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21: 15.9 Relations to Other Functions

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15.9.17 $$𝐅(\genfrac{}{}{0pt}{}{a,a+\frac{1}{2}}{c};z)=2^{c-1}{z}^{(1-c)/2}{(1-z)}^{-a+((c-1)/2)}{P}_{2a-c}^{1-c}\left(\frac{1}{\sqrt{1-z}}\right),$$ and .

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(iv), §15.9 and Ch.15

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15.9.21 $$𝐅(\genfrac{}{}{0pt}{}{a,1-a}{c};z)={\left(\frac{-z}{1-z}\right)}^{(1-c)/2}{P}_{-a}^{1-c}\left(1-2z\right),$$ .

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.9(iv)

Permalink:

http://dlmf.nist.gov/15.9.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(iv), §15.9 and Ch.15

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15.9.24 $$K\left(k\right)=\frac{\pi }{2}F(\genfrac{}{}{0pt}{}{\frac{1}{2},\frac{1}{2}}{1};k^2),$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind and $k$: integer

Referenced by:

§15.9(v)

Permalink:

http://dlmf.nist.gov/15.9.E24

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §15.9(v), §15.9 and Ch.15

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15.9.25 $$E\left(k\right)=\frac{\pi }{2}F(\genfrac{}{}{0pt}{}{-\frac{1}{2},\frac{1}{2}}{1};k^2),$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind and $k$: integer

Referenced by:

§15.9(v)

Permalink:

http://dlmf.nist.gov/15.9.E25

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §15.9(v), §15.9 and Ch.15

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15.9.26 $$D\left(k\right)=\frac{\pi }{4}F(\genfrac{}{}{0pt}{}{\frac{1}{2},\frac{3}{2}}{2};k^2).$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $D\left(k\right)$: complete elliptic integral of Legendre’s type and $k$: integer

Referenced by:

§15.9(v)

Permalink:

http://dlmf.nist.gov/15.9.E26

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §15.9(v), §15.9 and Ch.15

22: 10.65 Power Series

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10.65.3 $${\ker }_{n}x=\frac{1}{2}{(\frac{1}{2}x)}^{-n}\sum _{k=0}^{n-1}\frac{(n-k-1)!}{k!}\cos \left(\frac{3}{4}n\pi +\frac{1}{2}k\pi \right){(\frac{1}{4}{x}^2)}^k-\ln \left(\frac{1}{2}x\right){\mathrm{ber}}_{n}x+\frac{1}{4}\pi {\mathrm{bei}}_{n}x+\frac{1}{2}{(\frac{1}{2}x)}^n\sum _{k=0}^{\mathrm{\infty }}\frac{\psi \left(k+1\right)+\psi \left(n+k+1\right)}{k!(n+k)!}\cos \left(\frac{3}{4}n\pi +\frac{1}{2}k\pi \right){(\frac{1}{4}{x}^2)}^k,$$

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Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $n$: integer, $k$: nonnegative integer and $x$: real variable

A&S Ref:

9.9.11

Permalink:

http://dlmf.nist.gov/10.65.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.65(ii), §10.65 and Ch.10

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10.65.4 $${\mathrm{kei}}_{n}x=-\frac{1}{2}{(\frac{1}{2}x)}^{-n}\sum _{k=0}^{n-1}\frac{(n-k-1)!}{k!}\sin \left(\frac{3}{4}n\pi +\frac{1}{2}k\pi \right){(\frac{1}{4}{x}^2)}^k-\ln \left(\frac{1}{2}x\right){\mathrm{bei}}_{n}x-\frac{1}{4}\pi {\mathrm{ber}}_{n}x+\frac{1}{2}{(\frac{1}{2}x)}^n\sum _{k=0}^{\mathrm{\infty }}\frac{\psi \left(k+1\right)+\psi \left(n+k+1\right)}{k!(n+k)!}\sin \left(\frac{3}{4}n\pi +\frac{1}{2}k\pi \right){(\frac{1}{4}{x}^2)}^k.$$

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Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $n$: integer, $k$: nonnegative integer and $x$: real variable

