Symmetric holomorphic function
Modular lambda function in the complex plane.
In mathematics , the modular lambda function λ(τ)[ note 1] is a highly symmetric Holomorphic function on the complex upper half-plane . It is invariant under the fractional linear action of the congruence group Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curve X (2). Over any point τ, its value can be described as a cross ratio of the branch points of a ramified double cover of the projective line by the elliptic curve
C
/
⟨
1
,
τ
⟩
{\displaystyle \mathbb {C} /\langle 1,\tau \rangle }
, where the map is defined as the quotient by the [−1] involution.
The q-expansion, where
q
=
e
π
i
τ
{\displaystyle q=e^{\pi i\tau }}
is the nome , is given by:
λ
(
τ
)
=
16
q
−
128
q
2
+
704
q
3
−
3072
q
4
+
11488
q
5
−
38400
q
6
+
…
{\displaystyle \lambda (\tau )=16q-128q^{2}+704q^{3}-3072q^{4}+11488q^{5}-38400q^{6}+\dots }
. OEIS : A115977
By symmetrizing the lambda function under the canonical action of the symmetric group S 3 on X (2), and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group
SL
2
(
Z
)
{\displaystyle \operatorname {SL} _{2}(\mathbb {Z} )}
, and it is in fact Klein's modular j-invariant .
A plot of x→ λ(ix)
The function
λ
(
τ
)
{\displaystyle \lambda (\tau )}
is invariant under the group generated by[ 1]
τ
↦
τ
+
2
;
τ
↦
τ
1
−
2
τ
.
{\displaystyle \tau \mapsto \tau +2\ ;\ \tau \mapsto {\frac {\tau }{1-2\tau }}\ .}
The generators of the modular group act by[ 2]
τ
↦
τ
+
1
:
λ
↦
λ
λ
−
1
;
{\displaystyle \tau \mapsto \tau +1\ :\ \lambda \mapsto {\frac {\lambda }{\lambda -1}}\,;}
τ
↦
−
1
τ
:
λ
↦
1
−
λ
.
{\displaystyle \tau \mapsto -{\frac {1}{\tau }}\ :\ \lambda \mapsto 1-\lambda \ .}
Consequently, the action of the modular group on
λ
(
τ
)
{\displaystyle \lambda (\tau )}
is that of the anharmonic group , giving the six values of the cross-ratio :[ 3]
{
λ
,
1
1
−
λ
,
λ
−
1
λ
,
1
λ
,
λ
λ
−
1
,
1
−
λ
}
.
{\displaystyle \left\lbrace {\lambda ,{\frac {1}{1-\lambda }},{\frac {\lambda -1}{\lambda }},{\frac {1}{\lambda }},{\frac {\lambda }{\lambda -1}},1-\lambda }\right\rbrace \ .}
Relations to other functions [ edit ]
It is the square of the elliptic modulus,[ 4] that is,
λ
(
τ
)
=
k
2
(
τ
)
{\displaystyle \lambda (\tau )=k^{2}(\tau )}
. In terms of the Dedekind eta function
η
(
τ
)
{\displaystyle \eta (\tau )}
and theta functions ,[ 4]
λ
(
τ
)
=
(
2
η
(
τ
2
)
η
2
(
2
τ
)
η
3
(
τ
)
)
8
=
16
(
η
(
τ
/
2
)
η
(
2
τ
)
)
8
+
16
=
θ
2
4
(
τ
)
θ
3
4
(
τ
)
