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ModularLambda

Usage

ModularLambda[ ] 给出模lambda椭圆函数  .


Notes

• 数学函数 (参见 A.3.10节).
ModularLambda 只对  上半复平面由定义. 它对实数 无定义.
• 参数  是Weierstrass 半周期的比值 .
ModularLambda 根据  按照关系式  给出椭圆函数的参数  .
ModularLambdaEllipticTheta 的关系是  , 其中  等于  .
 在模态变换   的任何组合下保持不变.
• 参见3.2.11节 对椭圆函数参数约定的讨论.
• 参见Mathematica 全书: 3.2.11节.
Further Examples

The parameter  appearing in the definition of the incomplete elliptic integral  is called the modulus. For  fixed,  can be regarded as a map from (most of) the complex plane into the elliptic curve with half-periods  and  . Thus from the modulus we obtain the invariant  of the curve where the corresponding elliptic functions live. The modular function  reverses this correspondence, that is,  .

In[1]:=  

Out[1]=

The function  is invariant under the group of transformations of the complex upper half-plane generated by  and  .

In[2]:=  

Out[2]=