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11: 4.18 Inequalities

§4.18 Inequalities

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4.18.3 $$\cos x\le \frac{\sin x}{x}\le 1,$$ $0\le x\le \pi$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $x$: real variable

A&S Ref:

4.3.81

Referenced by:

§4.18

Permalink:

http://dlmf.nist.gov/4.18.E3

Encodings:

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See also:

Annotations for §4.18, §4.18 and Ch.4

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4.18.5 $$|\sinh y|\le |\sin z|\le \cosh y,$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\sin z$: sine function, $y$: real variable and $z$: complex variable

A&S Ref:

4.3.83

Referenced by:

§4.18

Permalink:

http://dlmf.nist.gov/4.18.E5

Encodings:

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See also:

Annotations for §4.18, §4.18 and Ch.4

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4.18.6 $$|\sinh y|\le |\cos z|\le \cosh y,$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $y$: real variable and $z$: complex variable

A&S Ref:

4.3.84

Permalink:

http://dlmf.nist.gov/4.18.E6

Encodings:

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See also:

Annotations for §4.18, §4.18 and Ch.4

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4.18.7 $$|\csc z|\le \mathrm{csch}|y|,$$

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Symbols:

$\csc z$: cosecant function, $\mathrm{csch}z$: hyperbolic cosecant function, $y$: real variable and $z$: complex variable

A&S Ref:

4.3.85

Permalink:

http://dlmf.nist.gov/4.18.E7

Encodings:

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See also:

Annotations for §4.18, §4.18 and Ch.4

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12: 22.11 Fourier and Hyperbolic Series

§22.11 Fourier and Hyperbolic Series

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13: 7.12 Asymptotic Expansions

… ►The asymptotic expansions of $C\left(z\right)$ and $S\left(z\right)$ are given by (7.5.3), (7.5.4), and … ►They are bounded by $|\csc \left(4\mathrm{ph}z\right)|$ times the first neglected terms when . …

14: 3.10 Continued Fractions

… ►For example, by converting the Maclaurin expansion of $\arctan z$ (4.24.3), we obtain a continued fraction with the same region of convergence ($\left|z\right|\le 1$, $z\ne \pm \mathrm{i}$), whereas the continued fraction (4.25.4) converges for all $z\in ℂ$ except on the branch cuts from $\mathrm{i}$ to $\mathrm{i}\mathrm{\infty }$ and $-\mathrm{i}$ to $-\mathrm{i}\mathrm{\infty }$. …

15: 8.21 Generalized Sine and Cosine Integrals

… ►For the corresponding expansions for $\mathrm{si}(a,z)$ and $\mathrm{ci}(a,z)$ apply (8.21.20) and (8.21.21). …

16: 22.5 Special Values

… ►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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$\mathrm{sn}(z,k)$ $\to$	$\sin z$	$\mathrm{cd}(z,k)$ $\to$	$\cos z$	$\mathrm{dc}(z,k)$ $\to$	$\sec z$	$\mathrm{ns}(z,k)$ $\to$	$\csc z$
$\mathrm{cn}(z,k)$ $\to$	$\cos z$	$\mathrm{sd}(z,k)$ $\to$	$\sin z$	$\mathrm{nc}(z,k)$ $\to$	$\sec z$	$\mathrm{ds}(z,k)$ $\to$	$\csc z$
…

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Table 22.5.4: Limiting forms of Jacobian elliptic functions as $k\to 1$.

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$\mathrm{sn}(z,k)$ $\to$	$\tanh z$	$\mathrm{cd}(z,k)$ $\to$	$1$	$\mathrm{dc}(z,k)$ $\to$	$1$	$\mathrm{ns}(z,k)$ $\to$	$\coth z$
$\mathrm{cn}(z,k)$ $\to$	$\mathrm{sech}z$	$\mathrm{sd}(z,k)$ $\to$	$\sinh z$	$\mathrm{nc}(z,k)$ $\to$	$\cosh z$	$\mathrm{ds}(z,k)$ $\to$	$\mathrm{csch}z$
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►Expansions for $K,K^{\prime }$ as $k\to 0$ or $1$ are given in §§19.5, 19.12. …

17: 6.20 Approximations

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Luke and Wimp (1963) covers $\mathrm{Ei}\left(x\right)$ for $x\le -4$ (20D), and $\mathrm{Si}\left(x\right)$ and $\mathrm{Ci}\left(x\right)$ for $x\ge 4$ (20D).

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Luke (1969b, pp. 41–42) gives Chebyshev expansions of $\mathrm{Ein}\left(ax\right)$, $\mathrm{Si}\left(ax\right)$, and $\mathrm{Cin}\left(ax\right)$ for $-1\le x\le 1$, $a\in ℂ$. The coefficients are given in terms of series of Bessel functions.

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Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric $U$-function (§13.2(i)) from which Chebyshev expansions near infinity for $E_1\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the $U$ functions. If the scheme can be used in backward direction.

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18: 10.35 Generating Function and Associated Series

… ►Jacobi–Anger expansions: for $z,\theta \in ℂ$, ►

10.35.2 $${\mathrm{e}}^{z\cos \theta }=I_0\left(z\right)+2\sum _{k=1}^{\mathrm{\infty }}{I}_k\left(z\right)\cos \left(k\theta \right),$$

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Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.6.34

Referenced by:

§10.35

Permalink:

http://dlmf.nist.gov/10.35.E2

Encodings:

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See also:

Annotations for §10.35 and Ch.10

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10.35.3 $${\mathrm{e}}^{z\sin \theta }=I_0\left(z\right)+2\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{I}_{2k+1}\left(z\right)\sin \left((2k+1)\theta \right)+2\sum _{k=1}^{\mathrm{\infty }}{(-1)}^k{I}_{2k}\left(z\right)\cos \left(2k\theta \right).$$

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Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\sin z$: sine function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.6.35

Referenced by:

§10.35

Permalink:

http://dlmf.nist.gov/10.35.E3

Encodings:

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See also:

Annotations for §10.35 and Ch.10

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$$\cosh z=I_0\left(z\right)+2I_2\left(z\right)+2I_4\left(z\right)+2I_6\left(z\right)+\mathrm{\dots },$$

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$$\sinh z=2I_1\left(z\right)+2I_3\left(z\right)+2I_5\left(z\right)+\mathrm{\dots }.$$

19: 6.13 Zeros

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6.13.2 $$c_k,s_k\sim \alpha +\frac{1}{\alpha }-\frac{16}{3}\frac{1}{{\alpha }^3}+\frac{1673}{15}\frac{1}{{\alpha }^5}-\frac{\mathrm{5\hspace{0.33em}07746}}{105}\frac{1}{{\alpha }^7}+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $c_k$: zeros of $\mathrm{Ci}\left(x\right)$ and $s_k$: zeros of $\mathrm{Si}\left(x\right)$

Referenced by:

§6.13, §6.18(iii)

Permalink:

http://dlmf.nist.gov/6.13.E2

Encodings:

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See also:

Annotations for §6.13 and Ch.6

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20: 3.11 Approximation Techniques

… ►In fact, (3.11.11) is the Fourier-series expansion of $f(\cos \theta )$; compare (3.11.6) and §1.8(i). …