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1: 31.2 Differential Equations

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§31.2(iii) Trigonometric Form

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2: 29.2 Differential Equations

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29.2.4 $$(1-k^2{\cos }^2\varphi )\frac{d^{2}w}{{d\varphi }^2}+k^2\cos \varphi \sin \varphi \frac{dw}{d\varphi }+(h-\nu (\nu +1)k^2{\cos }^2\varphi )w=0,$$

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Symbols:

$\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sin z$: sine function, $h$: real parameter, $k$: real parameter, $\nu$: real parameter and $\varphi$: change of variable

Permalink:

http://dlmf.nist.gov/29.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.2(ii), §29.2 and Ch.29

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3: 4.23 Inverse Trigonometric Functions

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§4.23(iv) Logarithmic Forms

… ►Care needs to be taken on the cuts, for example, if then $1/(x+\mathrm{i}0)=(1/x)-\mathrm{i}0$. …

4: 18.11 Relations to Other Functions

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Hermite

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5: 6.5 Further Interrelations

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6.5.3 $$\frac{1}{2}(\mathrm{Ei}\left(x\right)+E_1\left(x\right))=\mathrm{Shi}\left(x\right)=-\mathrm{i}\mathrm{Si}\left(\mathrm{i}x\right),$$

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Symbols:

$E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $\mathrm{Shi}\left(z\right)$: hyperbolic sine integral, $\mathrm{i}$: imaginary unit, $\mathrm{Si}\left(z\right)$: sine integral and $x$: real variable

A&S Ref:

5.2.22 (in modified form)

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

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6.5.4 $$\frac{1}{2}(\mathrm{Ei}\left(x\right)-E_1\left(x\right))=\mathrm{Chi}\left(x\right)=\mathrm{Ci}\left(\mathrm{i}x\right)-\frac{1}{2}\pi \mathrm{i}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $\mathrm{Chi}\left(z\right)$: hyperbolic cosine integral, $\mathrm{i}$: imaginary unit and $x$: real variable

A&S Ref:

5.2.24 (in modified form)

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

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6: 6.7 Integral Representations

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6.7.7 $${\int }_0^1\frac{{\mathrm{e}}^{-at}\sin \left(bt\right)}{t}dt=\mathrm{\Im }\mathrm{Ein}\left(a+\mathrm{i}b\right),$$ $a,b\in ℝ$,

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Symbols:

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

A&S Ref:

5.1.36 (modified form of)

Referenced by:

§6.7(i)

Permalink:

http://dlmf.nist.gov/6.7.E7

Encodings:

pMML, png, TeX

See also:

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

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6.7.8 $${\int }_0^1\frac{{\mathrm{e}}^{-at}(1-\cos \left(bt\right))}{t}dt=\mathrm{\Re }\mathrm{Ein}\left(a+\mathrm{i}b\right)-\mathrm{Ein}\left(a\right),$$ $a,b\in ℝ$.

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Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\cos z$: cosine function, $dx$: 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:

5.1.38 (modified form of)

Referenced by:

§6.7(i)

Permalink:

http://dlmf.nist.gov/6.7.E8

Encodings:

pMML, png, TeX

See also:

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

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

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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

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

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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

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7: 6.2 Definitions and Interrelations

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6.2.10 $$\mathrm{si}\left(z\right)=-{\int }_z^{\mathrm{\infty }}\frac{\sin t}{t}dt=\mathrm{Si}\left(z\right)-\frac{1}{2}\pi .$$

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Defines:

$\mathrm{si}\left(z\right)$: sine integral

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral and $z$: complex variable

A&S Ref:

5.2.5 (in modified form) 5.2.26 (in modified form)

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.2.E10

Encodings:

pMML, png, TeX

See also:

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

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6.2.13 $$\mathrm{Ci}\left(z\right)=-\mathrm{Cin}\left(z\right)+\ln z+\gamma .$$

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ci}\left(z\right)$: cosine integral, $\mathrm{Cin}\left(z\right)$: cosine integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

5.2.2 (in modified form)

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.2.E13

Encodings:

pMML, png, TeX

See also:

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

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8: 28.2 Definitions and Basic Properties

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28.2.1 $$w^{\prime \prime }+(a-2q\cos \left(2z\right))w=0.$$

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Symbols:

$\cos z$: cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

A&S Ref:

20.3.1 20.1.1 (in slightly different form)

Referenced by:

§1.18(vii), Figure 28.17.1, Figure 28.17.1, §28.17, §28.17, §28.2(ii), §28.2(ii), §28.2(iii), §28.2(iv), §28.20(i), §28.29(i), §28.32(i), §28.32(i), 1st item, 2nd item, §28.33(iii), item (a), item (c), item (c), §28.5(i), §28.8(iv), §28.8(iv), §28.8(iv), §30.2(iii)

Permalink:

http://dlmf.nist.gov/28.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(i), §28.2 and Ch.28

►With $\zeta ={\sin }^{2}z$ we obtain the algebraic form of Mathieu’s equation …With $\zeta =\cos z$ we obtain another algebraic form: … ►

28.2.16 $$\cos \left(\pi \nu \right)=w_{\text{I}}(\pi ;a,q)=w_{\text{I}}(\pi ;a,-q).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $q=h^2$: parameter, $\nu$: complex parameter, $a$: parameter and $w_{\text{I}}(z;a,q)$: fundamental solution

A&S Ref:

20.3.10 (in slightly different form)

Referenced by:

§28.12(i), §28.12(i), §28.12(ii), §28.2(iii), §28.2(iv), §28.2(v), item (c), §28.34(i)

Permalink:

http://dlmf.nist.gov/28.2.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(iii), §28.2 and Ch.28

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9: 22.5 Special Values

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§22.5(ii) Limiting Values of $k$

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

Table 22.5.3: Limiting forms of Jacobian elliptic functions as $k\to 0$.

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$\mathrm{sn}(z,k)$ $\to$	$\sin z$	$\mathrm{cd}(z,k)$ $\to$	$\cos z$	$\mathrm{dc}(z,k)$ $\to$	$\sec z$	$\mathrm{ns}(z,k)$ $\to$	$\csc z$
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Table 22.5.4: Limiting forms of Jacobian elliptic functions as $k\to 1$.

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$\mathrm{sn}(z,k)$ $\to$	$\tanh z$	$\mathrm{cd}(z,k)$ $\to$	$1$	$\mathrm{dc}(z,k)$ $\to$	$1$	$\mathrm{ns}(z,k)$ $\to$	$\coth z$
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10: 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. … ►The trigonometric functions may be computed from the definitions (4.14.1)–(4.14.7), and their inverses from the logarithmic forms in §4.23(iv), followed by (4.23.7)–(4.23.9). …