{\displaystyle \lambda (\tau )={\Bigg (}{\frac {{\sqrt {2}}\,\eta ({\tfrac {\tau }{2}})\eta ^{2}(2\tau )}{\eta ^{3}(\tau )}}{\Bigg )}^{8}={\frac {16}{\left({\frac {\eta (\tau /2)}{\eta (2\tau )}}\right)^{8}+16}}={\frac {\theta _{2}^{4}(\tau )}{\theta _{3}^{4}(\tau )}}}
and,
1
(
λ
(
τ
)
)
1
/
4
−
(
λ
(
τ
)
)
1
/
4
=
1
2
(
η
(
τ
4
)
η
(
τ
)
)
4
=
2
θ
4
2
(
τ
2
)
θ
2
2
(
τ
2
)
{\displaystyle {\frac {1}{{\big (}\lambda (\tau ){\big )}^{1/4}}}-{\big (}\lambda (\tau ){\big )}^{1/4}={\frac {1}{2}}\left({\frac {\eta ({\tfrac {\tau }{4}})}{\eta (\tau )}}\right)^{4}=2\,{\frac {\theta _{4}^{2}({\tfrac {\tau }{2}})}{\theta _{2}^{2}({\tfrac {\tau }{2}})}}}
where[ 5]
θ
2
(
τ
)
=
∑
n
=
−
∞
∞
e
π
i
τ
(
n
+
1
/
2
)
2
{\displaystyle \theta _{2}(\tau )=\sum _{n=-\infty }^{\infty }e^{\pi i\tau (n+1/2)^{2}}}
θ
3
(
τ
)
=
∑
n
=
−
∞
∞
e
π
i
τ
n
2
{\displaystyle \theta _{3}(\tau )=\sum _{n=-\infty }^{\infty }e^{\pi i\tau n^{2}}}
θ
4
(
τ
)
=
∑
n
=
−
∞
∞
(
−
1
)
n
e
π
i
τ
n
2
{\displaystyle \theta _{4}(\tau )=\sum _{n=-\infty }^{\infty }(-1)^{n}e^{\pi i\tau n^{2}}}
In terms of the half-periods of Weierstrass's elliptic functions , let
[
ω
1
,
ω
2
]
{\displaystyle [\omega _{1},\omega _{2}]}
be a fundamental pair of periods with
τ
=
ω
2
ω
1
{\displaystyle \tau ={\frac {\omega _{2}}{\omega _{1}}}}
.
e
1
=
℘
(
ω
1
2
)
,
e
2
=
℘
(
ω
2
2
)
,
e
3
=
℘
(
ω
1
+
ω
2
2
)
{\displaystyle e_{1}=\wp \left({\frac {\omega _{1}}{2}}\right),\quad e_{2}=\wp \left({\frac {\omega _{2}}{2}}\right),\quad e_{3}=\wp \left({\frac {\omega _{1}+\omega _{2}}{2}}\right)}
we have[ 4]
λ
=
e
3
−
e
2
e
1
−
e
2
.
{\displaystyle \lambda ={\frac {e_{3}-e_{2}}{e_{1}-e_{2}}}\,.}
Since the three half-period values are distinct, this shows that
λ
{\displaystyle \lambda }
does not take the value 0 or 1.[ 4]
The relation to the j-invariant is[ 6] [ 7]
j
(
τ
)
=
256
(
1
−
λ
(
1
−
λ
)
)
3
(
λ
(
1
−
λ
)
)
2
=
256
(
1
−
λ
+
λ
2
)
3
λ
2
(
1
−
λ
)
2
.
{\displaystyle j(\tau )={\frac {256(1-\lambda (1-\lambda ))^{3}}{(\lambda (1-\lambda ))^{2}}}={\frac {256(1-\lambda +\lambda ^{2})^{3}}{\lambda ^{2}(1-\lambda )^{2}}}\ .}
which is the j -invariant of the elliptic curve of Legendre form
y
2
=
x
(
x
−
1
)
(
x
−
λ
)
{\displaystyle y^{2}=x(x-1)(x-\lambda )}
Given
m
∈
C
∖
{
0
,
1
}
{\displaystyle m\in \mathbb {C} \setminus \{0,1\}}
, let
τ
=
i
K
{
1
−
m
}
K
{
m
}
{\displaystyle \tau =i{\frac {K\{1-m\}}{K\{m\}}}}
where
K
{\displaystyle K}
is the complete elliptic integral of the first kind with parameter
m
=
k
2
{\displaystyle m=k^{2}}
.
