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Statements
1
2
ζ
(
ζ
2
+
α
2
)
1
/
2
+
1
2
α
2
arcsinh
(
ζ
α
)
=
(
1
+
a
2
)
1
/
2
arctanh
(
x
(
1
+
a
2
x
2
+
a
2
)
1
/
2
)
-
arcsinh
(
x
a
)
,
1
2
𝜁
superscript
superscript
𝜁
2
superscript
𝛼
2
1
2
1
2
superscript
𝛼
2
hyperbolic-inverse-sine
𝜁
𝛼
superscript
1
superscript
𝑎
2
1
2
hyperbolic-inverse-tangent
𝑥
superscript
1
superscript
𝑎
2
superscript
𝑥
2
superscript
𝑎
2
1
2
hyperbolic-inverse-sine
𝑥
𝑎
{\displaystyle{\displaystyle\frac{1}{2}\zeta\left(\zeta^{2}+\alpha^{2}\right)^%
{1/2}+\frac{1}{2}\alpha^{2}\operatorname{arcsinh}\left(\frac{\zeta}{\alpha}%
\right)=\left(1+a^{2}\right)^{1/2}\operatorname{arctanh}\left(x\left(\frac{1+a%
^{2}}{x^{2}+a^{2}}\right)^{1/2}\right)-\operatorname{arcsinh}\left(\frac{x}{a}%
\right),}}
-
∞
<
ζ
<
∞
𝜁
{\displaystyle{\displaystyle-\infty<\zeta<\infty}}
arcsinh
z
hyperbolic-inverse-sine
𝑧
{\displaystyle{\displaystyle\operatorname{arcsinh}\NVar{z}}}
arctanh
z
hyperbolic-inverse-tangent
𝑧
{\displaystyle{\displaystyle\operatorname{arctanh}\NVar{z}}}
x
𝑥
{\displaystyle{\displaystyle x}}
a
𝑎
{\displaystyle{\displaystyle a}}
ζ
𝜁
{\displaystyle{\displaystyle\zeta}}
α
𝛼
{\displaystyle{\displaystyle\alpha}}
