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1: 4.16 Elementary Properties

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Table 4.16.2: Trigonometric functions: quarter periods and change of sign.

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$x$	$-\theta$	$\frac{1}{2}\pi \pm \theta$	$\pi \pm \theta$	$\frac{3}{2}\pi \pm \theta$	$2\pi \pm \theta$
…
$\csc x$	$-\csc \theta$	$\sec \theta$	$\mp \csc \theta$	$-\sec \theta$	$\pm \csc \theta$
$\sec x$	$\sec \theta$	$\mp \csc \theta$	$-\sec \theta$	$\pm \csc \theta$	$\sec \theta$
…

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Table 4.16.3: Trigonometric functions: interrelations. …

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	$\sin \theta =a$	$\cos \theta =a$	$\tan \theta =a$	$\csc \theta =a$	$\sec \theta =a$	$\cot \theta =a$
$\sin \theta$	$a$	${(1-a^2)}^{1/2}$	$a{(1+a^2)}^{-1/2}$	$a^{-1}$	$a^{-1}{(a^2-1)}^{1/2}$	${(1+a^2)}^{-1/2}$
$\cos \theta$	${(1-a^2)}^{1/2}$	$a$	${(1+a^2)}^{-1/2}$	$a^{-1}{(a^2-1)}^{1/2}$	$a^{-1}$	$a{(1+a^2)}^{-1/2}$
$\tan \theta$	$a{(1-a^2)}^{-1/2}$	$a^{-1}{(1-a^2)}^{1/2}$	$a$	${(a^2-1)}^{-1/2}$	${(a^2-1)}^{1/2}$	$a^{-1}$
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2: 10.34 Analytic Continuation

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10.34.3 $$I_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=(\mathrm{i}/\pi )\left(\pm {\mathrm{e}}^{m\nu \pi \mathrm{i}}{K}_{\nu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp {\mathrm{e}}^{(m\mp 1)\nu \pi \mathrm{i}}{K}_{\nu }\left(z\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $m$: integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.34, §10.40(i), §10.40(iii), §10.41(iv)

Permalink:

http://dlmf.nist.gov/10.34.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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10.34.4 $$K_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=\csc \left(\nu \pi \right)\left(\pm \sin \left(m\nu \pi \right)K_{\nu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp \sin \left((m\mp 1)\nu \pi \right)K_{\nu }\left(z\right)\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $m$: integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.34, §10.40(i)

Permalink:

http://dlmf.nist.gov/10.34.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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10.34.5 $$K_n\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={(-1)}^{mn}{K}_n\left(z\right)+{(-1)}^{n(m-1)-1}m\pi \mathrm{i}{I}_n\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $m$: integer, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.34.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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10.34.6 $$K_n\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=\pm {(-1)}^{n(m-1)}m{K}_n\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp {(-1)}^{nm}(m\mp 1)K_n\left(z\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $m$: integer, $n$: integer and $z$: complex variable

Referenced by:

§10.40(i)

Permalink:

http://dlmf.nist.gov/10.34.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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3: 4.24 Inverse Trigonometric Functions: Further Properties

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4.24.2 $$\arccos z={(2(1-z))}^{1/2}\left(1+\sum _{n=1}^{\mathrm{\infty }}\frac{1\cdot 3\cdot 5\mathrm{\cdots }(2n-1)}{2^{2n}(2n+1)n!}{(1-z)}^n\right),$$ $|1-z|\le 2$.

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Symbols:

$!$: factorial (as in $n!$), $\arccos z$: arccosine function, $n$: integer and $z$: complex variable

A&S Ref:

4.4.41 (which is stated differently and the constraint on $z$ is more restrictive.)

Permalink:

http://dlmf.nist.gov/4.24.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(i), §4.24 and Ch.4

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4.24.5 $$\arctan z=\frac{z}{z^2+1}\left(1+\frac{2}{3}\frac{z^2}{1+z^2}+\frac{2\cdot 4}{3\cdot 5}{\left(\frac{z^2}{1+z^2}\right)}^2+\mathrm{\cdots }\right),$$ $\mathrm{\Re }\left(z^2\right)>-\frac{1}{2}$,

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Symbols:

$\arctan z$: arctangent function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.4.42 (has an error in the conditions on $z$.)

Permalink:

http://dlmf.nist.gov/4.24.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(i), §4.24 and Ch.4

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4.24.10 $$\frac{d}{dz}\mathrm{arccsc}z=\mp \frac{1}{z{(z^2-1)}^{1/2}},$$ $\mathrm{\Re }z\gtrless 0$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arccsc}z$: arccosecant function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.4.57 (has an error.)

Referenced by:

§4.24(ii)

Permalink:

http://dlmf.nist.gov/4.24.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(ii), §4.24 and Ch.4

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4.24.14 $$\mathrm{Arccos}u\pm \mathrm{Arccos}v=\mathrm{Arccos}\left(uv\mp {((1-u^2)(1-v^2))}^{1/2}\right),$$

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Symbols:

$\mathrm{Arccos}z$: general arccosine function

A&S Ref:

4.4.33

Permalink:

http://dlmf.nist.gov/4.24.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

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4.24.16 $$\mathrm{Arcsin}u\pm \mathrm{Arccos}v=\mathrm{Arcsin}\left(uv\pm {((1-u^2)(1-v^2))}^{1/2}\right)=\mathrm{Arccos}\left(v{(1-u^2)}^{1/2}\mp u{(1-v^2)}^{1/2}\right),$$

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Symbols:

$\mathrm{Arccos}z$: general arccosine function and $\mathrm{Arcsin}z$: general arcsine function

A&S Ref:

4.4.35

Permalink:

http://dlmf.nist.gov/4.24.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

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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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4.38.4 $$\mathrm{arccosh}z={(2(z-1))}^{1/2}\left(1+\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{1\cdot 3\cdot 5\mathrm{\cdots }(2n-1)}{2^{2n}n!(2n+1)}{(z-1)}^n\right),$$ $\mathrm{\Re }z>0$, $|z-1|\le 2$.