Then
λ
(
τ
)
=
m
.
{\displaystyle \lambda (\tau )=m.}
The modular equation of degree
p
{\displaystyle p}
(where
p
{\displaystyle p}
is a prime number) is an algebraic equation in
λ
(
p
τ
)
{\displaystyle \lambda (p\tau )}
and
λ
(
τ
)
{\displaystyle \lambda (\tau )}
. If
λ
(
p
τ
)
=
u
8
{\displaystyle \lambda (p\tau )=u^{8}}
and
λ
(
τ
)
=
v
8
{\displaystyle \lambda (\tau )=v^{8}}
, the modular equations of degrees
p
=
2
,
3
,
5
,
7
{\displaystyle p=2,3,5,7}
are, respectively,[ 8]
(
1
+
u
4
)
2
v
8
−
4
u
4
=
0
,
{\displaystyle (1+u^{4})^{2}v^{8}-4u^{4}=0,}
u
4
−
v
4
+
2
u
v
(
1
−
u
2
v
2
)
=
0
,
{\displaystyle u^{4}-v^{4}+2uv(1-u^{2}v^{2})=0,}
u
6
−
v
6
+
5
u
2
v
2
(
u
2
−
v
2
)
+
4
u
v
(
1
−
u
4
v
4
)
=
0
,
{\displaystyle u^{6}-v^{6}+5u^{2}v^{2}(u^{2}-v^{2})+4uv(1-u^{4}v^{4})=0,}
(
1
−
u
8
)
(
1
−
v
8
)
−
(
1
−
u
v
)
8
=
0.
{\displaystyle (1-u^{8})(1-v^{8})-(1-uv)^{8}=0.}
The quantity
v
{\displaystyle v}
(and hence
u
{\displaystyle u}
) can be thought of as a holomorphic function on the upper half-plane
Im
τ
>
0
{\displaystyle \operatorname {Im} \tau >0}
:
v
=
∏
k
=
1
∞
tanh
(
k
−
1
/
2
)
π
i
τ
=
2
e
π
i
τ
/
8
∑
k
∈
Z
e
(
2
k
2
+
k
)
π
i
τ
∑
k
∈
Z
e
k
2
π
i
τ
=
2
e
π
i
τ
/
8
1
+
e
π
i
τ
1
+
e
π
i
τ
+
e
2
π
i
τ
1
+
e
2
π
i
τ
+
e
3
π
i
τ
1
+
e
3
π
i
τ
+
⋱
{\displaystyle {\begin{aligned}v&=\prod _{k=1}^{\infty }\tanh {\frac {(k-1/2)\pi i}{\tau }}={\sqrt {2}}e^{\pi i\tau /8}{\frac {\sum _{k\in \mathbb {Z} }e^{(2k^{2}+k)\pi i\tau }}{\sum _{k\in \mathbb {Z} }e^{k^{2}\pi i\tau }}}\\&={\cfrac {{\sqrt {2}}e^{\pi i\tau /8}}{1+{\cfrac {e^{\pi i\tau }}{1+e^{\pi i\tau }+{\cfrac {e^{2\pi i\tau }}{1+e^{2\pi i\tau }+{\cfrac {e^{3\pi i\tau }}{1+e^{3\pi i\tau }+\ddots }}}}}}}}\end{aligned}}}
Since
λ
(
i
)
=
1
/
2
{\displaystyle \lambda (i)=1/2}
, the modular equations can be used to give algebraic values of
λ
(
p
i
)
{\displaystyle \lambda (pi)}
for any prime
p
{\displaystyle p}
.[ note 2] The algebraic values of
λ
(
n
i
)
{\displaystyle \lambda (ni)}
are also given by[ 9] [ note 3]
λ
(
n
i
)
=
∏
k
=
1
n
/
2
sl
8
(
2
k
−
1
)
ϖ
2
n
(
n
even
)
{\displaystyle \lambda (ni)=\prod _{k=1}^{n/2}\operatorname {sl} ^{8}{\frac {(2k-1)\varpi }{2n}}\quad (n\,{\text{even}})}
λ
(
n
i
)
=
1
2
n
∏
k
=
1
n
−
1
(
1
−
sl
2
k
ϖ
n
)
2
(
n
odd
)
{\displaystyle \lambda (ni)={\frac {1}{2^{n}}}\prod _{k=1}^{n-1}\left(1-\operatorname {sl} ^{2}{\frac {k\varpi }{n}}\right)^{2}\quad (n\,{\text{odd}})}
where
sl
{\displaystyle \operatorname {sl} }
is the lemniscate sine and
ϖ
{\displaystyle \varpi }
is the lemniscate constant .
