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Statements
(
1
-
z
3
)
a
(
-
z
)
3
a
(
1
Γ
(
a
+
2
3
)
Γ
(
2
3
)
F
(
a
,
a
+
1
3
2
3
;
z
-
3
)
+
e
1
3
π
i
z
Γ
(
a
)
Γ
(
4
3
)
F
(
a
+
1
3
,
a
+
2
3
4
3
;
z
-
3
)
)
=
3
3
2
a
+
1
2
e
1
2
a
π
i
Γ
(
a
+
1
3
)
(
1
-
ζ
)
a
2
π
Γ
(
2
a
+
2
3
)
(
-
ζ
)
2
a
F
(
a
+
1
3
,
3
a
2
a
+
2
3
;
ζ
-
1
)
,
superscript
1
superscript
𝑧
3
𝑎
superscript
𝑧
3
𝑎
1
Euler-Gamma
𝑎
2
3
Euler-Gamma
2
3
Gauss-hypergeometric-F
𝑎
𝑎
1
3
2
3
superscript
𝑧
3
superscript
𝑒
1
3
𝜋
imaginary-unit
𝑧
Euler-Gamma
𝑎
Euler-Gamma
4
3
Gauss-hypergeometric-F
𝑎
1
3
𝑎
2
3
4
3
superscript
𝑧
3
superscript
3
3
2
𝑎
1
2
superscript
𝑒
1
2
𝑎
𝜋
imaginary-unit
Euler-Gamma
𝑎
1
3
superscript
1
𝜁
𝑎
2
𝜋
Euler-Gamma
2
𝑎
2
3
superscript
𝜁
2
𝑎
Gauss-hypergeometric-F
𝑎
1
3
3
𝑎
2
𝑎
2
3
superscript
𝜁
1
{\displaystyle{\displaystyle\frac{\left(1-z^{3}\right)^{a}}{\left(-z\right)^{3%
a}}\left(\frac{1}{\Gamma\left(a+\frac{2}{3}\right)\Gamma\left(\frac{2}{3}%
\right)}F\left({a,a+\frac{1}{3}\atop\frac{2}{3}};z^{-3}\right)+\frac{e^{\frac{%
1}{3}\pi\mathrm{i}}}{z\Gamma\left(a\right)\Gamma\left(\frac{4}{3}\right)}F%
\left({a+\frac{1}{3},a+\frac{2}{3}\atop\frac{4}{3}};z^{-3}\right)\right)=\frac%
{3^{\frac{3}{2}a+\frac{1}{2}}e^{\frac{1}{2}a\pi\mathrm{i}}\Gamma\left(a+\frac{%
1}{3}\right)(1-\zeta)^{a}}{2\pi\Gamma\left(2a+\frac{2}{3}\right)(-\zeta)^{2a}}%
F\left({a+\frac{1}{3},3a\atop 2a+\frac{2}{3}};\zeta^{-1}\right),}}
|
ph
(
-
z
)
|
<
1
3
π
phase
𝑧
1
3
𝜋
{\displaystyle{\displaystyle|\operatorname{ph}\left(-z\right)|<\frac{1}{3}\pi}}
|
z
|
>
1
𝑧
1
{\displaystyle{\displaystyle|z|>1}}
|
ph
(
-
z
)
|
<
1
3
π
ph
𝑧
1
3
𝜋
{\displaystyle{\displaystyle|\operatorname{ph}\left(-z\right)|<\frac{1}{3}\pi}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
F
(
a
,
b
;
c
;
z
)
Gauss-hypergeometric-F
𝑎
𝑏
𝑐
𝑧
{\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}
π
{\displaystyle{\displaystyle\pi}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
ph
phase
{\displaystyle{\displaystyle\operatorname{ph}}}
z
𝑧
{\displaystyle{\displaystyle z}}
a
𝑎
{\displaystyle{\displaystyle a}}
ζ
𝜁
{\displaystyle{\displaystyle\zeta}}
Identifiers
