hyperbolic cases

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1: 19.2 Definitions

… ►Also, if $k^2$ and ${\alpha }^2$ are real, then $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ is called a circular or hyperbolic case according as ${\alpha }^2({\alpha }^2-k^2)({\alpha }^2-1)$ is negative or positive. The circular and hyperbolic cases alternate in the four intervals of the real line separated by the points ${\alpha }^2=0,k^2,1$. … ►Formulas involving $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using $R_C(x,y)$. …

2: 19.20 Special Cases

… ►where $x,y,z$ may be permuted. ►When the variables are real and distinct, the various cases of $R_J(x,y,z,p)$ are called circular (hyperbolic) cases if $(p-x)(p-y)(p-z)$ is positive (negative), because they typically occur in conjunction with inverse circular (hyperbolic) functions. Cases encountered in dynamical problems are usually circular; hyperbolic cases include Cauchy principal values. … ►

19.20.17 $$(q+z)R_J(0,y,z,-q)=(p-z)R_J(0,y,z,p)-3R_F(0,y,z),$$ $p=z(y+q)/(z+q)$, $w=z/(z+q)$.

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind

Referenced by:

§19.20(iii)

Permalink:

http://dlmf.nist.gov/19.20.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(iii), §19.20 and Ch.19

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3: 19.7 Connection Formulas

… ►

§19.7(iii) Change of Parameter of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$

… ►If $k^2$ and ${\alpha }^2$ are real, then both integrals are circular cases or both are hyperbolic cases (see §19.2(ii)). ►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). …

4: 19.21 Connection Formulas

… ►

19.21.15 $$p{R}_J(0,y,z,p)+q{R}_J(0,y,z,q)=3R_F(0,y,z),$$ $pq=yz$.

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind

Permalink:

http://dlmf.nist.gov/19.21.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §19.21(iii), §19.21 and Ch.19

5: 19.6 Special Cases

… ►Circular and hyperbolic cases, including Cauchy principal values, are unified by using $R_C(x,y)$. …

6: 19.36 Methods of Computation

… ►The step from $n$ to $n+1$ is an ascending Landen transformation if $\theta =1$ (leading ultimately to a hyperbolic case of $R_C$) or a descending Gauss transformation if $\theta =-1$ (leading to a circular case of $R_C$). …

7: 22.5 Special Values

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

8: 13.20 Uniform Asymptotic Approximations for Large $\mu$

… ►(a) In the case ►

13.20.9 $$\zeta \sqrt{{\zeta }^2+{\alpha }^2}+{\alpha }^2\mathrm{arcsinh}\left(\frac{\zeta }{\alpha }\right)=\frac{X}{\mu }-\frac{2\kappa }{\mu }\ln \left(\frac{X+x-2\kappa }{2\sqrt{{\mu }^2-{\kappa }^2}}\right)-2\ln \left(\frac{\mu X+2{\mu }^2-\kappa x}{x\sqrt{{\mu }^2-{\kappa }^2}}\right).$$

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Symbols:

$\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\alpha$ and $X$

Permalink:

http://dlmf.nist.gov/13.20.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iii), §13.20 and Ch.13

►(b) In the case $\mu =\kappa$ … ►(In both cases (a) and (b) the $x$-interval $(0,\mathrm{\infty })$ is mapped one-to-one onto the $\zeta$-interval $(-\mathrm{\infty },\mathrm{\infty })$, with $x=0$ and $\mathrm{\infty }$ corresponding to $\zeta =-\mathrm{\infty }$ and $\mathrm{\infty }$, respectively.) … ►

13.20.13 $$\zeta \sqrt{{\zeta }^2-{\alpha }^2}-{\alpha }^2\mathrm{arccosh}\left(\frac{\zeta }{\alpha }\right)=\frac{X}{\mu }-\frac{2\kappa }{\mu }\ln \left(\frac{X+x-2\kappa }{2\sqrt{{\kappa }^2-{\mu }^2}}\right)-2\ln \left(\frac{\kappa x-\mu X-2{\mu }^2}{x\sqrt{{\kappa }^2-{\mu }^2}}\right),$$ $x\ge 2\kappa +2\sqrt{{\kappa }^2-{\mu }^2}$,

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\alpha$ and $X$

Permalink:

http://dlmf.nist.gov/13.20.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iv), §13.20 and Ch.13

…

9: 28.23 Expansions in Series of Bessel Functions

… ►

28.23.3 $${\mathrm{me}}_{\nu }^{\prime }(0,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h)=\mathrm{i}\tanh z\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n(\nu +2n)c_{2n}^{\nu }(h^2){𝒞}_{\nu +2n}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $c_{2m}(q)$: coefficients and ${𝒞}_{\mu }^{(j)}$: cylinder functions

Permalink:

http://dlmf.nist.gov/28.23.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

►valid for all $z$ when $j=1$, and for $\mathrm{\Re }z>0$ and $|\cosh z|>1$ when $j=2,3,4$. …valid for all $z$ when $j=1$, and for $\mathrm{\Re }z>0$ and $|\sinh z|>1$ when $j=2,3,4$. ►In the case when $\nu$ is an integer …When $j=2,3,4$ the series in the even-numbered equations converge for $\mathrm{\Re }z>0$ and $|\cosh z|>1$, and the series in the odd-numbered equations converge for $\mathrm{\Re }z>0$ and $|\sinh z|>1$. …

10: 4.43 Cubic Equations

… ►

4.43.2 $$z^3+pz+q=0$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $a$: real or complex constant, $z$: complex variable, $p$: real constant and $q$: real constant

Referenced by:

(4.43.1), §4.43

Permalink:

http://dlmf.nist.gov/4.43.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

Cases (a), (b), and (c) of of this equation have been corrected. Formerly, with constants $C$ and $D$ given in Eq. (4.43.1), they read

1. $A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=C$, when and $C\le 1$.

2. $A\cosh a$, $A\cosh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $A\cosh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\cosh \left(3a\right)=C$, when and $C>1$.

3. $B\sinh a$, $B\sinh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\sinh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\sinh \left(3a\right)=D$, when $p>0$.

Reported 2014-10-31 by Masataka Urago

See also:

Annotations for §4.43 and Ch.4

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