From Wikipedia, the free encyclopedia
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
Mathematical function
In mathematics , particularly q -analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions , while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant form when written in terms of the Ramanujan theta. The function is named after mathematician Srinivasa Ramanujan .
Definition
The Ramanujan theta function is defined as
f
(
a
,
b
)
=
∑
n
=
−
∞
∞
a
n
(
n
+
1
)
2
b
n
(
n
−
1
)
2
{\displaystyle f(a,b)=\sum _{n=-\infty }^{\infty }a^{\frac {n(n+1)}{2}}\;b^{\frac {n(n-1)}{2}}}
for |ab | < 1 . The Jacobi triple product identity then takes the form
f
(
a
,
b
)
=
(
−
a
;
a
b
)
∞
(
−
b
;
a
b
)
∞
(
a
b
;
a
b
)
∞
.
{\displaystyle f(a,b)=(-a;ab)_{\infty }\;(-b;ab)_{\infty }\;(ab;ab)_{\infty }.}
Here, the expression
(
a
;
q
)
n
{\displaystyle (a;q)_{n}}
denotes the q -Pochhammer symbol . Identities that follow from this include
φ
(
q
)
=
f
(
q
,
q
)
=
∑
n
=
−
∞
∞
q
n
2
=
(
−
q
;
q
2
)
∞
2
(
q
2
;
q
2
)
∞
{\displaystyle \varphi (q)=f(q,q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}={\left(-q;q^{2}\right)_{\infty }^{2}\left(q^{2};q^{2}\right)_{\infty }}}
and
ψ
(
q
)
=
f
(
q
,
q
3
)
=
∑
n
=
0
∞
q
n
(
n
+
1
)
2
=
(
q
2
;
q
2
)
∞
(
−
q
;
q
)
∞
{\displaystyle \psi (q)=f\left(q,q^{3}\right)=\sum _{n=0}^{\infty }q^{\frac {n(n+1)}{2}}={\left(q^{2};q^{2}\right)_{\infty }}{(-q;q)_{\infty }}}
and
f
(
−
q
)
=
f
(
−
q
,
−
q
2
)
=
∑
n
=
−
∞
∞
(
−
1
)
n
q
n
(
3
n
−
1
)
2
=
(
q
;
q
)
∞
{\displaystyle f(-q)=f\left(-q,-q^{2}\right)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{\frac {n(3n-1)}{2}}=(q;q)_{\infty }}
This last being the Euler function , which is closely related to the Dedekind eta function . The Jacobi theta function may be written in terms of the Ramanujan theta function as:
ϑ
(
w
,
q
)
=
f
(
q
w
2
,
q
w
−
2
)
{\displaystyle \vartheta (w,q)=f\left(qw^{2},qw^{-2}\right)}
Integral representations
We have the following integral representation for the full two-parameter form of Ramanujan's theta function:[ 1]
f
(
a
,
b
)
=
1
+
∫
0
∞
2
a
e
−
1
2
t
2
2
π
[
1
−
a
a
b
cosh
(
log
a
b
t
)
a
3
b
−
2
a
a
b
cosh
(
log
a
b
t
)
+
1
]
d
t
+
∫
0
∞
2
b
e
−
1
2
t
2
2
π
[
1
−
b
a
b
cosh
(
log
a
b
t
)
a
b
3
−
2
b
a
b
cosh
(
log
a
b
t
)
+
1
]
d
t
{\displaystyle f(a,b)=1+\int _{0}^{\infty }{\frac {2ae^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {1-a{\sqrt {ab}}\cosh \left({\sqrt {\log ab}}\,t\right)}{a^{3}b-2a{\sqrt {ab}}\cosh \left({\sqrt {\log ab}}\,t\right)+1}}\right]dt+\int _{0}^{\infty }{\frac {2be^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {1-b{\sqrt {ab}}\cosh \left({\sqrt {\log ab}}\,t\right)}{ab^{3}-2b{\sqrt {ab}}\cosh \left({\sqrt {\log ab}}\,t\right)+1}}\right]dt}