A&S Ref:

9.9.11

Permalink:

http://dlmf.nist.gov/10.65.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.65(ii), §10.65 and Ch.10

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§10.65(iii) Cross-Products and Sums of Squares

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23: 10.21 Zeros

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10.21.24 $$\theta \left(-2^{\frac{1}{3}}\alpha \right)=\pi t,$$

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Defines:

$\alpha$: coefficients (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter and $\theta \left(z\right)$: Airy phase function

Permalink:

http://dlmf.nist.gov/10.21.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(vii), §10.21 and Ch.10

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10.21.29 $$\varphi \left(-2^{\frac{1}{3}}{\alpha }^{\prime }\right)=\pi t,$$

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Defines:

${\alpha }^{\prime }$: coefficients (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter and $\varphi \left(z\right)$: Airy phase function

Permalink:

http://dlmf.nist.gov/10.21.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(vii), §10.21 and Ch.10

… ►

§10.21(x) Cross-Products

… ►Higher coefficients in the asymptotic expansions in this subsection can be obtained by expressing the cross-products in terms of the modulus and phase functions (§10.18), and then reverting the asymptotic expansion for the difference of the phase functions. … ►For information on the zeros of the derivatives of Riccati–Bessel functions, and also on zeros of their cross-products, see Boyer (1969). …

24: 4.24 Inverse Trigonometric Functions: Further Properties

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4.24.4 $$\arctan z=\pm \frac{\pi }{2}-\frac{1}{z}+\frac{1}{3z^3}-\frac{1}{5z^5}+\mathrm{\cdots },$$ $\mathrm{\Re }z\gtrless 0$, $|z|\ge 1$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.4.42 (has an error. $\frac{\pi }{2}$ should have a negative sign when .)

Referenced by:

§4.45(i)

Permalink:

http://dlmf.nist.gov/4.24.E4

Encodings:

pMML, png, TeX

See also:

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

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25: Bibliography D

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Strong and ratio asymptotics for Laguerre polynomials revisited. J. Math. Anal. Appl. 403 (2), pp. 477–486.

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External Links:

ISSN 0022-247X, Document, FullText, MathReview (Pablo Manuel Román), ZentralBlatt

Referenced by:

§18.15(iv), §18.15(iv).

Encodings:

BibTeX

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Texas Instruments, Inc..

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Notes:

Symbolic and extended-precision numerical computing system.

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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Computation of the incomplete gamma function ratios and their inverses. ACM Trans. Math. Software 12 (4), pp. 377–393.

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External Links:

ISSN 0098-3500, Document, ZentralBlatt

Referenced by:

§8.12, §8.25(iii).

Encodings:

BibTeX

►

Algorithm 654: Fortran subroutines for computing the incomplete gamma function ratios and their inverses. ACM Trans. Math. Software 13 (3), pp. 318–319.

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Notes:

Single-precision Fortran, maximum accuracy 14D.

External Links:

ISSN 0098-3500, Document, GAMS (module 654; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Algorithm 708: Significant digit computation of the incomplete beta function ratios. ACM Trans. Math. Software 18 (3), pp. 360–373.

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Notes:

Single-precision Fortran, maximum accuracy 14D.

External Links:

ISSN 0098-3500, Document, GAMS (module 708; package TOMS), ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

…

26: 33.2 Definitions and Basic Properties

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33.2.5 $$C_{\mathrm{\ell }}\left(\eta \right)=\frac{2^{\mathrm{\ell }}{\mathrm{e}}^{-\pi \eta /2}|\mathrm{\Gamma }\left(\mathrm{\ell }+1+\mathrm{i}\eta \right)|}{(2\mathrm{\ell }+1)!}.$$

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Defines:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter

A&S Ref:

14.1.7

Referenced by:

§33.10(ii), §33.10(iii), §33.16(i)

Permalink:

http://dlmf.nist.gov/33.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §33.2(ii), §33.2 and Ch.33

