relations to trigonometric functions

SearchAdvancedHelp

(0.020 seconds)

11—20 of 110 matching pages

11: 19.25 Relations to Other Functions

§19.25 Relations to Other Functions

… ►

§19.25(iv) Theta Functions

… ► … ►

§19.25(vii) Hypergeometric Function

… ►

12: 13.18 Relations to Other Functions

§13.18 Relations to Other Functions

►

§13.18(i) Elementary Functions

… ►

§13.18(iv) Parabolic Cylinder Functions

… ►

§13.18(v) Orthogonal Polynomials

… ►

Laguerre Polynomials

…

13: 19.10 Relations to Other Functions

§19.10 Relations to Other Functions

►

§19.10(i) Theta and Elliptic Functions

►For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. … ►

§19.10(ii) Elementary Functions

… ►For relations to the Gudermannian function $\mathrm{gd}\left(x\right)$ and its inverse ${\mathrm{gd}}^{-1}\left(x\right)$ (§4.23(viii)), see (19.6.8) and …

14: 19.6 Special Cases

… ►

19.6.8 $$F(\varphi ,1)=(\sin \varphi )R_C(1,{\cos }^2\varphi )={\mathrm{gd}}^{-1}\left(\varphi \right).$$

ⓘ

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\sin z$: sine function and $\varphi$: real or complex argument

Referenced by:

§19.10(ii), §19.10(ii)

Permalink:

http://dlmf.nist.gov/19.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.6(ii), §19.6 and Ch.19

…

15: 22.16 Related Functions

… ►

22.16.8 $$\mathrm{am}(x,k)=\mathrm{gd}x-\frac{1}{4}k_{}^{\prime }{}_{}{}^2(x-\sinh x\cosh x)\mathrm{sech}x+O\left(k_{}^{\prime }{}_{}{}^4\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{gd}x$: Gudermannian function, $\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.4

Permalink:

http://dlmf.nist.gov/22.16.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

…

16: 7.2 Definitions

… ►

§7.2(i) Error Functions

… ► $\mathrm{erf}z$, $\mathrm{erfc}z$, and $w\left(z\right)$ are entire functions of $z$, as is $F\left(z\right)$ in the next subsection. ►

Values at Infinity

… ► $ℱ\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$ are entire functions of $z$, as are $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$ in the next subsection. … ►

§7.2(iv) Auxiliary Functions

…

17: 28.32 Mathematical Applications

… ►If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient. … ►This leads to integral equations and an integral relation between the solutions of Mathieu’s equation (setting $\zeta =\mathrm{i}\xi$, $z=\eta$ in (28.32.3)). … ►approaches the same value when $\zeta$ tends to the endpoints of $ℒ$. … ►Two conditions are used to determine $A,B$. …

18: 24.15 Related Sequences of Numbers

§24.15 Related Sequences of Numbers

►

§24.15(i) Genocchi Numbers

… ►

§24.15(ii) Tangent Numbers

►

24.15.3 $$\tan t=\sum _{n=0}^{\mathrm{\infty }}{T}_n\frac{t^n}{n!},$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\tan z$: tangent function, $n$: integer, $t$: real or complex and $T_n$: tangent numbers

Referenced by:

§24.19(i)

Permalink:

http://dlmf.nist.gov/24.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(ii), §24.15 and Ch.24

… ►

§24.15(iii) Stirling Numbers

…

19: 18.35 Pollaczek Polynomials

… ►

18.35.7 $${(1-z{\mathrm{e}}^{\mathrm{i}\theta })}^{-\lambda +\mathrm{i}{\tau }_{a,b}(\theta )}{(1-z{\mathrm{e}}^{-\mathrm{i}\theta })}^{-\lambda -\mathrm{i}{\tau }_{a,b}(\theta )}=\sum _{n=0}^{\mathrm{\infty }}{P}_n^{(\lambda )}(\cos \theta ;a,b)z^n,$$ , .

ⓘ

Symbols:

$P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $n$: nonnegative integer and ${\tau }_{a,b}(\theta )$

Proved:

Ismail (2009, (5.4.8))(proved)

Permalink:

http://dlmf.nist.gov/18.35.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.35(iii), §18.35 and Ch.18

…

20: 32.11 Asymptotic Approximations for Real Variables

… ►where … ►Any nontrivial real solution of (32.11.4) that satisfies (32.11.5) is asymptotic to $k\mathrm{Ai}\left(x\right)$, for some nonzero real $k$, where $\mathrm{Ai}$ denotes the Airy function (§9.2). … ►where … ►The connection formulas relating (32.11.25) and (32.11.26) are … ►Now suppose $x\to -\mathrm{\infty }$. …