The special cases of Ramanujan's theta functions given by φ (q ) := f (q , q ) OEIS : A000122 and ψ (q ) := f (q , q 3 ) OEIS : A010054 [ 2] also have the following integral representations:[ 1]
φ
(
q
)
=
1
+
∫
0
∞
e
−
1
2
t
2
2
π
[
4
q
(
1
−
q
2
cosh
(
2
log
q
t
)
)
q
4
−
2
q
2
cosh
(
2
log
q
t
)
+
1
]
d
t
ψ
(
q
)
=
∫
0
∞
2
e
−
1
2
t
2
2
π
[
1
−
q
cosh
(
log
q
t
)
q
−
2
q
cosh
(
log
q
t
)
+
1
]
d
t
{\displaystyle {\begin{aligned}\varphi (q)&=1+\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {4q\left(1-q^{2}\cosh \left({\sqrt {2\log q}}\,t\right)\right)}{q^{4}-2q^{2}\cosh \left({\sqrt {2\log q}}\,t\right)+1}}\right]dt\\[6pt]\psi (q)&=\int _{0}^{\infty }{\frac {2e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {1-{\sqrt {q}}\cosh \left({\sqrt {\log q}}\,t\right)}{q-2{\sqrt {q}}\cosh \left({\sqrt {\log q}}\,t\right)+1}}\right]dt\end{aligned}}}
This leads to several special case integrals for constants defined by these functions when q := e −kπ (cf. theta function explicit values ). In particular, we have that [ 1]
φ
(
e
−
k
π
)
=
1
+
∫
0
∞
e
−
1
2
t
2
2
π
[
4
e
k
π
(
e
2
k
π
−
cos
(
2
π
k
t
)
)
e
4
k
π
−
2
e
2
k
π
cos
(
2
π
k
t
)
+
1
]
d
t
π
1
4
Γ
(
3
4
)
=
1
+
∫
0
∞
e
−
1
2
t
2
2
π
[
4
e
π
(
e
2
π
−
cos
(
2
π
t
)
)
e
4
π
−
2
e
2
π
cos
(
2
π
t
)
+
1
]
d
t
π
1
4
Γ
(
3
4
)
⋅
2
+
2
2
=
1
+
∫
0
∞
e
−
1
2
t
2
2
π
[
4
e
2
π
(
e
4
π
−
cos
(
2
π
t
)
)
e
8
π
−
2
e
4
π
cos
(
2
π
t
)
+
1
]
d
t
π
1
4
Γ
(
3
4
)
⋅
1
+
3
2
1
4
3
3
8
=
1
+
∫
0
∞
e
−
1
2
t
2
2
π
[
4
e
3
π
(
e
6
π
−
cos
(
6
π
t
)
)
e
12
π
−
2
e
6
π
cos
(
6
π
t
)
+
1
]
d
t
π
1
4
Γ
(
3
4
)
⋅
5
+
2
5
5
3
4
=
1
+
∫
0
∞
e
−
1
2
t
2
2
π
[
4
e
5
π
(
e
10
π
−
cos
(
10
π
t
)
)
e
20
π
−
2
e
10
π
cos
(
10
π
t
)
+
1
]
d
t
{\displaystyle {\begin{aligned}\varphi \left(e^{-k\pi }\right)&=1+\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {4e^{k\pi }\left(e^{2k\pi }-\cos \left({\sqrt {2\pi k}}\,t\right)\right)}{e^{4k\pi }-2e^{2k\pi }\cos \left({\sqrt {2\pi k}}\,t\right)+1}}\right]dt\\[6pt]{\frac {\pi ^{\frac {1}{4}}}{\Gamma \left({\frac {3}{4}}\right)}}&=1+\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {4e^{\pi }\left(e^{2\pi }-\cos \left({\sqrt {2\pi }}\,t\right)\right)}{e^{4\pi }-2e^{2\pi }\cos \left({\sqrt {2\pi }}\,t\right)+1}}\right]dt\\[6pt]{\frac {\pi ^{\frac {1}{4}}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}&=1+\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {4e^{2\pi }\left(e^{4\pi }-\cos \left(2{\sqrt {\pi }}\,t\right)\right)}{e^{8\pi }-2e^{4\pi }\cos \left(2{\sqrt {\pi }}\,t\right)+1}}\right]dt\\[6pt]{\frac {\pi ^{\frac {1}{4}}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {\sqrt {1+{\sqrt {3}}}}{2^{\frac {1}{4}}3^{\frac {3}{8}}}}&=1+\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {4e^{3\pi }\left(e^{6\pi }-\cos \left({\sqrt {6\pi }}\,t\right)\right)}{e^{12\pi }-2e^{6\pi }\cos \left({\sqrt {6\pi }}\,t\right)+1}}\right]dt\\[6pt]{\frac {\pi ^{\frac {1}{4}}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {\sqrt {5+2{\sqrt {5}}}}{5^{\frac {3}{4}}}}&=1+\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {4e^{5\pi }\left(e^{10\pi }-\cos \left({\sqrt {10\pi }}\,t\right)\right)}{e^{20\pi }-2e^{10\pi }\cos \left({\sqrt {10\pi }}\,t\right)+1}}\right]dt\end{aligned}}}