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Symbols:

$!$: factorial (as in $n!$), $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{\Re }$: real part, $n$: integer and $z$: complex variable

Referenced by:

§4.38(i)

Permalink:

http://dlmf.nist.gov/4.38.E4

Encodings:

pMML, png, TeX

See also:

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

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4.38.7 $$\mathrm{arctanh}z=\frac{z}{1-z^2}\left(1+\frac{2}{3}\frac{z^2}{z^2-1}+\frac{2\cdot 4}{3\cdot 5}{\left(\frac{z^2}{z^2-1}\right)}^2+\mathrm{\cdots }\right),$$ ,

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Symbols:

$\mathrm{arctanh}z$: inverse hyperbolic tangent function, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§4.38(i)

Permalink:

http://dlmf.nist.gov/4.38.E7

Encodings:

pMML, png, TeX

See also:

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

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4.38.12 $$\frac{d}{dz}\mathrm{arccsch}z=\mp \frac{1}{z{(1+z^2)}^{1/2}},$$ $\mathrm{\Re }z\gtrless 0$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arccsch}z$: inverse hyperbolic cosecant function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.6.40

Permalink:

http://dlmf.nist.gov/4.38.E12

Encodings:

pMML, png, TeX

See also:

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

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5: 25.10 Zeros

… ►The product representation (25.2.11) implies $\zeta \left(s\right)\ne 0$ for $\mathrm{\Re }s>1$. …The functional equation (25.4.1) implies $\zeta \left(-2n\right)=0$ for $n=1,2,3,\mathrm{\dots }$. … ►is chosen to make $Z(t)$ real, and $\mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}t\right)$ assumes its principal value. …Because $Z(t)$ changes sign infinitely often, $\zeta \left(\frac{1}{2}+\mathrm{i}t\right)$ has infinitely many zeros with $t$ real. … ►More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …

6: 4.37 Inverse Hyperbolic Functions

… ►In (4.37.2) the integration path may not pass through either of the points $\pm 1$, and the function ${(t^2-1)}^{1/2}$ assumes its principal value when $t\in (1,\mathrm{\infty })$. …In (4.37.3) the integration path may not intersect $\pm 1$. …$\mathrm{Arcsinh}z$ and $\mathrm{Arccsch}z$ have branch points at $z=\pm \mathrm{i}$; the other four functions have branch points at $z=\pm 1$. … ►On the part of the cuts from $-1$ to $1$ … ►On the part of the cut from $-\mathrm{\infty }$ to $-1$ …

7: 4.23 Inverse Trigonometric Functions

… ►In (4.23.1) and (4.23.2) the integration paths may not pass through either of the points $t=\pm 1$. The function ${(1-t^2)}^{1/2}$ assumes its principal value when $t\in (-1,1)$; elsewhere on the integration paths the branch is determined by continuity. … ►Care needs to be taken on the cuts, for example, if then $1/(x+\mathrm{i}0)=(1/x)-\mathrm{i}0$. … ►For example, from the heading and last entry in the penultimate column we have $\mathrm{arcsec}a=\mathrm{arccot}\left({(a^2-1)}^{-1/2}\right)$. … ►Equivalently, and again when , …

8: Bibliography B

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A note on integrals involving parabolic cylinder functions. SIAM J. Appl. Math. 16 (1), pp. 71–74.

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External Links:

Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§12.12.

Encodings:

BibTeX

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Rainbow over Woolsthorpe Manor. Notes and Records Roy. Soc. London 36 (1), pp. 3–11 (1 plate).

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External Links:

ISSN 0035-9149, Document, MathReview

Referenced by:

Sidebar 9.SB1.

Encodings:

BibTeX

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A cubic counterpart of Jacobi’s identity and the AGM. Trans. Amer. Math. Soc. 323 (2), pp. 691–701.

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External Links:

ISSN 0002-9947, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§15.8(v).

Encodings:

BibTeX

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More than 41% of the zeros of the zeta function are on the critical line. Acta Arith. 150 (1), pp. 35–64.

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External Links:

ISSN 0065-1036, Document, Link, MathReview (Timothy S. Trudgian), ZentralBlatt

Referenced by:

§25.10(ii), §25.10(ii), Erratum (V1.1.4) for Subsection 25.10(ii).

Encodings:

BibTeX

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A program to calculate a general recoupling coefficient. Comput. Phys. Comm. 1 (4), pp. 241–250.

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Notes:

Single-precision Fortran

External Links:

Document, Code

Referenced by:

3rd item.

Encodings:

BibTeX

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9: 16.17 Definition

§16.17 Definition

… ►Then the Meijer $G$-function is defined via the Mellin–Barnes integral representation: … ►When more than one of Cases (i), (ii), and (iii) is applicable the same value is obtained for the Meijer $G$-function. ►Assume $p\le q$, no two of the bottom parameters $b_j$, $j=1,\mathrm{\dots },m$, differ by an integer, and $a_j-b_k$ is not a positive integer when $j=1,2,\mathrm{\dots },n$ and $k=1,2,\mathrm{\dots },m$. Then …

10: Possible Errors in DLMF

… ►One source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the icon) for links to defining formula. …Errors in the printed Handbook may already have been corrected in the online version; please consult Errata. …