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33.2.6 $$C_{\mathrm{\ell }}\left(\eta \right)=\frac{2^{\mathrm{\ell }}{\left((2\pi \eta /({\mathrm{e}}^{2\pi \eta }-1)){\prod }_{k=1}^{\mathrm{\ell }}({\eta }^2+k^2)\right)}^{1/2}}{(2\mathrm{\ell }+1)!}.$$

ⓘ

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter

Referenced by:

§33.13, §33.16(i)

Permalink:

http://dlmf.nist.gov/33.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.2(ii), §33.2 and Ch.33

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33.2.7 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )={(\mp \mathrm{i})}^{\mathrm{\ell }}{\mathrm{e}}^{(\pi \eta /2)\pm \mathrm{i}{\sigma }_{\mathrm{\ell }}\left(\eta \right)}{W}_{\mp \mathrm{i}\eta ,\mathrm{\ell }+\frac{1}{2}}\left(\mp 2\mathrm{i}\rho \right),$$

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Defines:

$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions

Symbols:

${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Referenced by:

§13.18(vi), §33.23(vi)

Permalink:

http://dlmf.nist.gov/33.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §33.2(iii), §33.2 and Ch.33

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33.2.9 $${\theta }_{\mathrm{\ell }}(\eta ,\rho )=\rho -\eta \ln \left(2\rho \right)-\frac{1}{2}\mathrm{\ell }\pi +{\sigma }_{\mathrm{\ell }}\left(\eta \right),$$

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Defines:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions

Symbols:

${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

A&S Ref:

14.5.5

Referenced by:

§33.10(i), §33.11

Permalink:

http://dlmf.nist.gov/33.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §33.2(iii), §33.2 and Ch.33

… ►

§33.2(iv) Wronskians and Cross-Product

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27: 2.11 Remainder Terms; Stokes Phenomenon

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2.11.1 $$I(m)={\int }_0^{\pi }\frac{\cos \left(mt\right)}{t^2+1}dt,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $I(m)$: integral

Permalink:

http://dlmf.nist.gov/2.11.E1

Encodings:

pMML, png, TeX

See also:

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

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2.11.2 $$I(m)\sim {(-1)}^m\sum _{s=1}^{\mathrm{\infty }}\frac{q_s(\pi )}{m^{2s}},$$ $m\to \mathrm{\infty }$,

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $I(m)$: integral and $q_s(t)$: coefficients

Referenced by:

§2.11(i)

Permalink:

http://dlmf.nist.gov/2.11.E2

Encodings:

pMML, png, TeX

See also:

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

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2.11.16 $$c(\theta )=\sqrt{2(1+{\mathrm{e}}^{\mathrm{i}\theta }+\mathrm{i}(\theta -\pi ))},$$

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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, $\theta$: radius and $c(\theta )$

Permalink:

http://dlmf.nist.gov/2.11.E16

Encodings:

pMML, png, TeX

See also:

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

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2.11.18 $$h_0(\theta ,\alpha )=\frac{{\mathrm{e}}^{\mathrm{i}\alpha (\pi -\theta )}}{1+{\mathrm{e}}^{-\mathrm{i}\theta }}-\frac{\mathrm{i}}{c(\theta )}.$$

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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, $\theta$: radius, $c(\theta )$ and $h_{2s}(\theta ,\alpha )$: functions

Permalink:

http://dlmf.nist.gov/2.11.E18

Encodings:

pMML, png, TeX

See also:

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

… ►As these lines are crossed exponentially-small contributions, such as that in (2.11.7), are “switched on” smoothly, in the manner of the graph in Figure 2.11.1. …

28: Bibliography L

… ►

Ratios of Bessel functions and roots of $\alpha J_{\nu }(x)+x{J}_{\nu }^{\prime }(x)=0$ . J. Math. Anal. Appl. 240 (1), pp. 174–204.

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External Links:

ISSN 0022-247X, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.14, §10.15, §10.21(xii).

Encodings:

BibTeX

… ►

A Numerical Library in C for Scientists and Engineers. CRC Press, Boca Raton, FL.

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Notes:

With 1 IBM-PC floppy disk (3.5 inch; HD) containing a large general purpose mathematical software collection written in C including a collection of special function classes. The functions are computed for real variables only. (See Lau (2004) for Java version).

