# 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 $(q+z)R_{J}\left(0,y,z,-q\right)=(p-z)R_{J}\left(0,y,z,p\right)-3R_{F}\left(0,y% ,z\right),$ $p=z(y+q)/(z+q)$, $w=z/(z+q)$.
##### 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 $pR_{J}\left(0,y,z,p\right)+qR_{J}\left(0,y,z,q\right)=3R_{F}\left(0,y,z\right),$ $pq=yz$.
##### 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 $n$ 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 $R_{C}$). …
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 …
###### Case$p=q$
The special case $a_{1}=1$, $p=q=2$ is discussed in Kim (1972).