Definition and computation of lambda-star [ edit ]
The function
λ
∗
(
x
)
{\displaystyle \lambda ^{*}(x)}
[ 10] (where
x
∈
R
+
{\displaystyle x\in \mathbb {R} ^{+}}
) gives the value of the elliptic modulus
k
{\displaystyle k}
, for which the complete elliptic integral of the first kind
K
(
k
)
{\displaystyle K(k)}
and its complementary counterpart
K
(
1
−
k
2
)
{\displaystyle K({\sqrt {1-k^{2}}})}
are related by following expression:
K
[
1
−
λ
∗
(
x
)
2
]
K
[
λ
∗
(
x
)
]
=
x
{\displaystyle {\frac {K\left[{\sqrt {1-\lambda ^{*}(x)^{2}}}\right]}{K[\lambda ^{*}(x)]}}={\sqrt {x}}}
The values of
λ
∗
(
x
)
{\displaystyle \lambda ^{*}(x)}
can be computed as follows:
λ
∗
(
x
)
=
θ
2
2
(
i
x
)
θ
3
2
(
i
x
)
{\displaystyle \lambda ^{*}(x)={\frac {\theta _{2}^{2}(i{\sqrt {x}})}{\theta _{3}^{2}(i{\sqrt {x}})}}}
λ
∗
(
x
)
=
[
∑
a
=
−
∞
∞
exp
[
−
(
a
+
1
/
2
)
2
π
x
]
]
2
[
∑
a
=
−
∞
∞
exp
(
−
a
2
π
x
)
]
−
2
{\displaystyle \lambda ^{*}(x)=\left[\sum _{a=-\infty }^{\infty }\exp[-(a+1/2)^{2}\pi {\sqrt {x}}]\right]^{2}\left[\sum _{a=-\infty }^{\infty }\exp(-a^{2}\pi {\sqrt {x}})\right]^{-2}}
λ
∗
(
x
)
=
[
∑
a
=
−
∞
∞
sech
[
(
a
+
1
/
2
)
π
x
]
]
[
∑
a
=
−
∞
∞
sech
(
a
π
x
)
]
−
1
{\displaystyle \lambda ^{*}(x)=\left[\sum _{a=-\infty }^{\infty }\operatorname {sech} [(a+1/2)\pi {\sqrt {x}}]\right]\left[\sum _{a=-\infty }^{\infty }\operatorname {sech} (a\pi {\sqrt {x}})\right]^{-1}}
The functions
λ
∗
{\displaystyle \lambda ^{*}}
and
λ
{\displaystyle \lambda }
are related to each other in this way:
λ
∗
(
x
)
=
λ
(
i
x
)
{\displaystyle \lambda ^{*}(x)={\sqrt {\lambda (i{\sqrt {x}})}}}
Properties of lambda-star [ edit ]
Every
λ
∗
{\displaystyle \lambda ^{*}}
value of a positive rational number is a positive algebraic number :
λ
∗
(
x
∈
Q
+
)
∈
A
+
.