External Links:

ISBN 0-8493-7376-X, MathReview, ZentralBlatt

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, H. T. Lau (2004).

Encodings:

BibTeX

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Comparison of a pair of upper bounds for a ratio of gamma functions. Math. Balkanica (N.S.) 16 (1-4), pp. 195–202.

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External Links:

ISSN 0205-3217, MathReview, ZentralBlatt

Referenced by:

§5.6(i).

Encodings:

BibTeX

…

29: 4.37 Inverse Hyperbolic Functions

… ►The principal values (or principal branches) of the inverse $\sinh$, $\cosh$, and $\tanh$ are obtained by introducing cuts in the $z$-plane as indicated in Figure 4.37.1(i)-(iii), and requiring the integration paths in (4.37.1)–(4.37.3) not to cross these cuts. … ►

4.37.11 $$\mathrm{arccosh}\left(-z\right)=\pm \pi \mathrm{i}+\mathrm{arccosh}z,$$ $\mathrm{\Im }z\gtrless 0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.6.12 (has an error even in the tenth printing.)

Referenced by:

§4.37(iii)

Permalink:

http://dlmf.nist.gov/4.37.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iii), §4.37 and Ch.4

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4.37.14 $$\mathrm{arcsech}\left(-z\right)=\mp \pi \mathrm{i}+\mathrm{arcsech}z,$$ $\mathrm{\Im }z\gtrless 0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arcsech}z$: inverse hyperbolic secant function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iii), §4.37 and Ch.4

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4.37.20 $$\mathrm{arccosh}\left(\mathrm{i}y\right)=\pm \frac{1}{2}\pi \mathrm{i}+\ln \left({(y^2+1)}^{1/2}\pm y\right),$$ $y\gtrless 0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function and $y$: real variable

Referenced by:

§4.37(iv)

Permalink:

http://dlmf.nist.gov/4.37.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

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4.37.29 $$w=\mathrm{Arcsinh}z={(-1)}^k\mathrm{arcsinh}z+k\pi \mathrm{i},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\mathrm{Arcsinh}z$: general inverse hyperbolic sine function, $k$: integer and $z$: complex variable

A&S Ref:

4.6.8

Permalink:

http://dlmf.nist.gov/4.37.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(v), §4.37 and Ch.4

…

30: 4.23 Inverse Trigonometric Functions

… ►The principal values (or principal branches) of the inverse sine, cosine, and tangent are obtained by introducing cuts in the $z$-plane as indicated in Figures 4.23.1(i) and 4.23.1(ii), and requiring the integration paths in (4.23.1)–(4.23.3) not to cross these cuts. … ►

4.23.11 $$\arccos \left(-z\right)=\pi -\arccos z.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\arccos z$: arccosine function and $z$: complex variable

A&S Ref:

4.4.15

Permalink:

http://dlmf.nist.gov/4.23.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(iii), §4.23 and Ch.4

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4.23.14 $$\mathrm{arcsec}\left(-z\right)=\pi -\mathrm{arcsec}z.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arcsec}z$: arcsecant function and $z$: complex variable

A&S Ref:

4.4.18

Permalink:

http://dlmf.nist.gov/4.23.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(iii), §4.23 and Ch.4

… ►

4.23.16 $$\arccos z=\frac{1}{2}\pi -\arcsin z,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\arccos z$: arccosine function, $\arcsin z$: arcsine function and $z$: complex variable

A&S Ref:

4.4.2

Referenced by:

§4.15(iii), §4.23(iv)

Permalink:

http://dlmf.nist.gov/4.23.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(iii), §4.23 and Ch.4

►

4.23.17 $$\mathrm{arcsec}z=\frac{1}{2}\pi -\mathrm{arccsc}z.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arccsc}z$: arccosecant function, $\mathrm{arcsec}z$: arcsecant function and $z$: complex variable

A&S Ref:

4.4.9

Permalink:

http://dlmf.nist.gov/4.23.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(iii), §4.23 and Ch.4

…