trigonometric functions

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31: Guide to Searching the DLMF

… ►

Table 1: Query Examples

► ►►►►►

Query	Matching records contain
…
trigonometric	the word ”trigonometric” or any of the various trigonometric functions such as $\sin$, $\cos$, $\tan$, and $\cot$.
trig$	any word that matches the pattern “trig$”, that is, that starts with “trig”, such as “trigonometric” and “trigonometry”, including the various trigonometric functions such as $\sin$ and $\cos$.
trigonometric^2 + trig$^2	any sum of the squares of two trigonometric functions such as ${\sin }^{2}z+{\cos }^{2}z$.
…

►

… ►

All the inverse trigonometric functions (arcsin vs. Arcsin, etc.).

… ►For example, you may want equations that contain trigonometric functions, but you don’t care which trigonometric function. …

32: 4.40 Integrals

… ►

4.40.1 $$\int \sinh xdx=\cosh x,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral and $x$: real variable

A&S Ref:

4.5.77

Permalink:

http://dlmf.nist.gov/4.40.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

►

4.40.2 $$\int \cosh xdx=\sinh x,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral and $x$: real variable

A&S Ref:

4.5.78

Permalink:

http://dlmf.nist.gov/4.40.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

►

4.40.3 $$\int \tanh xdx=\ln \left(\cosh x\right).$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.5.79

Permalink:

http://dlmf.nist.gov/4.40.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

►

4.40.4 $$\int \mathrm{csch}xdx=\ln \left(\tanh \left(\frac{1}{2}x\right)\right),$$ .

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{csch}z$: hyperbolic cosecant function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.5.80

Permalink:

http://dlmf.nist.gov/4.40.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

… ►

4.40.6 $$\int \coth xdx=\ln \left(\sinh x\right),$$ .

ⓘ

Symbols:

$dx$: differential of $x$ , $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.5.82

Permalink:

http://dlmf.nist.gov/4.40.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

…

33: 28.11 Expansions in Series of Mathieu Functions

… ►

28.11.4 $$\cos 2mz=\sum _{n=0}^{\mathrm{\infty }}{A}_{2m}^{2n}(q){\mathrm{ce}}_{2n}(z,q),$$ $m\ne 0$,

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

►

28.11.5 $$\cos (2m+1)z=\sum _{n=0}^{\mathrm{\infty }}{A}_{2m+1}^{2n+1}(q){\mathrm{ce}}_{2n+1}(z,q),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

►

28.11.6 $$\sin (2m+1)z=\sum _{n=0}^{\mathrm{\infty }}{B}_{2m+1}^{2n+1}(q){\mathrm{se}}_{2n+1}(z,q),$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

►

28.11.7 $$\sin (2m+2)z=\sum _{n=0}^{\mathrm{\infty }}{B}_{2m+2}^{2n+2}(q){\mathrm{se}}_{2n+2}(z,q).$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

34: 28.23 Expansions in Series of Bessel Functions

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28.23.2 $${\mathrm{me}}_{\nu }(0,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n{c}_{2n}^{\nu }(h^2){𝒞}_{\nu +2n}^{(j)}(2h\cosh z),$$

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $c_{2m}(q)$: coefficients and ${𝒞}_{\mu }^{(j)}$: cylinder functions

Permalink:

http://dlmf.nist.gov/28.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

►

28.23.3 $${\mathrm{me}}_{\nu }^{\prime }(0,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h)=\mathrm{i}\tanh z\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n(\nu +2n)c_{2n}^{\nu }(h^2){𝒞}_{\nu +2n}^{(j)}(2h\cosh z),$$

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $c_{2m}(q)$: coefficients and ${𝒞}_{\mu }^{(j)}$: cylinder functions

Permalink:

http://dlmf.nist.gov/28.23.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

… ►

28.23.6 $${\mathrm{Mc}}_{2m}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m}(0,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}{A}_{2\mathrm{\ell }}^{2m}(h^2){𝒞}_{2\mathrm{\ell }}^{(j)}(2h\cosh z),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

A&S Ref:

20.6.12

Referenced by:

§28.23

Permalink:

http://dlmf.nist.gov/28.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

… ►

28.23.10 $${\mathrm{Ms}}_{2m+1}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{se}}_{2m+1}^{\prime }(0,h^2)\right)}^{-1}\tanh z\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}(2\mathrm{\ell }+1)B_{2\mathrm{\ell }+1}^{2m+1}(h^2){𝒞}_{2\mathrm{\ell }+1}^{(j)}(2h\cosh z),$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

… ►

28.23.12 $${\mathrm{Ms}}_{2m+2}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{se}}_{2m+2}^{\prime }(0,h^2)\right)}^{-1}\tanh z\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}(2\mathrm{\ell }+2)B_{2\mathrm{\ell }+2}^{2m+2}(h^2){𝒞}_{2\mathrm{\ell }+2}^{(j)}(2h\cosh z),$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

