trigonometric functions

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41: 19.11 Addition Theorems

… ►

19.11.6_5 $$R_C(\gamma -\delta ,\gamma )=\frac{-1}{\sqrt{\delta }}\arctan \left(\frac{\sqrt{\delta }\sin \theta \sin \varphi \sin \psi }{{\alpha }^2-1-{\alpha }^2\cos \theta \cos \varphi \cos \psi }\right).$$

ⓘ

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $\arctan z$: arctangent function, $\sin z$: sine function, $\varphi$: real or complex argument, ${\alpha }^2$: real or complex parameter, $\theta$: amplitude, $\psi$: angle, $\gamma$ and $\delta$

Referenced by:

§19.11(i), Erratum (V1.0.28) for Equations (15.2.3_5), (19.11.6_5)

Permalink:

http://dlmf.nist.gov/19.11.E6_5

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.28):

This equation was given a number.

See also:

Annotations for §19.11(i), §19.11 and Ch.19

… ►

19.11.8 $$E(\varphi ,k)=E\left(k\right)-E(\theta ,k)+k^2\sin \theta \sin \varphi ,$$

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Symbols:

$E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $\theta$: amplitude

Permalink:

http://dlmf.nist.gov/19.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(ii), §19.11 and Ch.19

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19.11.9 $$\tan \theta =1/(k^{\prime }\tan \varphi ).$$

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Symbols:

$\tan z$: tangent function, $\varphi$: real or complex argument, $k^{\prime }$: complementary modulus and $\theta$: amplitude

Permalink:

http://dlmf.nist.gov/19.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(ii), §19.11 and Ch.19

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19.11.13 $$E(\psi ,k)=2E(\theta ,k)-k^2{\sin }^2\theta \sin \psi ,$$

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Symbols:

$E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $k$: real or complex modulus, $\theta$: amplitude and $\psi$: angle

Permalink:

http://dlmf.nist.gov/19.11.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(iii), §19.11 and Ch.19

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19.11.14 $$\sin \psi =(\sin 2\theta )\mathrm{\Delta }(\theta )/(1-k^2{\sin }^4\theta ),$$

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Defines:

$\psi$: angle (locally)

Symbols:

$\sin z$: sine function, $k$: real or complex modulus, $\theta$: amplitude and $\mathrm{\Delta }(\theta )$

Permalink:

http://dlmf.nist.gov/19.11.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(iii), §19.11 and Ch.19

…

42: 23.12 Asymptotic Approximations

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23.12.1 $$\mathrm{\wp }\left(z\right)=\frac{{\pi }^2}{4{\omega }_1^2}\left(-\frac{1}{3}+{\csc }^2\left(\frac{\pi z}{2{\omega }_1}\right)+8\left(1-\cos \left(\frac{\pi z}{{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $𝕃$: lattice, $q$: nome, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§23.12

Permalink:

http://dlmf.nist.gov/23.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.12 and Ch.23

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23.12.2 $$\zeta \left(z\right)=\frac{{\pi }^2}{4{\omega }_1^2}\left(\frac{z}{3}+\frac{2{\omega }_1}{\pi }\cot \left(\frac{\pi z}{2{\omega }_1}\right)-8\left(z-\frac{{\omega }_1}{\pi }\sin \left(\frac{\pi z}{{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\sin z$: sine function, $𝕃$: lattice, $q$: nome, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§23.12, Erratum (V1.0.23) for Equation (23.12.2)

Permalink:

http://dlmf.nist.gov/23.12.E2

Encodings:

pMML, png, TeX

Correction (effective with 1.0.23):

The factor of 2, previously omitted from the denominator of the argument of the $\cot$ function has been inserted.

Suggested 2019-05-27 by Blagoje Oblak

See also:

Annotations for §23.12 and Ch.23

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23.12.3 $$\sigma \left(z\right)=\frac{2{\omega }_1}{\pi }\exp \left(\frac{{\pi }^2{z}^2}{24{\omega }_1^2}\right)\sin \left(\frac{\pi z}{2{\omega }_1}\right)\left(1-\left(\frac{{\pi }^2{z}^2}{{\omega }_1^2}-4{\sin }^2\left(\frac{\pi z}{2{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\sin z$: sine function, $𝕃$: lattice, $q$: nome, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Permalink:

http://dlmf.nist.gov/23.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.12 and Ch.23

…

43: 10.15 Derivatives with Respect to Order

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10.15.2 $$\frac{\partial Y_{\nu }\left(z\right)}{\partial \nu }=\cot \left(\nu \pi \right)\left(\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }-\pi Y_{\nu }\left(z\right)\right)-\csc \left(\nu \pi \right)\frac{\partial J_{-\nu }\left(z\right)}{\partial \nu }-\pi J_{\nu }\left(z\right).$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.65