and that
ψ
(
e
−
k
π
)
=
∫
0
∞
e
−
1
2
t
2
2
π
[
cos
(
k
π
t
)
−
e
k
π
2
cos
(
k
π
t
)
−
cosh
k
π
2
]
d
t
π
1
4
Γ
(
3
4
)
⋅
e
π
8
2
5
8
=
∫
0
∞
e
−
1
2
t
2
2
π
[
cos
(
π
t
)
−
e
π
2
cos
(
π
t
)
−
cosh
π
2
]
d
t
π
1
4
Γ
(
3
4
)
⋅
e
π
4
2
5
4
=
∫
0
∞
e
−
1
2
t
2
2
π
[
cos
(
2
π
t
)
−
e
π
cos
(
2
π
t
)
−
cosh
π
]
d
t
π
1
4
Γ
(
3
4
)
⋅
1
+
2
4
e
π
16
2
7
16
=
∫
0
∞
e
−
1
2
t
2
2
π
[
cos
(
π
2
t
)
−
e
π
4
cos
(
π
2
t
)
−
cosh
π
4
]
d
t
{\displaystyle {\begin{aligned}\psi \left(e^{-k\pi }\right)&=\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {\cos \left({\sqrt {k\pi }}\,t\right)-e^{\frac {k\pi }{2}}}{\cos \left({\sqrt {k\pi }}\,t\right)-\cosh {\frac {k\pi }{2}}}}\right]dt\\[6pt]{\frac {\pi ^{\frac {1}{4}}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {e^{\frac {\pi }{8}}}{2^{\frac {5}{8}}}}&=\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {\cos \left({\sqrt {\pi }}\,t\right)-e^{\frac {\pi }{2}}}{\cos \left({\sqrt {\pi }}\,t\right)-\cosh {\frac {\pi }{2}}}}\right]dt\\[6pt]{\frac {\pi ^{\frac {1}{4}}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {e^{\frac {\pi }{4}}}{2^{\frac {5}{4}}}}&=\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {\cos \left({\sqrt {2\pi }}\,t\right)-e^{\pi }}{\cos \left({\sqrt {2\pi }}\,t\right)-\cosh \pi }}\right]dt\\[6pt]{\frac {\pi ^{\frac {1}{4}}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {{\sqrt[{4}]{1+{\sqrt {2}}}}\,e^{\frac {\pi }{16}}}{2^{\frac {7}{16}}}}&=\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {\cos \left({\sqrt {\frac {\pi }{2}}}\,t\right)-e^{\frac {\pi }{4}}}{\cos \left({\sqrt {\frac {\pi }{2}}}\,t\right)-\cosh {\frac {\pi }{4}}}}\right]dt\end{aligned}}}
Application in string theory
The Ramanujan theta function is used to determine the critical dimensions in bosonic string theory , superstring theory and M-theory .
References
Bailey, W. N. (1935). Generalized Hypergeometric Series . Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 32. Cambridge: Cambridge University Press.
Gasper, George; Rahman, Mizan (2004). Basic Hypergeometric Series . Encyclopedia of Mathematics and Its Applications. Vol. 96 (2nd ed.). Cambridge: Cambridge University Press. ISBN 0-521-83357-4 .
"Ramanujan function" , Encyclopedia of Mathematics , EMS Press , 2001 [1994]
Kaku, Michio (1994). Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension . Oxford: Oxford University Press. ISBN 0-19-286189-1 .
Weisstein, Eric W. "Ramanujan Theta Functions" . MathWorld .