# 模λ函数

## 维基百科，自由的百科全书

${\displaystyle \lambda \left(\tau \right)=16q-128q^{2}+704q^{3}-3072q^{4}+11488q^{5}-38400q^{6}+\dots }$，其中${\displaystyle q=e^{\pi i\tau ))$

## 模性质

${\displaystyle \tau \mapsto \tau +2\ ;\ \tau \mapsto {\frac {\tau }{1-2\tau ))\ .}$

${\displaystyle \tau \mapsto \tau +1\ :\ \lambda \mapsto {\frac {\lambda }{\lambda -1))\,;}$
${\displaystyle \tau \mapsto -{\frac {1}{\tau ))\ :\ \lambda \mapsto 1-\lambda \ .}$

## 与其他椭圆函数的关联

λ函数为亚可比模量（Jacobi modulus）的平方[3]:108，即${\displaystyle \lambda (\tau )=k^{2}(\tau )}$；亦可以戴德金η函数Θ函数表达：

${\displaystyle \lambda (\tau )={\Bigg (}{\frac ((\sqrt {2))\,\eta ({\tfrac {\tau }{2)))\eta ^{2}(2\tau )}{\eta ^{3}(\tau ))){\Bigg )}^{8}={\frac {16}{\left({\frac {\eta (\tau /2)}{\eta (2\tau )))\right)^{8}+16))={\frac {\theta _{2}^{4}(0,\tau )}{\theta _{3}^{4}(0,\tau )))}$
${\displaystyle {\frac {1}((\big (}\lambda (\tau ){\big )}^{1/4))}-{\big (}\lambda (\tau ){\big )}^{1/4}={\frac {1}{2))\left({\frac {\eta ({\tfrac {\tau }{4)))}{\eta (\tau )))\right)^{4}=2\,{\frac {\theta _{4}^{2}(0,{\tfrac {\tau }{2)))}{\theta _{2}^{2}(0,{\tfrac {\tau }{2)))))}$

${\displaystyle \theta _{2}(0,\tau )=\sum _{n=-\infty }^{\infty }q^{\left({n+{\frac {1}{2))}\right)^{2))}$
${\displaystyle \theta _{3}(0,\tau )=\sum _{n=-\infty }^{\infty }q^{n^{2))}$
${\displaystyle \theta _{4}(0,\tau )=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n^{2))}$
${\displaystyle q=e^{\pi i\tau ))$

λ函数亦可以魏尔斯特拉斯椭圆函数在定义其的格子的棱边中点和面心处的函数值表达；若令${\displaystyle [\omega _{1},\omega _{2}]}$为满足${\displaystyle \tau ={\frac {\omega _{2)){\omega _{1))))$的基本周期二元组：

${\displaystyle e_{1}=\wp \left({\frac {\omega _{1)){2))\right),e_{2}=\wp \left({\frac {\omega _{2)){2))\right),e_{3}=\wp \left({\frac {\omega _{1}+\omega _{2)){2))\right)}$

${\displaystyle \lambda ={\frac {e_{3}-e_{2)){e_{1}-e_{2))}\,.}$

${\displaystyle j(\tau )={\frac {256(1-\lambda (1-\lambda ))^{3)){(\lambda (1-\lambda ))^{2))}={\frac {256(1-\lambda +\lambda ^{2})^{3)){\lambda ^{2}(1-\lambda )^{2))}\ .}$

## 椭圆模量

λ*(x)函数的色相环复变函数图形，绘制范围在实部-3至3内、虚部-3至3内
λ*(x)函数的色相环复变函数图形，绘制范围在实部-1至1内、虚部-1至1内

${\displaystyle {\frac {K\left[{\sqrt {1-\lambda ^{*}(x)^{2))}\right]}{K[\lambda ^{*}(x)]))={\sqrt {x))}$

λ*(x)函数的函数值可透过下列式子计算：

${\displaystyle \lambda ^{*}(x)={\frac {\vartheta _{2}^{2}[0;\exp(-\pi {\sqrt {x)))]}{\vartheta _{3}^{2}[0;\exp(-\pi {\sqrt {x)))]))}$
${\displaystyle \lambda ^{*}(x)=\left[\sum _{a=-\infty }^{\infty }\exp[-(a+1/2)^{2}\pi {\sqrt {x))]\right]^{2}\left[\sum _{a=-\infty }^{\infty }\exp(-a^{2}\pi {\sqrt {x)))\right]^{-2))$
${\displaystyle \lambda ^{*}(x)=\left[\sum _{a=-\infty }^{\infty }\operatorname {sech} [(a+1/2)\pi {\sqrt {x))]\right]\left[\sum _{a=-\infty }^{\infty }\operatorname {sech} (a\pi {\sqrt {x)))\right]^{-1))$

${\displaystyle \lambda ^{*}(x)={\sqrt {\lambda (i{\sqrt {x)))))}$

## 参考文献

1. ^ 日本数学会. 数学百科辞典. 科学出版社. 1984.
2. ^ Weisstein, Eric W. (编). Elliptic Lambda Function. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. （英语）.
3. Chandrasekharan, K., Elliptic Functions, Grundlehren der mathematischen Wissenschaften 281, Springer-Verlag: 108–121, 1985, ISBN 3-540-15295-4, Zbl 0575.33001
4. ^ Rankin, Robert A., Modular Forms and Functions, Cambridge University Press: 226–228, 1977, ISBN 0-521-21212-X, Zbl 0376.10020
5. ^ Selberg, A.; Chowla, S. "On Epstein's Zeta-Function.". J. reine angew. Math. 1967, 227: 86–110.