…

35: 4.37 Inverse Hyperbolic Functions

… ►

4.37.4 $$\mathrm{Arccsch}z=\mathrm{Arcsinh}\left(1/z\right),$$

ⓘ

Defines:

$\mathrm{Arccsch}z$: general inverse hyperbolic cosecant function

Symbols:

$\mathrm{Arcsinh}z$: general inverse hyperbolic sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E4

Encodings:

pMML, png, TeX

See also:

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

… ►Each is two-valued on the corresponding cut(s), and each is real on the part of the real axis that remains after deleting the intersections with the corresponding cuts. … ►

4.37.7 $$\mathrm{arccsch}z=\mathrm{arcsinh}\left(1/z\right),$$

ⓘ

Defines:

$\mathrm{arccsch}z$: inverse hyperbolic cosecant function

Symbols:

$\mathrm{arcsinh}z$: inverse hyperbolic sine function and $z$: complex variable

A&S Ref:

4.6.4

Referenced by:

§4.37(iv), §4.45(i), §4.45(ii)

Permalink:

http://dlmf.nist.gov/4.37.E7

Encodings:

pMML, png, TeX

See also:

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

►

4.37.8 $$\mathrm{arcsech}z=\mathrm{arccosh}\left(1/z\right).$$

ⓘ

Defines:

$\mathrm{arcsech}z$: inverse hyperbolic secant function

Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function and $z$: complex variable

A&S Ref:

4.6.5

Permalink:

http://dlmf.nist.gov/4.37.E8

Encodings:

pMML, png, TeX

See also:

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

►

4.37.9 $$\mathrm{arccoth}z=\mathrm{arctanh}\left(1/z\right),$$ $z\ne \pm 1$.

ⓘ

Defines:

$\mathrm{arccoth}z$: inverse hyperbolic cotangent function

Symbols:

$\mathrm{arctanh}z$: inverse hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.6.6

Referenced by:

§4.37(iv), §4.45(i), §4.45(ii)

Permalink:

http://dlmf.nist.gov/4.37.E9

Encodings:

pMML, png, TeX

See also:

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

…

36: 14.19 Toroidal (or Ring) Functions

… ►

14.19.4 $$P_{n-\frac{1}{2}}^m\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(n+m+\frac{1}{2}\right){(\sinh \xi )}^m}{2^m{\pi }^{1/2}\mathrm{\Gamma }\left(n-m+\frac{1}{2}\right)\mathrm{\Gamma }\left(m+\frac{1}{2}\right)}{\int }_0^{\pi }\frac{{(\sin \varphi )}^{2m}}{{(\cosh \xi +\cos \varphi \sinh \xi )}^{n+m+(1/2)}}d\varphi ,$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer and $\xi >0$: variable

A&S Ref:

8.11.2

Permalink:

http://dlmf.nist.gov/14.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.19(iii), §14.19 and Ch.14

►

14.19.5 $${𝑸}_{n-\frac{1}{2}}^m\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(n+\frac{1}{2}\right)}{\mathrm{\Gamma }\left(n+m+\frac{1}{2}\right)\mathrm{\Gamma }\left(n-m+\frac{1}{2}\right)}{\int }_0^{\mathrm{\infty }}\frac{\cosh \left(mt\right)}{{(\cosh \xi +\cosh t\sinh \xi )}^{n+(1/2)}}dt,$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $m$: nonnegative integer, $n$: nonnegative integer and $\xi >0$: variable

A&S Ref:

8.11.4 (modified and corrected)

Permalink:

http://dlmf.nist.gov/14.19.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.19(iii), §14.19 and Ch.14

… ►

14.19.6 $${𝑸}_{-\frac{1}{2}}^{\mu }\left(\cosh \xi \right)+2\sum _{n=1}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\mu +n+\frac{1}{2}\right)}{\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right)}{𝑸}_{n-\frac{1}{2}}^{\mu }\left(\cosh \xi \right)\cos \left(n\varphi \right)=\frac{{\left(\frac{1}{2}\pi \right)}^{1/2}{\left(\sinh \xi \right)}^{\mu }}{{\left(\cosh \xi -\cos \varphi \right)}^{\mu +(1/2)}},$$ $\mathrm{\Re }\mu >-\frac{1}{2}$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $\mu$: general order and $\xi >0$: variable

Permalink:

http://dlmf.nist.gov/14.19.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.19(iv), §14.19 and Ch.14

… ►

14.19.7 $$P_{n-\frac{1}{2}}^m\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(n+m+\frac{1}{2}\right)}{\mathrm{\Gamma }\left(n-m+\frac{1}{2}\right)}{\left(\frac{2}{\pi \sinh \xi }\right)}^{1/2}{𝑸}_{m-\frac{1}{2}}^n\left(\coth \xi \right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $m$: nonnegative integer, $n$: nonnegative integer and $\xi >0$: variable

