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Statements
D
(
ϕ
,
k
)
=
∫
0
ϕ
sin
2
θ
d
θ
1
-
k
2
sin
2
θ
=
∫
0
sin
ϕ
t
2
d
t
1
-
t
2
1
-
k
2
t
2
=
(
F
(
ϕ
,
k
)
-
E
(
ϕ
,
k
)
)
/
k
2
.
elliptic-integral-third-kind-D
italic-ϕ
𝑘
superscript
subscript
0
italic-ϕ
2
𝜃
𝜃
1
superscript
𝑘
2
2
𝜃
superscript
subscript
0
italic-ϕ
superscript
𝑡
2
𝑡
1
superscript
𝑡
2
1
superscript
𝑘
2
superscript
𝑡
2
elliptic-integral-first-kind-F
italic-ϕ
𝑘
elliptic-integral-second-kind-E
italic-ϕ
𝑘
superscript
𝑘
2
{\displaystyle{\displaystyle D\left(\phi,k\right)=\int_{0}^{\phi}\frac{{\sin^{%
2}}\theta\mathrm{d}\theta}{\sqrt{1-k^{2}{\sin^{2}}\theta}}=\int_{0}^{\sin\phi}%
\frac{t^{2}\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}=(F\left(\phi,k%
\right)-E\left(\phi,k\right))/k^{2}.}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
F
(
ϕ
,
k
)
elliptic-integral-first-kind-F
italic-ϕ
𝑘
{\displaystyle{\displaystyle F\left(\NVar{\phi},\NVar{k}\right)}}
E
(
ϕ
,
k
)
elliptic-integral-second-kind-E
italic-ϕ
𝑘
{\displaystyle{\displaystyle E\left(\NVar{\phi},\NVar{k}\right)}}
∫
{\displaystyle{\displaystyle\int}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
ϕ
italic-ϕ
{\displaystyle{\displaystyle\phi}}
k
𝑘
{\displaystyle{\displaystyle k}}
Identifiers
