雅可比椭圆函数

雅可比椭圆函数有十二种，各对映到某个矩形的顶点连线。此诸顶点记作 ${\displaystyle s\,}$  ${\displaystyle c\,}$  ${\displaystyle d\,}$  ${\displaystyle n\,}$ 。

视此矩形为复数平面的一部分，${\displaystyle s\,}$  是原点，${\displaystyle c\,}$  是实轴上的一点 ${\displaystyle K,d\,}$  是 ${\displaystyle K+{\rm {i}}K'\,}$ ，${\displaystyle n\,}$  是 ${\displaystyle {\rm {i}}K'\,}$ 。${\displaystyle K\,}$  与 ${\displaystyle {\rm {i}}K'\,}$  称作四分之一周期。

十二个椭圆函数分别记为 ${\displaystyle {\rm {sc}},{\rm {sd}},{\rm {sn}},{\rm {cs}},{\rm {cd}},{\rm {cn}},{\rm {ds}},{\rm {dc}},{\rm {dn}},{\rm {ns}},{\rm {nc}},{\rm {nd}}\,}$ 。为方便起见，取变数 ${\displaystyle p,q\,}$ 意指矩形上的任一对顶点，则函数 ${\displaystyle pq\,}$  是唯一满足以下性质的周期亚纯函数

• ${\displaystyle p\,}$  是单零点，${\displaystyle q\,}$  是单极点。
• ${\displaystyle pq\,}$  在 ${\displaystyle {\vec {pq}}}$  方向的周期等于 ${\displaystyle p,q\,}$  距离的两倍。对另两个从 ${\displaystyle p\,}$ 出发的方向，${\displaystyle pq\,}$ 亦满足同样性质。
• ${\displaystyle pq\,}$  在顶点 ${\displaystyle p\,}$  ${\displaystyle q\,}$  的展式首项系数均为一。

表列如次：

一般而言，须以平行四边形代替上述矩形，以考虑更一般的周期。

以上定义略显抽象，更具体的定义是将之表为某类椭圆积分(第一类不完全椭圆积分)之逆。设

${\displaystyle u=\int _{0}^{\phi }{\frac {d\theta }{\sqrt {1-m\sin ^{2}\theta }}}.}$ 

椭圆正弦函数 sn u 定义为

${\displaystyle \operatorname {sn} \;u=\sin \phi \,}$ 

而椭圆余弦函数 cn u 定义为

${\displaystyle \operatorname {cn} \;u=\cos \phi }$ 

同理，椭圆德尔塔函数有

${\displaystyle \operatorname {dn} \;u={\sqrt {1-m\sin ^{2}\phi }}.\,}$ 

这里的 ${\displaystyle m\in \mathbb {R} }$  是自由变元，通常取 ${\displaystyle 0\leq m\leq 1}$ 。

剩下的九种椭圆函数能由这三种构造。

雅可比椭圆函数的反函数可以像三角函数与反三角函数那样被定义。因为椭圆函数往往是椭圆积分之逆，这些反函数也都可以用勒让德椭圆积分来描述。如同反三角函数一样，雅可比椭圆函数的反函数也是多值的，因此需要支割线。以下是部分反函数的积分表达：

• ${\displaystyle \mathrm {arcsn} \,(z,k)=\int _{0}^{z}{\frac {\mathrm {d} t}{\sqrt {(1-t^{2})(1-k^{2}t^{2})}}}}$ 
• ${\displaystyle \mathrm {arccn} \,(z,k)=\int _{z}^{1}{\frac {\mathrm {d} t}{\sqrt {(1-t^{2})(1-k^{2}+k^{2}t^{2})}}}}$ 
• ${\displaystyle \mathrm {arcdn} \,(z,k)=\int _{z}^{1}{\frac {\mathrm {d} t}{\sqrt {(1-t^{2})(t^{2}+k^{2}-1)}}}}$ 

• ${\displaystyle \mathrm {arcns} \,(z,k)=\int _{0}^{\infty }{\frac {\mathrm {d} t}{\sqrt {(t^{2}-1)(t^{2}-k^{2})}}}}$ 
• ${\displaystyle \mathrm {arcnc} \,(z,k)=\int _{1}^{z}{\frac {\mathrm {d} t}{\sqrt {(t^{2}-1)(1-k^{2})(k^{2}+t^{2})}}}}$ 
• ${\displaystyle \mathrm {arcnd} \,(z,k)=\int _{1}^{z}{\frac {\mathrm {d} t}{\sqrt {(t^{2}-1)(1-(1-k^{2})t^{2})}}}}$ 

雅可比椭圆函数也可以用Θ函数来定义。如果我们把${\displaystyle \vartheta (0;\tau )}$ 简写为${\displaystyle \vartheta }$ ，把${\displaystyle \vartheta _{01}(0;\tau ),\vartheta _{10}(0;\tau ),\vartheta _{11}(0;\tau )}$ 分别简写为${\displaystyle \vartheta _{01},\vartheta _{10},\vartheta _{11}}$ （Theta常数），那么椭圆模k是${\displaystyle k=({\vartheta _{10} \over \vartheta })^{2}}$ 。如果我们设${\displaystyle u=\pi \vartheta ^{2}z}$ ，我们便有：

${\displaystyle {\mbox{sn}}(u;k)=-{\vartheta \vartheta _{11}(z;\tau ) \over \vartheta _{10}\vartheta _{01}(z;\tau )}}$ 

${\displaystyle {\mbox{cn}}(u;k)={\vartheta _{01}\vartheta _{10}(z;\tau ) \over \vartheta _{10}\vartheta _{01}(z;\tau )}}$ 