{\displaystyle \lambda ^{*}(x\in \mathbb {Q} ^{+})\in \mathbb {A} ^{+}.}
K
(
λ
∗
(
x
)
)
{\displaystyle K(\lambda ^{*}(x))}
and
E
(
λ
∗
(
x
)
)
{\displaystyle E(\lambda ^{*}(x))}
(the complete elliptic integral of the second kind ) can be expressed in closed form in terms of the gamma function for any
x
∈
Q
+
{\displaystyle x\in \mathbb {Q} ^{+}}
, as Selberg and Chowla proved in 1949.[ 11] [ 12]
The following expression is valid for all
n
∈
N
{\displaystyle n\in \mathbb {N} }
:
n
=
∑
a
=
1
n
dn
[
2
a
n
K
[
λ
∗
(
1
n
)
]
;
λ
∗
(
1
n
)
]
{\displaystyle {\sqrt {n}}=\sum _{a=1}^{n}\operatorname {dn} \left[{\frac {2a}{n}}K\left[\lambda ^{*}\left({\frac {1}{n}}\right)\right];\lambda ^{*}\left({\frac {1}{n}}\right)\right]}
where
dn
{\displaystyle \operatorname {dn} }
is the Jacobi elliptic function delta amplitudinis with modulus
k
{\displaystyle k}
.
By knowing one
λ
∗
{\displaystyle \lambda ^{*}}
value, this formula can be used to compute related
λ
∗
{\displaystyle \lambda ^{*}}
values:[ 9]
λ
∗
(
n
2
x
)
=
λ
∗
(
x
)
n
∏
a
=
1
n
sn
{
2
a
−
1
n
K
[
λ
∗
(
x
)
]
;
λ
∗
(
x
)
}
2
{\displaystyle \lambda ^{*}(n^{2}x)=\lambda ^{*}(x)^{n}\prod _{a=1}^{n}\operatorname {sn} \left\{{\frac {2a-1}{n}}K[\lambda ^{*}(x)];\lambda ^{*}(x)\right\}^{2}}
where
n
∈
N
{\displaystyle n\in \mathbb {N} }
and
sn
{\displaystyle \operatorname {sn} }
is the Jacobi elliptic function sinus amplitudinis with modulus
k
{\displaystyle k}
.
Further relations:
λ
∗
(
x
)
2
+
λ
∗
(
1
/
x
)
2
=
1
{\displaystyle \lambda ^{*}(x)^{2}+\lambda ^{*}(1/x)^{2}=1}
[
λ
∗
(
x
)
+
1
]
[
λ
∗
(
4
/
x
)
+
1
]
=
2
{\displaystyle [\lambda ^{*}(x)+1][\lambda ^{*}(4/x)+1]=2}
λ
∗
(
4
x
)
=
1
−
1
−
λ
∗
(
x
)
2
1
+
1
−
λ
∗
(
x
)
2
=
tan
{
1
2
arcsin
[
λ
∗
(
x
)
]
}
2
{\displaystyle \lambda ^{*}(4x)={\frac {1-{\sqrt {1-\lambda ^{*}(x)^{2}}}}{1+{\sqrt {1-\lambda ^{*}(x)^{2}}}}}=\tan \left\{{\frac {1}{2}}\arcsin[\lambda ^{*}(x)]\right\}^{2}}
λ
∗
(
x
)
−
λ
∗
(
9
x
)
=
2
[
λ
∗
(
x
)
λ
∗
(
9
x
)
]
1
/
4
−
2
[
λ
∗
(
x
)
λ
∗
(
9
x
)
]
3
/
4
{\displaystyle \lambda ^{*}(x)-\lambda ^{*}(9x)=2[\lambda ^{*}(x)\lambda ^{*}(9x)]^{1/4}-2[\lambda ^{*}(x)\lambda ^{*}(9x)]^{3/4}}
a
6
−
f
6
=
2
a
f
+
2
a
5
f
5
(
a
=
[
2
λ
∗
(
x
)
1
−
λ
∗
(
x
)
2
]
1
/
12
)
(
f
=
[
2
λ
∗
(
25
x
)
1
−
λ
∗
(
25
x
)
2
]
1
/
12
)
a
8
+
b
8
−
7
a
4
b
4
=
2
2
