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Statements
el3
(
x
,
k
c
,
p
)
=
∫
0
arctan
x
d
θ
(
cos
2
θ
+
p
sin
2
θ
)
cos
2
θ
+
k
c
2
sin
2
θ
=
Π
(
arctan
x
,
1
-
p
,
k
)
,
Bulirsch-integral-el-3
𝑥
subscript
𝑘
𝑐
𝑝
superscript
subscript
0
𝑥
𝜃
2
𝜃
𝑝
2
𝜃
2
𝜃
superscript
subscript
𝑘
𝑐
2
2
𝜃
elliptic-integral-third-kind-Pi
𝑥
1
𝑝
𝑘
{\displaystyle{\displaystyle\mathrm{el3}\left(x,k_{c},p\right)=\int_{0}^{%
\operatorname{arctan}x}\frac{\mathrm{d}\theta}{({\cos^{2}}\theta+p{\sin^{2}}%
\theta)\sqrt{{\cos^{2}}\theta+k_{c}^{2}{\sin^{2}}\theta}}=\Pi\left(%
\operatorname{arctan}x,1-p,k\right),}}
x
2
≠
-
1
/
p
superscript
𝑥
2
1
𝑝
{\displaystyle{\displaystyle x^{2}\neq-1/p}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
Π
(
ϕ
,
α
2
,
k
)
elliptic-integral-third-kind-Pi
italic-ϕ
superscript
𝛼
2
𝑘
{\displaystyle{\displaystyle\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}%
\right)}}
∫
{\displaystyle{\displaystyle\int}}
arctan
z
𝑧
{\displaystyle{\displaystyle\operatorname{arctan}\NVar{z}}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
k
𝑘
{\displaystyle{\displaystyle k}}
p
𝑝
{\displaystyle{\displaystyle p}}
k
c
subscript
𝑘
𝑐
{\displaystyle{\displaystyle k_{c}}}
x
𝑥
{\displaystyle{\displaystyle x}}
Identifiers