Referenced by:

(10.15.1), §10.15

Permalink:

http://dlmf.nist.gov/10.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

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10.15.6 $${\frac{\partial J_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\frac{1}{2}}=\sqrt{\frac{2}{\pi x}}\left(\mathrm{Ci}\left(2x\right)\sin x-\mathrm{Si}\left(2x\right)\cos x\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.1.41 (corrected)

Referenced by:

§10.15, §10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

►

10.15.7 $${\frac{\partial J_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =-\frac{1}{2}}=\sqrt{\frac{2}{\pi x}}\left(\mathrm{Ci}\left(2x\right)\cos x+\mathrm{Si}\left(2x\right)\sin x\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.1.42 (corrected)

Referenced by:

§10.15, §10.38

Permalink:

http://dlmf.nist.gov/10.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

►

10.15.8 $${\frac{\partial Y_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\frac{1}{2}}=\sqrt{\frac{2}{\pi x}}\left(\mathrm{Ci}\left(2x\right)\cos x+\left(\mathrm{Si}\left(2x\right)-\pi \right)\sin x\right),$$

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.1.43 (corrected)

Referenced by:

§10.15

Permalink:

http://dlmf.nist.gov/10.15.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

►

10.15.9 $${\frac{\partial Y_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =-\frac{1}{2}}=-\sqrt{\frac{2}{\pi x}}\left(\mathrm{Ci}\left(2x\right)\sin x-\left(\mathrm{Si}\left(2x\right)-\pi \right)\cos x\right).$$

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.1.44 (corrected)

Referenced by:

§10.15, Ch.10

Permalink:

http://dlmf.nist.gov/10.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

…

44: 22.5 Special Values

… ►In these cases the elliptic functions degenerate into elementary trigonometric and hyperbolic functions, respectively. …

45: 14.25 Integral Representations

… ►

14.25.1 $$P_{\nu }^{-\mu }\left(z\right)=\frac{{\left(z^2-1\right)}^{\mu /2}}{2^{\nu }\mathrm{\Gamma }\left(\mu -\nu \right)\mathrm{\Gamma }\left(\nu +1\right)}{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\nu +1}}{{(z+\cosh t)}^{\nu +\mu +1}}dt,$$ $\mathrm{\Re }\mu >\mathrm{\Re }\nu >-1$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.25

Permalink:

http://dlmf.nist.gov/14.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.25 and Ch.14

►

14.25.2 $${𝑸}_{\nu }^{\mu }\left(z\right)=\frac{{\pi }^{1/2}{\left(z^2-1\right)}^{\mu /2}}{2^{\mu }\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{\left(z+{(z^2-1)}^{1/2}\cosh t\right)}^{\nu +\mu +1}}dt,$$ $\mathrm{\Re }\left(\nu +1\right)>\mathrm{\Re }\mu >-\frac{1}{2}$,

ⓘ

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, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.25

Permalink:

http://dlmf.nist.gov/14.25.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.25 and Ch.14

…

46: 14.5 Special Values

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14.5.11 $${𝖯}_{\nu }^{1/2}\left(\cos \theta \right)={\left(\frac{2}{\pi \sin \theta }\right)}^{1/2}\cos \left(\left(\nu +\frac{1}{2}\right)\theta \right),$$

ⓘ

Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable

A&S Ref:

8.6.12

Permalink:

http://dlmf.nist.gov/14.5.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(iii), §14.5 and Ch.14

… ►

14.5.13 $${𝖰}_{\nu }^{1/2}\left(\cos \theta \right)=-{\left(\frac{\pi }{2\sin \theta }\right)}^{1/2}\sin \left(\left(\nu +\frac{1}{2}\right)\theta \right),$$

ⓘ

Symbols:

: Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable

A&S Ref:

8.6.13

Permalink:

http://dlmf.nist.gov/14.5.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(iii), §14.5 and Ch.14

►

14.5.14 $${𝖰}_{\nu }^{-1/2}\left(\cos \theta \right)={\left(\frac{\pi }{2\sin \theta }\right)}^{1/2}\frac{\cos \left(\left(\nu +\frac{1}{2}\right)\theta \right)}{\nu +\frac{1}{2}}.$$

ⓘ

Symbols:

: Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable

A&S Ref:

8.6.15 (corrected)

Referenced by:

Erratum (V1.0.16) for Equation (14.5.14)

Permalink:

http://dlmf.nist.gov/14.5.E14

Encodings:

pMML, png, TeX

Errata (effective with 1.0.16):

Originally this equation was incorrect because of a minus sign in front of the right-hand side.