a
b
+
2
2
a
7
b
7
(
a
=
[
2
λ
∗
(
x
)
1
−
λ
∗
(
x
)
2
]
1
/
12
)
(
b
=
[
2
λ
∗
(
49
x
)
1
−
λ
∗
(
49
x
)
2
]
1
/
12
)
a
12
−
c
12
=
2
2
(
a
c
+
a
3
c
3
)
(
1
+
3
a
2
c
2
+
a
4
c
4
)
(
2
+
3
a
2
c
2
+
2
a
4
c
4
)
(
a
=
[
2
λ
∗
(
x
)
1
−
λ
∗
(
x
)
2
]
1
/
12
)
(
c
=
[
2
λ
∗
(
121
x
)
1
−
λ
∗
(
121
x
)
2
]
1
/
12
)
(
a
2
−
d
2
)
(
a
4
+
d
4
−
7
a
2
d
2
)
[
(
a
2
−
d
2
)
4
−
a
2
d
2
(
a
2
+
d
2
)
2
]
=
8
a
d
+
8
a
13
d
13
(
a
=
[
2
λ
∗
(
x
)
1
−
λ
∗
(
x
)
2
]
1
/
12
)
(
d
=
[
2
λ
∗
(
169
x
)
1
−
λ
∗
(
169
x
)
2
]
1
/
12
)
{\displaystyle {\begin{aligned}&a^{6}-f^{6}=2af+2a^{5}f^{5}\,&\left(a=\left[{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}\right]^{1/12}\right)&\left(f=\left[{\frac {2\lambda ^{*}(25x)}{1-\lambda ^{*}(25x)^{2}}}\right]^{1/12}\right)\\&a^{8}+b^{8}-7a^{4}b^{4}=2{\sqrt {2}}ab+2{\sqrt {2}}a^{7}b^{7}\,&\left(a=\left[{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}\right]^{1/12}\right)&\left(b=\left[{\frac {2\lambda ^{*}(49x)}{1-\lambda ^{*}(49x)^{2}}}\right]^{1/12}\right)\\&a^{12}-c^{12}=2{\sqrt {2}}(ac+a^{3}c^{3})(1+3a^{2}c^{2}+a^{4}c^{4})(2+3a^{2}c^{2}+2a^{4}c^{4})\,&\left(a=\left[{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}\right]^{1/12}\right)&\left(c=\left[{\frac {2\lambda ^{*}(121x)}{1-\lambda ^{*}(121x)^{2}}}\right]^{1/12}\right)\\&(a^{2}-d^{2})(a^{4}+d^{4}-7a^{2}d^{2})[(a^{2}-d^{2})^{4}-a^{2}d^{2}(a^{2}+d^{2})^{2}]=8ad+8a^{13}d^{13}\,&\left(a=\left[{\frac {2\lambda ^{*}(x)}{1-\lambda ^{*}(x)^{2}}}\right]^{1/12}\right)&\left(d=\left[{\frac {2\lambda ^{*}(169x)}{1-\lambda ^{*}(169x)^{2}}}\right]^{1/12}\right)\end{aligned}}}
Ramanujan's class invariants[ edit ]
Ramanujan's class invariants
G
n
{\displaystyle G_{n}}
and
g
n
{\displaystyle g_{n}}
are defined as[ 13]
G
n
=
2
−
1
/
4
e
π
n
/
24
∏
k
=
0
∞
(
1
+
e
−
(
2
k
+
1
)
π
n
)
,
{\displaystyle G_{n}=2^{-1/4}e^{\pi {\sqrt {n}}/24}\prod _{k=0}^{\infty }\left(1+e^{-(2k+1)\pi {\sqrt {n}}}\right),}
g
n
=
2
−
1
/
4
e
π
n
/
24
∏
k
=
0
∞
(
1
−
e
−
(
2
k
+
1
)
π
n
)
,
{\displaystyle g_{n}=2^{-1/4}e^{\pi {\sqrt {n}}/24}\prod _{k=0}^{\infty }\left(1-e^{-(2k+1)\pi {\sqrt {n}}}\right),}
where
n
∈
Q
+
{\displaystyle n\in \mathbb {Q} ^{+}}
. For such
n
{\displaystyle n}
, the class invariants are algebraic numbers. For example
g
58
=
5
+
29
2
,
g
190
=
(
5
+
2
)
(
10
+
3
)
.
