模λ函數

模λ函數在由下式生成的模群的主同余子群Γ(2)的作用下保持不变：[3]^:115

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

模群自身的生成元则以如下方式作用于模λ函数之上：[3]^:109

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

${\displaystyle \tau \mapsto -{\frac {1}{\tau }}\$ :\ \lambda \mapsto 1-\lambda \ .}

λ函數為亞可比模量（Jacobi modulus）的平方[3]^:108，即${\displaystyle \lambda (\tau )=k^{2}(\tau )}$；亦可以戴德金η函數與Θ函數表达：

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

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

其中：[3]^:63

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

${\displaystyle \theta _{3}(0,\tau )=\sum _{n=-\infty }^{\infty }q^{n^{2}}}$

${\displaystyle \theta _{4}(0,\tau )=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n^{2}}}$

${\displaystyle q=e^{\pi i\tau }}$

λ函數亦可以魏爾斯特拉斯橢圓函數在定义其的格子的棱边中点和面心处的函数值表达；若令${\displaystyle [\omega _{1},\omega _{2}]}$為满足${\displaystyle \tau ={\frac {\omega _{2}}{\omega _{1}}}}$的基本週期二元組：

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

則有：[3]^:108

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

魏尔斯特拉斯函数在上述三点的值各不相同，這意味著λ函數取不到值0或1。[3]^:108

其與克萊因j函數的關係為：[3]^:117[4]

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

λ*(x)函數的色相環複變函數圖形，繪製範圍在實部-3至3內、虛部-3至3內

λ*(x)函數的色相環複變函數圖形，繪製範圍在實部-1至1內、虛部-1至1內

有一個與模λ函數相關的函數：λ*(x)函數，其給出了橢圓模量k的值。第一類完全橢圓積分K(k)與其互補對應的${\displaystyle K\left({\sqrt {1-k^{2}}}\right)}$關係如下：

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

λ*(x)函數的函數值可透過下列式子計算：

${\displaystyle \lambda ^{*}(x)={\frac {\vartheta _{2}^{2}[0;\exp(-\pi {\sqrt {x}})]}{\vartheta _{3}^{2}[0;\exp(-\pi {\sqrt {x}})]}}}$

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

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

其中${\displaystyle \vartheta }$為Θ函數。

此外λ函數與λ*(x)函數存在下列關聯：

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

所有的有理數r，${\displaystyle K\left(\lambda ^{*}(r)\right)}$與${\displaystyle E\left(\lambda ^{*}(r)\right)}$都可以視為橢圓積分的奇異值，可透過有限的伽馬函數表示[5]。

• 模形式