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31: 4.30 Elementary Properties

… ►

Table 4.30.1: Hyperbolic functions: interrelations. All square roots have their principal values when the functions are real, nonnegative, and finite.

	$\sinh\theta=a$	$\cosh\theta=a$	$\tanh\theta=a$	$\operatorname{csch}\theta=a$	$\operatorname{sech}\theta=a$	$\coth\theta=a$
…

►

32: 19.3 Graphics

… ►

See accompanying text — Figure 19.3.2: $R_{C}\left(x,1\right)$ and the Cauchy principal value of $R_{C}\left(x,-1\right)$ for $0\leq x\leq 5$ . … Magnify

… ►

See accompanying text — Figure 19.3.5: $\Pi\left(\alpha^{2},k\right)$ as a function of $k^{2}$ and $\alpha^{2}$ for $-2\leq k^{2}<1$ , $-2\leq\alpha^{2}\leq 2$ . Cauchy principal values are shown when $\alpha^{2}>1$ . … Magnify 3D Help

►

See accompanying text — Figure 19.3.6: $\Pi\left(\phi,2,k\right)$ as a function of $k^{2}$ and ${\sin}^{2}\phi$ for $-1\leq k^{2}\leq 3$ , $0\leq{\sin}^{2}\phi<1$ . Cauchy principal values are shown when ${\sin}^{2}\phi>\frac{1}{2}$ . …If ${\sin}^{2}\phi=1$ ( $>k^{2}$ ), then the function reduces to $\Pi\left(2,k\right)$ with Cauchy principal value $K\left(k\right)-\Pi\left(\frac{1}{2}k^{2},k\right)$ , which tends to $-\infty$ as $k^{2}\to 1-$ . …If ${\sin}^{2}\phi=1/k^{2}$ ( $<1$ ), then by (19.7.4) it reduces to $\Pi\left(2/k^{2},1/k\right)/k$ , $k^{2}\neq 2$ , with Cauchy principal value $(K\left(1/k\right)-\Pi\left(\frac{1}{2},1/k\right))/k$ , $1<k^{2}<2$ , by (19.6.5). … Magnify 3D Help

…

33: 10.2 Definitions

… ►

§10.2(ii) Standard Solutions

… ►The principal branch of

J_{\nu}\left(z\right)

corresponds to the principal value of

(\tfrac{1}{2}z)^{\nu}

(§4.2(iv)) and is analytic in the

z

-plane cut along the interval

(-\infty,0]

. … ►The principal branches correspond to principal values of the square roots in (10.2.5) and (10.2.6), again with a cut in the

z

-plane along the interval

(-\infty,0]

. … ► … ►Except where indicated otherwise, it is assumed throughout the DLMF that the symbols

J_{\nu}\left(z\right)

,

Y_{\nu}\left(z\right)

,

{H^{(1)}_{\nu}}\left(z\right)

, and

{H^{(2)}_{\nu}}\left(z\right)

denote the principal values of these functions. …

34: 19.2 Definitions

… ►The principal values of

K\left(k\right)

and

E\left(k\right)

are even functions. … ►If

-\infty <mrow> <mrow> <mo>−</mo> <mi mathvariant=

, then the integral in (19.2.11) is a Cauchy principal value. … ►where the Cauchy principal value is taken if

y<0

. … ►In (19.2.18)–(19.2.22) the inverse trigonometric and hyperbolic functions assume their principal values (§§4.23(ii) and 4.37(ii)). …The Cauchy principal value is hyperbolic: …

35: 15.2 Definitions and Analytical Properties

… ►The branch obtained by introducing a cut from

1

to

+\infty

on the real

z

-axis, that is, the branch in the sector

|\operatorname{ph}\left(1-z\right)|\leq\pi

, is the principal branch (or principal value) of

F\left(a,b;c;z\right)

. … ►again with analytic continuation for other values of

z

, and with the principal branch defined in a similar way. …

36: 14.28 Sums

… ►where the branches of the square roots have their principal values when

z_{1},z_{2}\in(1,\infty)

and are continuous when

z_{1},z_{2}\in\mathbb{C}\setminus(0,1]

. …

37: 23.18 Modular Transformations

… ►where the square root has its principal value and …

38: 23.8 Trigonometric Series and Products

… ►where in (23.8.4) the terms in

n

and

-n

are to be bracketed together (the Eisenstein convention or principal value: see Weil (1999, p. 6) or Walker (1996, p. 3)). …

39: 28.28 Integrals, Integral Representations, and Integral Equations

… ►

28.28.34

\dfrac{1}{\pi^{2}}\pvint_{0}^{2\pi}\dfrac{\operatorname{me}_{\nu}'\left(t,h^{2% }\right)\operatorname{me}_{-\nu-2m-1}\left(t,h^{2}\right)}{\sin t}\,\mathrm{d}% t=(-1)^{m+1}\mathrm{i}h\alpha^{(0)}_{\nu,m}\operatorname{D}_{1}\left(\nu,\nu+2% m+1,0\right),

ⓘ

Symbols:: $\operatorname{D}_{\NVar{j}}\left(\NVar{\nu},\NVar{\mu},\NVar{z}\right)$ : cross-products of modified Mathieu functions and their derivatives, $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\pvint_{\NVar{a}}^{\NVar{b}}$ : Cauchy principal value, $\sin\NVar{z}$ : sine function, $m$ : integer, $h$ : parameter, $\nu$ : complex parameter and $\alpha^{(s)}_{\nu,m}$
Permalink:: http://dlmf.nist.gov/28.28.E34
Encodings:: pMML, png, TeX
See also:: Annotations for §28.28(iii), §28.28 and Ch.28

►where the integral is a Cauchy principal value (§1.4(v)). …

40: 19.17 Graphics

… ►

See accompanying text — Figure 19.17.6: Cauchy principal value of $R_{J}\left(x,y,1,-0.5\right)$ for $0\leq x\leq 1$ , $y=0,\,0.1,\,0.5,\,1$ . … Magnify

►

See accompanying text — Figure 19.17.7: Cauchy principal value of $R_{J}\left(0.5,y,1,p\right)$ for $y=0,\,0.01,\,0.05,\,0.2,\,1$ , $-1\leq p<0$ . … Magnify

►

See accompanying text — Figure 19.17.8: $R_{J}\left(0,y,1,p\right)$ , $0\leq y\leq 1$ , $-1\leq p\leq 2$ . Cauchy principal values are shown when $p<0$ . … Magnify 3D Help

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