{\displaystyle g_{58}={\sqrt {\frac {5+{\sqrt {29}}}{2}}},\quad g_{190}={\sqrt {({\sqrt {5}}+2)({\sqrt {10}}+3)}}.}
Identities with the class invariants include[ 14]
G
n
=
G
1
/
n
,
g
n
=
1
g
4
/
n
,
g
4
n
=
2
1
/
4
g
n
G
n
.
{\displaystyle G_{n}=G_{1/n},\quad g_{n}={\frac {1}{g_{4/n}}},\quad g_{4n}=2^{1/4}g_{n}G_{n}.}
The class invariants are very closely related to the Weber modular functions
f
{\displaystyle {\mathfrak {f}}}
and
f
1
{\displaystyle {\mathfrak {f}}_{1}}
. These are the relations between lambda-star and the class invariants:
G
n
=
sin
{
2
arcsin
[
λ
∗
(
n
)
]
}
−
1
/
12
=
1
/
[
2
λ
∗
(
n
)
12
1
−
λ
∗
(
n
)
2
24
]
{\displaystyle G_{n}=\sin\{2\arcsin[\lambda ^{*}(n)]\}^{-1/12}=1{\Big /}\left[{\sqrt[{12}]{2\lambda ^{*}(n)}}{\sqrt[{24}]{1-\lambda ^{*}(n)^{2}}}\right]}
g
n
=
tan
{
2
arctan
[
λ
∗
(
n
)
]
}
−
1
/
12
=
[
1
−
λ
∗
(
n
)
2
]
/
[
2
λ
∗
(
n
)
]
12
{\displaystyle g_{n}=\tan\{2\arctan[\lambda ^{*}(n)]\}^{-1/12}={\sqrt[{12}]{[1-\lambda ^{*}(n)^{2}]/[2\lambda ^{*}(n)]}}}
λ
∗
(
n
)
=
tan
{
1
2
arctan
[
g
n
−
12
]
}
=
g
n
24
+
1
−
g
n
12
{\displaystyle \lambda ^{*}(n)=\tan \left\{{\frac {1}{2}}\arctan[g_{n}^{-12}]\right\}={\sqrt {g_{n}^{24}+1}}-g_{n}^{12}}
Little Picard theorem [ edit ]
The lambda function is used in the original proof of the Little Picard theorem , that an entire non-constant function on the complex plane cannot omit more than one value. This theorem was proved by Picard in 1879.[ 15] Suppose if possible that f is entire and does not take the values 0 and 1. Since λ is holomorphic, it has a local holomorphic inverse ω defined away from 0,1,∞. Consider the function z → ω(f (z )). By the Monodromy theorem this is holomorphic and maps the complex plane C to the upper half plane. From this it is easy to construct a holomorphic function from C to the unit disc, which by Liouville's theorem must be constant.[ 16]
The function
τ
↦
16
/
λ
(
2
τ
)
−
8
{\displaystyle \tau \mapsto 16/\lambda (2\tau )-8}
is the normalized Hauptmodul for the group
Γ
0
(
4
)
{\displaystyle \Gamma _{0}(4)}
, and its q -expansion
q
−
1
+
20
q
−
62
q
3
+
…
{\displaystyle q^{-1}+20q-62q^{3}+\dots }
, OEIS : A007248 where
q
=
e
2
π
i
τ
{\displaystyle q=e^{2\pi i\tau }}
, is the graded character of any element in conjugacy class 4C of the monster group acting on the monster vertex algebra .