Reported 2017-04-10 by André Greiner-Petter

See also:

Annotations for §14.5(iii), §14.5 and Ch.14

… ►

14.5.17 $${𝑸}_{\nu }^{\pm 1/2}\left(\cosh \xi \right)={\left(\frac{\pi }{2\sinh \xi }\right)}^{1/2}\frac{\exp \left(-\left(\nu +\frac{1}{2}\right)\xi \right)}{\mathrm{\Gamma }\left(\nu +\frac{3}{2}\right)}.$$

ⓘ

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, $\exp z$: exponential function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\nu$: general degree and $\xi >0$: variable

A&S Ref:

8.6.10 (corrected) 8.6.11 (corrected)

Permalink:

http://dlmf.nist.gov/14.5.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(iii), §14.5 and Ch.14

… ►

14.5.18 $${𝖯}_{\nu }^{-\nu }\left(\cos \theta \right)=\frac{{(\sin \theta )}^{\nu }}{2^{\nu }\mathrm{\Gamma }\left(\nu +1\right)},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable

A&S Ref:

8.6.17

Permalink:

http://dlmf.nist.gov/14.5.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(iv), §14.5 and Ch.14

…

47: 36.13 Kelvin’s Ship-Wave Pattern

… ►

36.13.1 $$z(\varphi ,\rho )={\int }_{-\pi /2}^{\pi /2}\cos \left(\rho \frac{\cos \left(\theta +\varphi \right)}{{\cos }^2\theta }\right)d\theta ,$$

ⓘ

Defines:

$z(\varphi ,\rho )$: wave height approximant (locally)

Symbols:

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

Permalink:

http://dlmf.nist.gov/36.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §36.13 and Ch.36

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36.13.3 $$f(\theta ,\varphi )=-\frac{\cos \left(\theta +\varphi \right)}{{\cos }^2\theta }.$$

ⓘ

Symbols:

$\cos z$: cosine function

Permalink:

http://dlmf.nist.gov/36.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.13 and Ch.36

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36.13.5 $$|\varphi |={\varphi }_c=\arcsin \left(\frac{1}{3}\right)={19}^{\circ }.47122.$$

ⓘ

Symbols:

$\arcsin z$: arcsine function

Permalink:

http://dlmf.nist.gov/36.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.13 and Ch.36

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36.13.8 $$z(\rho ,\varphi )=2\pi \left({\rho }^{-1/3}u(\varphi )\cos \left(\rho \tilde{f}(\varphi )\right)\mathrm{Ai}\left(-{\rho }^{2/3}\mathrm{\Delta }(\varphi )\right)(1+O\left(1/\rho \right))+{\rho }^{-2/3}v(\varphi )\sin \left(\rho \tilde{f}(\varphi )\right){\mathrm{Ai}}^{\prime }\left(-{\rho }^{2/3}\mathrm{\Delta }(\varphi )\right)(1+O\left(1/\rho \right))\right),$$ $\rho \to \mathrm{\infty }$.

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z(\varphi ,\rho )$: wave height approximant, $u(\varphi )$: function and $v(\varphi )$: function

Referenced by:

Figure 36.13.1, Figure 36.13.1

Permalink:

http://dlmf.nist.gov/36.13.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.13 and Ch.36

… ►

►►

Figure 36.13.1: Kelvin’s ship wave pattern, computed from the uniform asymptotic approximation (36.13.8), as a function of $x=\rho \cos \varphi$, $y=\rho \sin \varphi$. Magnify

…

48: 6.7 Integral Representations

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6.7.10 $$\mathrm{Ein}\left(z\right)-\mathrm{Cin}\left(z\right)={\int }_0^{\pi /2}{\mathrm{e}}^{-z\cos t}\sin \left(z\sin t\right)dt,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\cos z$: cosine function, $\mathrm{Cin}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\sin z$: sine function and $z$: complex variable

A&S Ref:

5.2.11 (modified form of)

Permalink:

http://dlmf.nist.gov/6.7.E10

Encodings:

pMML, png, TeX

See also:

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

►

6.7.11 $${\int }_0^1\frac{(1-{\mathrm{e}}^{-at})\cos \left(bt\right)}{t}dt=\mathrm{\Re }\mathrm{Ein}\left(a+\mathrm{i}b\right)-\mathrm{Cin}\left(b\right),$$ $a,b\in ℝ$.

