# lemniscate constants

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## 5 matching pages

##### 1: 19.20 Special Cases
$R_{F}\left(0,0,z\right)=\infty.$
The first lemniscate constant is given by …
19.20.21 $R_{D}\left(x,x,z\right)=\frac{3}{z-x}\left(R_{C}\left(z,x\right)-\frac{1}{% \sqrt{z}}\right),$ $x\neq z$, $xz\neq 0$.
The second lemniscate constant is given by …
##### 2: Bibliography T
• J. Todd (1975) The lemniscate constants. Comm. ACM 18 (1), pp. 14–19.
• ##### 3: 19.3 Graphics Figure 19.3.6: Π ⁡ ( ϕ , 2 , k ) as a function of k 2 and sin 2 ⁡ ϕ for - 1 ≤ k 2 ≤ 3 , 0 ≤ sin 2 ⁡ ϕ < 1 . …Its value tends to - ∞ as k 2 → 1 + by (19.6.6), and to the negative of the second lemniscate constant (see (19.20.22)) as k 2 ( = csc 2 ⁡ ϕ ) → 2 - . Magnify 3D Help
##### 4: 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 $\pi/4$. …
##### 5: 22.18 Mathematical Applications
###### Lemniscate
In polar coordinates, $x=r\cos\phi$, $y=r\sin\phi$, the lemniscate is given by $r^{2}=\cos\left(2\phi\right)$, $0\leq\phi\leq 2\pi$. …
22.18.7 $ax^{2}y^{2}+b(x^{2}y+xy^{2})+c(x^{2}+y^{2})+2dxy+e(x+y)+f=0,$
in which $a,b,c,d,e,f$ are real constants, can be achieved in terms of single-valued functions. …