trigonometric arguments

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1: 4.15 Graphics

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§4.15(i) Real Arguments

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§4.15(iii) Complex Arguments: Surfaces

… ►The corresponding surfaces for $\arccos \left(x+\mathrm{i}y\right)$, $\mathrm{arccot}\left(x+\mathrm{i}y\right)$, $\mathrm{arcsec}\left(x+\mathrm{i}y\right)$ can be visualized from Figures 4.15.9, 4.15.11, 4.15.13 with the aid of equations (4.23.16)–(4.23.18).

2: 4.45 Methods of Computation

… ►The inverses $\mathrm{arcsinh}$, $\mathrm{arccosh}$, and $\mathrm{arctanh}$ can be computed from the logarithmic forms given in §4.37(iv), with real arguments. …

3: 10.22 Integrals

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Trigonometric Arguments

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4: 23.12 Asymptotic Approximations

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

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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:

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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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5: 4.21 Identities

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§4.21(iii) Multiples of the Argument

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6: 4.29 Graphics

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§4.29(i) Real Arguments

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Figure 4.29.6: Principal values of $\mathrm{arccsch}x$ and $\mathrm{arcsech}x$. … Magnify

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§4.29(ii) Complex Arguments

►The conformal mapping $w=\sinh z$ is obtainable from Figure 4.15.7 by rotating both the $w$-plane and the $z$-plane through an angle $\frac{1}{2}\pi$, compare (4.28.8). ►The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in §4.15(iii) for the trigonometric and inverse trigonometric functions. …

7: 19.7 Connection Formulas

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$\sin \theta ={\displaystyle \frac{\sqrt{1+k^2}\sin \varphi }{\sqrt{1+k^2{\sin }^2\varphi }}},$

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Imaginary-Argument Transformation

►With $\sinh \varphi =\tan \psi$, … ►The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ when ${\alpha }^2>{\csc }^2\varphi$ (see (19.6.5) for the complete case). Let $c={\csc }^2\varphi \ne {\alpha }^2$. …

8: 19.11 Addition Theorems

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19.11.2 $$E(\theta ,k)+E(\varphi ,k)=E(\psi ,k)+k^2\sin \theta \sin \varphi \sin \psi .$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/19.11.E2

Encodings:

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See also:

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

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

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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:

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Clarification (effective with 1.0.28):

This equation was given a number.

See also:

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

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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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9: 19.14 Reduction of General Elliptic Integrals

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19.14.5 $${\sin }^2\varphi =\frac{\gamma -\alpha }{U^2+\gamma },$$

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Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Permalink:

http://dlmf.nist.gov/19.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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19.14.7 $${\sin }^2\varphi =\frac{(\gamma -\alpha )x^2}{a_1{a}_2+\gamma x^2}.$$

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Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Permalink:

http://dlmf.nist.gov/19.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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19.14.8 $${\sin }^2\varphi =\frac{\gamma -\alpha }{b_1{b}_2{y}^2+\gamma }.$$

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Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Permalink:

http://dlmf.nist.gov/19.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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19.14.9 $${\sin }^2\varphi =\frac{(\gamma -\alpha )(x^2-y^2)}{\gamma (x^2-y^2)-a_1(a_2+b_2{x}^2)}.$$

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Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Permalink:

http://dlmf.nist.gov/19.14.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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19.14.10 $${\sin }^2\varphi =\frac{(\gamma -\alpha )(y^2-x^2)}{\gamma (y^2-x^2)-a_1(a_2+b_2{y}^2)}.$$

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Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Referenced by:

§19.15, §19.29(iii)

Permalink:

http://dlmf.nist.gov/19.14.E10

Encodings:

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See also:

Annotations for §19.14(i), §19.14 and Ch.19

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10: 4.35 Identities

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4.35.14 $$2\sinh u\sinh v=\cosh \left(u+v\right)-\cosh \left(u-v\right),$$

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Symbols:

$\cosh z$: hyperbolic cosine function and $\sinh z$: hyperbolic sine function

A&S Ref:

4.5.38

Permalink:

http://dlmf.nist.gov/4.35.E14

Encodings:

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See also:

Annotations for §4.35(ii), §4.35 and Ch.4

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4.35.15 $$2\cosh u\cosh v=\cosh \left(u+v\right)+\cosh \left(u-v\right),$$

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Symbols:

$\cosh z$: hyperbolic cosine function

A&S Ref:

4.5.39

Permalink:

http://dlmf.nist.gov/4.35.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §4.35(ii), §4.35 and Ch.4

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§4.35(iii) Multiples of the Argument

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4.35.34 $$\sinh z=\sinh x\cos y+\mathrm{i}\cosh x\sin y,$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable, $y$: real variable and $z$: complex variable

A&S Ref:

4.5.49

Permalink:

http://dlmf.nist.gov/4.35.E34

Encodings:

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See also:

Annotations for §4.35(iv), §4.35 and Ch.4

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4.35.35 $$\cosh z=\cosh x\cos y+\mathrm{i}\sinh x\sin y,$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable, $y$: real variable and $z$: complex variable

A&S Ref:

4.5.50

Permalink:

http://dlmf.nist.gov/4.35.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §4.35(iv), §4.35 and Ch.4

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