ⓘ

Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\cos z$: cosine function, $\mathrm{Cin}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part and $ℝ$: real line

A&S Ref:

5.2.33 (modified form of)

Permalink:

http://dlmf.nist.gov/6.7.E11

Encodings:

pMML, png, TeX

See also:

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

… ►

6.7.13 $$\mathrm{f}\left(z\right)={\int }_0^{\mathrm{\infty }}\frac{\sin t}{t+z}dt={\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}}{t^2+1}dt,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\sin z$: sine function, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.12(ii), §6.7(iii)

Permalink:

http://dlmf.nist.gov/6.7.E13

Encodings:

pMML, png, TeX

See also:

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

… ►

6.7.15 $$\mathrm{f}\left(z\right)=2{\int }_0^{\mathrm{\infty }}{K}_0\left(2\sqrt{zt}\right)\cos tdt,$$

ⓘ

Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.7(iii)

Permalink:

http://dlmf.nist.gov/6.7.E15

Encodings:

pMML, png, TeX

See also:

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

►

6.7.16 $$\mathrm{g}\left(z\right)=2{\int }_0^{\mathrm{\infty }}{K}_0\left(2\sqrt{zt}\right)\sin tdt.$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.7(iii)

Permalink:

http://dlmf.nist.gov/6.7.E16

Encodings:

pMML, png, TeX

See also:

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

…

49: 9.8 Modulus and Phase

… ►

9.8.1 $$\mathrm{Ai}\left(x\right)=M\left(x\right)\sin \theta \left(x\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $M\left(z\right)$: Airy modulus function, $\theta \left(z\right)$: Airy phase function, $\sin z$: sine function and $x$: real variable

Source:

Olver (1997b, (2.01), p. 394)

Referenced by:

(9.8.13), (9.9.3)

Permalink:

http://dlmf.nist.gov/9.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(i), §9.8 and Ch.9

►

9.8.2 $$\mathrm{Bi}\left(x\right)=M\left(x\right)\cos \theta \left(x\right),$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\cos z$: cosine function, $M\left(z\right)$: Airy modulus function, $\theta \left(z\right)$: Airy phase function and $x$: real variable

Source:

Olver (1997b, (2.01), p. 394)

Referenced by:

(9.8.13), (9.9.3)

Permalink:

http://dlmf.nist.gov/9.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(i), §9.8 and Ch.9

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9.8.4 $$\theta \left(x\right)=\arctan \left(\mathrm{Ai}\left(x\right)/\mathrm{Bi}\left(x\right)\right).$$

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Defines:

$\theta \left(z\right)$: Airy phase function

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\arctan z$: arctangent function, $x$: real variable and $z$: complex variable

Source:

Olver (1997b, (2.02), p. 395)

Referenced by:

(9.11.19), (9.8.11), (9.8.15), (9.8.17), (9.8.19)

Permalink:

http://dlmf.nist.gov/9.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(i), §9.8 and Ch.9

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9.8.5 $${\mathrm{Ai}}^{\prime }\left(x\right)=N\left(x\right)\sin \varphi \left(x\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $N\left(z\right)$: Airy modulus function, $\varphi \left(z\right)$: Airy phase function, $\sin z$: sine function and $x$: real variable

Source:

Olver (1997b, (2.08), p. 396); with different notation

Referenced by:

(9.8.13), (9.9.4)

Permalink:

http://dlmf.nist.gov/9.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(i), §9.8 and Ch.9

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9.8.6 $${\mathrm{Bi}}^{\prime }\left(x\right)=N\left(x\right)\cos \varphi \left(x\right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\cos z$: cosine function, $N\left(z\right)$: Airy modulus function, $\varphi \left(z\right)$: Airy phase function and $x$: real variable

Source:

Olver (1997b, (2.08), p. 396); with different notation

Referenced by:

(9.8.13), (9.9.4)

Permalink:

http://dlmf.nist.gov/9.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(i), §9.8 and Ch.9

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50: 10.64 Integral Representations

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10.64.1 $${\mathrm{ber}}_n\left(x\sqrt{2}\right)=\frac{{(-1)}^n}{\pi }{\int }_0^{\pi }\cos \left(x\sin t-nt\right)\cosh \left(x\sin t\right)dt,$$

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Symbols:

${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $n$: integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/10.64.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.64, §10.64 and Ch.10

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10.64.2 $${\mathrm{bei}}_n\left(x\sqrt{2}\right)=\frac{{(-1)}^n}{\pi }{\int }_0^{\pi }\sin \left(x\sin t-nt\right)\sinh \left(x\sin t\right)dt.$$

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Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $n$: integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/10.64.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.64, §10.64 and Ch.10

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