circular cases

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1: 19.7 Connection Formulas

… ►

§19.7(iii) Change of Parameter of $\Pi\left(\phi,\alpha^{2},k\right)$

… ►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

\Pi\left(\phi,\alpha^{2},k\right)

when

\alpha^{2}>{\csc}^{2}\phi

(see (19.6.5) for the complete case). …

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}\left(x,y,z,p\right)

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}\left(0,y,z,-q\right)=(p-z)R_{J}\left(0,y,z,p\right)-3R_{F}\left(0,y% ,z\right),

p=z(y+q)/(z+q)

w=z/(z+q)

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : 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

…

3: 19.21 Connection Formulas

… ►The case

z=1

shows that the product of the two lemniscate constants, (19.20.2) and (19.20.22), is

\pi/4

. … ►

19.21.15

pR_{J}\left(0,y,z,p\right)+qR_{J}\left(0,y,z,q\right)=3R_{F}\left(0,y,z\right),

pq=yz

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : 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

4: 19.2 Definitions

… ►Also, if

k^{2}

and

\alpha^{2}

are real, then

\Pi\left(\phi,\alpha^{2},k\right)

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

. ►The cases with

\phi=\pi/2

are the complete integrals: … ►Formulas involving

\Pi\left(\phi,\alpha^{2},k\right)

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}\left(x,y\right)

. …

5: 10.42 Zeros

… ►The distribution of the zeros of

K_{n}\left(nz\right)

in the sector

-\tfrac{3}{2}\pi\leq\operatorname{ph}z\leq\tfrac{1}{2}\pi

in the cases

n=1,5,10

is obtained on rotating Figures 10.21.2, 10.21.4, 10.21.6, respectively, through an angle

-\tfrac{1}{2}\pi

so that in each case the cut lies along the positive imaginary axis. …

6: 10.21 Zeros

… ►The zeros of any cylinder function or its derivative are simple, with the possible exceptions of

z=0

in the case of the functions, and

z=0,\pm\nu

in the case of the derivatives. … ►All of these zeros are simple, provided that

\nu\geq-1

in the case of

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

, and

\nu\geq-\tfrac{1}{2}

in the case of

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

. … ►An error bound is included for the case

\nu\geq\tfrac{3}{2}

. … ►where, in the case of (10.21.48), …and, in the case of (10.21.49), …

7: 10.70 Zeros

… ►

\mbox{zeros of $\operatorname{ber}_{\nu}x$}\sim\sqrt{2}(t-f(t)),

t=(m-\tfrac{1}{2}\nu-\tfrac{3}{8})\pi

►

\mbox{zeros of $\operatorname{bei}_{\nu}x$}\sim\sqrt{2}(t-f(t)),

t=(m-\tfrac{1}{2}\nu+\tfrac{1}{8})\pi

►

\mbox{zeros of $\operatorname{ker}_{\nu}x$}\sim\sqrt{2}(t+f(-t)),

t=(m-\tfrac{1}{2}\nu-\tfrac{5}{8})\pi

►

\mbox{zeros of $\operatorname{kei}_{\nu}x$}\sim\sqrt{2}(t+f(-t)),

t=(m-\tfrac{1}{2}\nu-\tfrac{1}{8})\pi

►In the case

\nu=0

, numerical tabulations (Abramowitz and Stegun (1964, Table 9.12)) indicate that each of (10.70.2) corresponds to the

m

th zero of the function on the left-hand side. …

8: 19.36 Methods of Computation

… ►The step from

n

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}

). …

9: 19.11 Addition Theorems

… ►In the case of

\theta,\phi\in[0,\pi/2)

and

0\leq k^{2}\leq\alpha^{2}<\min\left(1,\left(1-\cos\theta\cos\phi\cos\psi\right% )^{-1}\right)

, we can use … ►

§19.11(ii) Case $\psi=\pi/2$

…

10: 16.11 Asymptotic Expansions

… ►

Case $p=q+1$

… ►

Case $p=q$

… ►The special case

a_{1}=1

p=q=2

is discussed in Kim (1972). ►

Case $p=q-1$

… ►

Case $p\leq q-2$

…