Ramanujans thetafunktion är en generalisering av Jacobis thetafunktion . Funktionen är uppkallad efter Srinivasa Ramanujan .
Ramanujans thetafunktion definieras som
f
(
a
,
b
)
=
∑
n
=
−
∞
∞
a
n
(
n
+
1
)
/
2
b
n
(
n
−
1
)
/
2
{\displaystyle f(a,b)=\sum _{n=-\infty }^{\infty }a^{n(n+1)/2}\;b^{n(n-1)/2}}
då |ab | < 1. Jacobis trippelprodukt tar då formen
f
(
a
,
b
)
=
f
(
b
,
a
)
=
(
−
a
;
a
b
)
∞
(
−
b
;
a
b
)
∞
(
a
b
;
a
b
)
∞
{\displaystyle f(a,b)=f(b,a)=(-a;ab)_{\infty }\;(-b;ab)_{\infty }\;(ab;ab)_{\infty }}
där
(
a
;
q
)
n
{\displaystyle (a;q)_{n}}
q-Pochhammersymbolen . Tre viktiga specialfall av Ramanujans thetafunktion är
f
(
q
,
q
)
=
∑
n
=
−
∞
∞
q
n
2
=
(
−
q
;
q
2
)
∞
2
(
q
2
;
q
2
)
∞
{\displaystyle f(q,q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}={(-q;q^{2})_{\infty }^{2}(q^{2};q^{2})_{\infty }}}
och
f
(
q
,
q
3
)
=
∑
n
=
0
∞
q
n
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n
+
1
)
/
2
=
(
q
2
;
q
2
)
∞
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−
q
;
q
)
∞
{\displaystyle f(q,q^{3})=\sum _{n=0}^{\infty }q^{n(n+1)/2}={(q^{2};q^{2})_{\infty }}{(-q;q)_{\infty }}}
och
f
(
−
q
)
:=
f
(
−
q
,
−
q
2
)
=
∑
n
=
−
∞
∞
(
−
1
)
n
q
n
(
3
n
−
1
)
/
2
=
(
q
;
q
)
∞
{\displaystyle f(-q):=f(-q,-q^{2})=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n(3n-1)/2}=(q;q)_{\infty }}
φ
(
e
−
π
)
=
π
4
Γ
(
3
4
)
{\displaystyle \varphi \left(e^{-\pi }\right)={\frac {\sqrt[{4}]{\pi }}{\Gamma ({\frac {3}{4}})}}}
φ
(
e
−
2
π
)
=
6
π
+
4
2
π
4
2
Γ
(
3
4
)
{\displaystyle \varphi \left(e^{-2\pi }\right)={\frac {\sqrt[{4}]{6\pi +4{\sqrt {2}}\pi }}{2\Gamma ({\frac {3}{4}})}}}
φ
(
e
−
3
π
)
=
27
π
+
18
3
π
4
3
Γ
(
3
4
)
{\displaystyle \varphi \left(e^{-3\pi }\right)={\frac {\sqrt[{4}]{27\pi +18{\sqrt {3}}\pi }}{3\Gamma ({\frac {3}{4}})}}}
φ
(
e
−
4
π
)
=
8
π
4
+
2
π
4
4
Γ
(
3
4
)
{\displaystyle \varphi \left(e^{-4\pi }\right)={\frac {{\sqrt[{4}]{8\pi }}+2{\sqrt[{4}]{\pi }}}{4\Gamma ({\frac {3}{4}})}}}
φ
(
e
−
5
π
)
=
225
π
+
100
5
π
4
5
Γ
(
3
4
)
{\displaystyle \varphi \left(e^{-5\pi }\right)={\frac {\sqrt[{4}]{225\pi +100{\sqrt {5}}\pi }}{5\Gamma ({\frac {3}{4}})}}}
φ
(
e
−
6
π
)
=
3
2
+
3
3
4
+
2
3
−
27
4
+
1728
4
−
4
3
⋅
243
π
2
8
6
1
+
6
−
2
−
3
6
Γ
(
3
4
)
{\displaystyle \varphi \left(e^{-6\pi }\right)={\frac {{\sqrt[{3}]{3{\sqrt {2}}+3{\sqrt[{4}]{3}}+2{\sqrt {3}}-{\sqrt[{4}]{27}}+{\sqrt[{4}]{1728}}-4}}\cdot {\sqrt[{8}]{243{\pi }^{2}}}}{6{\sqrt[{6}]{1+{\sqrt {6}}-{\sqrt {2}}-{\sqrt {3}}}}{\Gamma ({\frac {3}{4}})}}}}
W.N. Bailey, Generalized Hypergeometric Series , (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge.
George Gasper and Mizan Rahman, Basic Hypergeometric Series, 2nd Edition , (2004), Encyclopedia of Mathematics and Its Applications, 96 , Cambridge University Press, Cambridge. ISBN 0-521-83357-4 .
Hazewinkel, Michiel, red. (2001), ”Ramanujan function” , Encyclopedia of Mathematics ISBN 978-1556080104 Weisstein, Eric W. , "Ramanujan Theta Functions ", MathWorld (engelska) Speciella funktioner Gamma- och relaterade funktioner Zeta- och L -funktioner Besselfunktioner och relaterade funktioner Elliptiska funktioner och thetafunktioner Hypergeometriska funktioner Ortogonala polynom Andra funktioner
![{\displaystyle \varphi \left(e^{-\pi }\right)={\frac {\sqrt[{4}]{\pi }}{\Gamma ({\frac {3}{4}})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fe9ada64c9690314e9f709e5f8b5005824a8b90)
![{\displaystyle \varphi \left(e^{-2\pi }\right)={\frac {\sqrt[{4}]{6\pi +4{\sqrt {2}}\pi }}{2\Gamma ({\frac {3}{4}})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e386e5d82e663d5cd67806a082bce188d747c8)
![{\displaystyle \varphi \left(e^{-3\pi }\right)={\frac {\sqrt[{4}]{27\pi +18{\sqrt {3}}\pi }}{3\Gamma ({\frac {3}{4}})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf3ba8bb29f02c3e46c859e4e0ccf10e1e35e688)
![{\displaystyle \varphi \left(e^{-4\pi }\right)={\frac {{\sqrt[{4}]{8\pi }}+2{\sqrt[{4}]{\pi }}}{4\Gamma ({\frac {3}{4}})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d99dc9be809c76d1491b7b2d61c3f3e17d558652)
![{\displaystyle \varphi \left(e^{-5\pi }\right)={\frac {\sqrt[{4}]{225\pi +100{\sqrt {5}}\pi }}{5\Gamma ({\frac {3}{4}})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0f9b03659f7a692a99a38394f114da38df3d4a0)
![{\displaystyle \varphi \left(e^{-6\pi }\right)={\frac {{\sqrt[{3}]{3{\sqrt {2}}+3{\sqrt[{4}]{3}}+2{\sqrt {3}}-{\sqrt[{4}]{27}}+{\sqrt[{4}]{1728}}-4}}\cdot {\sqrt[{8}]{243{\pi }^{2}}}}{6{\sqrt[{6}]{1+{\sqrt {6}}-{\sqrt {2}}-{\sqrt {3}}}}{\Gamma ({\frac {3}{4}})}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbe75042d22ec00bb32b3e38a5d706ff7050b136)