Permalink:

http://dlmf.nist.gov/14.19.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §14.19(v), §14.19 and Ch.14

►

14.19.8 $${𝑸}_{n-\frac{1}{2}}^m\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(m-n+\frac{1}{2}\right)}{\mathrm{\Gamma }\left(m+n+\frac{1}{2}\right)}{\left(\frac{\pi }{2\sinh \xi }\right)}^{1/2}{P}_{m-\frac{1}{2}}^n\left(\coth \xi \right).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $m$: nonnegative integer, $n$: nonnegative integer and $\xi >0$: variable

Permalink:

http://dlmf.nist.gov/14.19.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §14.19(v), §14.19 and Ch.14

37: 28.32 Mathematical Applications

… ►

28.32.3 $$\frac{{\partial }^{2}V}{{\partial \xi }^2}+\frac{{\partial }^{2}V}{{\partial \eta }^2}+\frac{1}{2}{c}^2{k}^2(\cosh \left(2\xi \right)-\cos \left(2\eta \right))V=0.$$

ⓘ

Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\eta$: variable, $\xi$: variable, $k$: parameter and $V(\xi ,\eta )$: solution

Referenced by:

§28.32(i), §28.33(ii)

Permalink:

http://dlmf.nist.gov/28.32.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(i), §28.32 and Ch.28

… ►

28.32.4 $$\frac{{\partial }^{2}K}{{\partial z}^2}-\frac{{\partial }^{2}K}{{\partial \zeta }^2}=2q\left(\cos \left(2z\right)-\cos \left(2\zeta \right)\right)K.$$

ⓘ

Symbols:

$\cos z$: cosine function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $q=h^2$: parameter, $z$: complex variable, $\zeta$: variable and $K(z,\zeta )$: solution

Permalink:

http://dlmf.nist.gov/28.32.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(i), §28.32 and Ch.28

…

38: 6.15 Sums

… ►

6.15.1 $$\sum _{n=1}^{\mathrm{\infty }}\mathrm{Ci}\left(\pi n\right)=\frac{1}{2}(\ln 2-\gamma ),$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

►

6.15.2 $$\sum _{n=1}^{\mathrm{\infty }}\frac{\mathrm{si}\left(\pi n\right)}{n}=\frac{1}{2}\pi (\ln \pi -1),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{si}\left(z\right)$: sine integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

►

6.15.3 $$\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\mathrm{Ci}\left(2\pi n\right)=1-\ln 2-\frac{1}{2}\gamma ,$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

►

6.15.4 $$\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{\mathrm{si}\left(2\pi n\right)}{n}=\pi (\frac{3}{2}\ln 2-1).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{si}\left(z\right)$: sine integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

…

39: 4.43 Cubic Equations

§4.43 Cubic Equations

… ►

4.43.2 $$z^3+pz+q=0$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $a$: real or complex constant, $z$: complex variable, $p$: real constant and $q$: real constant

Referenced by:

(4.43.1), §4.43

Permalink:

http://dlmf.nist.gov/4.43.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

Cases (a), (b), and (c) of of this equation have been corrected. Formerly, with constants $C$ and $D$ given in Eq. (4.43.1), they read

1. $A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=C$, when and $C\le 1$.

2. $A\cosh a$, $A\cosh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $A\cosh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\cosh \left(3a\right)=C$, when and $C>1$.

3. $B\sinh a$, $B\sinh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\sinh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\sinh \left(3a\right)=D$, when $p>0$.

Reported 2014-10-31 by Masataka Urago

See also:

Annotations for §4.43 and Ch.4

…

40: 6.2 Definitions and Interrelations

… ►This is also true of the functions $\mathrm{Ci}\left(z\right)$ and $\mathrm{Chi}\left(z\right)$ defined in §6.2(ii). … ► $\mathrm{Si}\left(z\right)$ is an odd entire function. …$\mathrm{Cin}\left(z\right)$ is an even entire function. … ►

6.2.19 $$\mathrm{Si}\left(z\right)=\frac{1}{2}\pi -\mathrm{f}\left(z\right)\cos z-\mathrm{g}\left(z\right)\sin z,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{Si}\left(z\right)$: sine integral and $z$: complex variable

A&S Ref:

5.2.8

Referenced by:

§6.12(ii)

Permalink:

http://dlmf.nist.gov/6.2.E19

Encodings:

pMML, png, TeX

See also:

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

►

6.2.20 $$\mathrm{Ci}\left(z\right)=\mathrm{f}\left(z\right)\sin z-\mathrm{g}\left(z\right)\cos z.$$

ⓘ

Symbols:

$\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\sin z$: sine function, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

A&S Ref:

5.2.9

Referenced by:

§6.12(ii)

Permalink:

http://dlmf.nist.gov/6.2.E20

Encodings:

pMML, png, TeX

See also:

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

…