^ Chandrasekharan (1985) p.115
^ Chandrasekharan (1985) p.109
^ Chandrasekharan (1985) p.110
^ a b c d Chandrasekharan (1985) p.108
^ Chandrasekharan (1985) p.63
^ Chandrasekharan (1985) p.117
^ Rankin (1977) pp.226–228
^ Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7 . p. 103–109, 134
^ a b Jacobi, Carl Gustav Jacob (1829). Fundamenta nova theoriae functionum ellipticarum (in Latin). p. 42
^ Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience. ISBN 0-471-83138-7 . p. 152
^ Chowla, S.; Selberg, A. (1949). "On Epstein's Zeta Function (I)" . Proceedings of the National Academy of Sciences . 35 (7): 373. doi :10.1073/PNAS.35.7.371 . PMC 1063041 . S2CID 45071481 .
^ Chowla, S.; Selberg, A. "On Epstein's Zeta-Function" . EuDML . pp. 86–110.
^ Berndt, Bruce C.; Chan, Heng Huat; Zhang, Liang-Cheng (6 June 1997). "Ramanujan's class invariants, Kronecker's limit formula, and modular equations" . Transactions of the American Mathematical Society . 349 (6): 2125–2173.
^ Eymard, Pierre; Lafon, Jean-Pierre (1999). Autour du nombre Pi (in French). HERMANN. ISBN 2705614435 . p. 240
^ Chandrasekharan (1985) p.121
^ Chandrasekharan (1985) p.118
^
λ
(
τ
)
{\displaystyle \lambda (\tau )}
is not a modular function (per the Wikipedia definition), but every modular function is a rational function in
λ
(
τ
)
{\displaystyle \lambda (\tau )}
. Some authors use a non-equivalent definition of "modular functions".
^ For any prime power , we can iterate the modular equation of degree
p
{\displaystyle p}
. This process can be used to give algebraic values of
λ
(
n
i
)
{\displaystyle \lambda (ni)}
for any
n
∈
N
.
{\displaystyle n\in \mathbb {N} .}
^
sl
a
ϖ
{\displaystyle \operatorname {sl} a\varpi }
is algebraic for every
a
∈
Q
.
{\displaystyle a\in \mathbb {Q} .}
Abramowitz, Milton ; Stegun, Irene A. , eds. (1972), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , New York: Dover Publications , ISBN 978-0-486-61272-0 , Zbl 0543.33001
Chandrasekharan, K. (1985), Elliptic Functions , Grundlehren der mathematischen Wissenschaften, vol. 281, Springer-Verlag , pp. 108–121, ISBN 3-540-15295-4 , Zbl 0575.33001
Conway, John Horton ; Norton, Simon (1979), "Monstrous moonshine", Bulletin of the London Mathematical Society , 11 (3): 308–339, doi :10.1112/blms/11.3.308 , MR 0554399 , Zbl 0424.20010
Rankin, Robert A. (1977), Modular Forms and Functions , Cambridge University Press , ISBN 0-521-21212-X , Zbl 0376.10020
Reinhardt, W. P.; Walker, P. L. (2010), "Elliptic Modular Function" , in Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions , Cambridge University Press, ISBN 978-0-521-19225-5 , MR 2723248 .
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 139 and 298, 1987.
Conway, J. H. and Norton, S. P. "Monstrous Moonshine." Bull. London Math. Soc. 11, 308-339, 1979.
Selberg, A. and Chowla, S. "On Epstein's Zeta-Function." J. reine angew. Math. 227, 86-110, 1967.