mathrm{e}}^{\mathrm{i}a}{E}_1\left(x+\mathrm{i}a\right)-{\mathrm{e}}^{-\mathrm{i}a}{E}_1\left(x-\mathrm{i}a\right)\right),$$ $a>0$, $x\in ℝ$,

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Symbols:

$dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\int$: integral, $ℝ$: real line and $x$: real variable

A&S Ref:

5.1.43 (modified form of)

Permalink:

http://dlmf.nist.gov/6.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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6.7.6 $${\int }_x^{\mathrm{\infty }}\frac{t{\mathrm{e}}^{-t}}{a^2+t^2}dt=\frac{1}{2}\left({\mathrm{e}}^{\mathrm{i}a}{E}_1\left(x+\mathrm{i}a\right)+{\mathrm{e}}^{-\mathrm{i}a}{E}_1\left(x-\mathrm{i}a\right)\right),$$ $a>0$, $x\in ℝ$.

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Symbols:

$dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\int$: integral, $ℝ$: real line and $x$: real variable

A&S Ref:

5.1.44 (modified form of)

Referenced by:

§6.7(i)

Permalink:

http://dlmf.nist.gov/6.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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9: 3.9 Acceleration of Convergence

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3.9.5 $$\ln 2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\mathrm{\cdots }=\frac{1}{1\cdot 2^1}+\frac{1}{2\cdot 2^2}+\frac{1}{3\cdot 2^3}+\mathrm{\cdots },$$

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Symbols:

$\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/3.9.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §3.9(ii), §3.9 and Ch.3

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§3.9(v) Levin’s and Weniger’s Transformations

►We give a special form of Levin’s transformation in which the sequence $s=\{s_n\}$ of partial sums $s_n={\sum }_{j=0}^n{a}_j$ is transformed into: …Sequences that are accelerated by Levin’s transformation include logarithmically convergent sequences, i. …

10: 6.2 Definitions and Interrelations

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6.2.2 $$E_1\left(z\right)={\mathrm{e}}^{-z}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t}}{t+z}dt,$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

5.1.1 (in modified form)

Referenced by:

§2.5(iii), §6.18(i), §6.7(iii)

Permalink:

http://dlmf.nist.gov/6.2.E2

Encodings:

pMML, png, TeX

See also:

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

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6.2.7 $$\mathrm{Ei}\left(\pm x\right)=-\mathrm{Ein}\left(\mp x\right)+\ln x+\gamma .$$

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

5.1.40 (in modified form)

Permalink:

http://dlmf.nist.gov/6.2.E7

Encodings:

pMML, png, TeX

See also:

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

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6.2.13 $$\mathrm{Ci}\left(z\right)=-\mathrm{Cin}\left(z\right)+\ln z+\gamma .$$

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ci}\left(z\right)$: cosine integral, $\mathrm{Cin}\left(z\right)$: cosine integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

5.2.2 (in modified form)

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.2.E13

Encodings:

pMML, png, TeX

See also:

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

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