${\displaystyle {\mbox{dn}}(u;k)={\vartheta _{01}\vartheta (z;\tau ) \over \vartheta \vartheta _{01}(z;\tau )}}$ 

${\displaystyle \operatorname {cn} ^{2}+\operatorname {sn} ^{2}=1,\,}$ 

${\displaystyle \operatorname {dn} ^{2}+k^{2}\operatorname {sn} ^{2}=1.\,}$ 

由此可见 (cn,sn,dn) 描出射影空间 ${\displaystyle \mathbb {P} ^{3}(\mathbb {C} )}$  中两个二次曲面之交，这同构于一条椭圆曲线。曲线上的群运算由下列加法公式描述：

${\displaystyle \operatorname {cn} (x+y)={\operatorname {cn} (x)\;\operatorname {cn} (y)-\operatorname {sn} (x)\;\operatorname {sn} (y)\;\operatorname {dn} (x)\;\operatorname {dn} (y) \over {1-k^{2}\;\operatorname {sn} ^{2}(x)\;\operatorname {sn} ^{2}(y)}},}$ 

${\displaystyle \operatorname {sn} (x+y)={\operatorname {sn} (x)\;\operatorname {cn} (y)\;\operatorname {dn} (y)+\operatorname {sn} (y)\;\operatorname {cn} (x)\;\operatorname {dn} (x) \over {1-k^{2}\;\operatorname {sn} ^{2}(x)\;\operatorname {sn} ^{2}(y)}},}$ 

${\displaystyle \operatorname {dn} (x+y)={\operatorname {dn} (x)\;\operatorname {dn} (y)-k^{2}\;\operatorname {sn} (x)\;\operatorname {sn} (y)\;\operatorname {cn} (x)\;\operatorname {cn} (y) \over {1-k^{2}\;\operatorname {sn} ^{2}(x)\;\operatorname {sn} ^{2}(y)}}.}$ 

${\displaystyle -\operatorname {dn} ^{2}(u)+(1-k^{2})=-k^{2}\;\operatorname {cn} ^{2}(u)=k^{2}\;\operatorname {sn} ^{2}(u)-k^{2}}$ 

${\displaystyle (k^{2}-1)\;\operatorname {nd} ^{2}(u)+(1-k^{2})=k^{2}(k^{2}-1)\;\operatorname {sd} ^{2}(u)=k^{2}\;\operatorname {cd} ^{2}(u)-k^{2}}$ 

${\displaystyle (1-k^{2})\;\operatorname {sc} ^{2}(u)+(1-k^{2})=(1-k^{2})\;\operatorname {nc} ^{2}(u)=\operatorname {dc} ^{2}(u)-k^{2}}$ 

${\displaystyle \operatorname {cs} ^{2}(u)+(1-k^{2})=\operatorname {ds} ^{2}(u)=\operatorname {ns} ^{2}(u)-k^{2}}$ 

三个基本的雅可比椭圆函数的导数为：

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\,\mathrm {sn} \,(z;k)=\mathrm {cn} \,(z;k)\,\mathrm {dn} \,(z;k),}$ 

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\,\mathrm {cn} \,(z;k)=-\mathrm {sn} \,(z;k)\,\mathrm {dn} \,(z;k),}$ 

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\,\mathrm {dn} \,(z;k)=-k^{2}\mathrm {sn} \,(z;k)\,\mathrm {cn} \,(z;k).}$ 

根据以上的加法定理，可知它们是以下非线性常微分方程的解：

• ${\displaystyle \mathrm {sn} \,(x;k)}$ 是微分方程${\displaystyle {\frac {\mathrm {d} ^{2}y}{\mathrm {d} x^{2}}}+(1+k^{2})y-2k^{2}y^{3}=0,}$ 和${\displaystyle \left({\frac {\mathrm {d} y}{\mathrm {d} x}}\right)^{2}=(1-y^{2})(1-k^{2}y^{2})}$ 的解；

• ${\displaystyle \mathrm {cn} \,(x;k)}$ 是微分方程${\displaystyle {\frac {\mathrm {d} ^{2}y}{\mathrm {d} x^{2}}}+(1-2k^{2})y+2k^{2}y^{3}=0,}$ 和${\displaystyle \left({\frac {\mathrm {d} y}{\mathrm {d} x}}\right)^{2}=(1-y^{2})(1-k^{2}+k^{2}y^{2})}$ 的解；

• ${\displaystyle \mathrm {dn} \,(x;k)}$ 是微分方程${\displaystyle {\frac {\mathrm {d} ^{2}y}{\mathrm {d} x^{2}}}-(2-k^{2})y+2y^{3}=0,}$ 和${\displaystyle \left({\frac {\mathrm {d} y}{\mathrm {d} x}}\right)^{2}=(y^{2}-1)(1-k^{2}-y^{2})}$ 的解。

• Abramowitz, Milton; Stegun, Irene A. eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. 1972.   引文使用过时参数coauthors (帮助); 使用|accessdate=需要含有|url= (帮助) 见 第16章 （页面存档备份，存于互联网档案馆）

• Naum Illyich Akhiezer, Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island ISBN 0-8218-4532-2
• E. T. Whittaker and G. N. Watson A Course of Modern Analysis, (1940, 1996) Cambridge University Press. ISBN 0-521-58807-3

• Alfred George Greenhill, The applications of elliptic functions (London, New York, Macmillan, 1892)
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• A. C. Dixon The elementary properties of the elliptic functions, with examples (Macmillan, 1894)