雅可比橢圓函數

雅可比矩形

雅可比橢圓函數有十二種，各對映到某個矩形的頂點連線。此諸頂點記作 ${\displaystyle s\,}$ ${\displaystyle c\,}$ ${\displaystyle d\,}$ ${\displaystyle n\,}$。

視此矩形為複數平面的一部分，${\displaystyle s\,}$ 是原點，${\displaystyle c\,}$ 是實軸上的一點 ${\displaystyle K,d\,}$ 是 ${\displaystyle K+{\rm {i}}K'\,}$，${\displaystyle n\,}$ 是 ${\displaystyle {\rm {i}}K'\,}$。${\displaystyle K\,}$ 與 ${\displaystyle {\rm {i}}K'\,}$ 稱作四分之一週期。

十二個橢圓函數分別記為 ${\displaystyle {\rm {sc}},{\rm {sd}},{\rm {sn}},{\rm {cs}},{\rm {cd}},{\rm {cn}},{\rm {ds}},{\rm {dc}},{\rm {dn}},{\rm {ns}},{\rm {nc}},{\rm {nd}}\,}$。為方便起見，取變數 ${\displaystyle p,q\,}$意指矩形上的任一對頂點，則函數 ${\displaystyle pq\,}$ 是唯一滿足以下性質的週期亞純函數

• ${\displaystyle p\,}$ 是單零點，${\displaystyle q\,}$ 是單極點。
• ${\displaystyle pq\,}$ 在 ${\displaystyle {\vec {pq}}}$ 方向的週期等於 ${\displaystyle p,q\,}$ 距離的兩倍。對另兩個從 ${\displaystyle p\,}$出發的方向，${\displaystyle pq\,}$亦滿足同樣性質。
• ${\displaystyle pq\,}$ 在頂點 ${\displaystyle p\,}$ ${\displaystyle q\,}$ 的展式首項係數均為一。

表列如次：

函數	週期	零點	極點	留數
${\displaystyle \mathrm {sn} \,(z;k)}$	${\displaystyle 4\,K,\ 2\,\mathrm {i} K'}$	${\displaystyle 2mK+2\,n\,\mathrm {i} \,K'}$	${\displaystyle 2\,mK+(2n+1)\,\mathrm {i} \,K'}$	${\displaystyle (-1)^{m}{\frac {1}{k}}}$
${\displaystyle \mathrm {cn} \,(z;k)}$	${\displaystyle 4\,K,\ 2\,(K+\mathrm {i} K')}$	${\displaystyle (2m+1)\,K+2\,n\,\mathrm {i} \,K'}$	${\displaystyle 2\,mK+(2n+1)\,\mathrm {i} \,K'}$	${\displaystyle (-1)^{m+n}{\frac {1}{{\rm {i}}k}}}$
${\displaystyle \mathrm {dn} \,(z;k)}$	${\displaystyle 2\,K,\ 4\,\mathrm {i} K'}$	${\displaystyle (2\,m+1)\,K+2\,(n+1)\,\mathrm {i} \,K'}$	${\displaystyle 2\,mK+(2n+1)\,\mathrm {i} \,K'}$	${\displaystyle (-1)^{n-1}{\rm {i}}\,}$
${\displaystyle n\,}$ 與 ${\displaystyle m\,}$ 是整數

一般而言，須以平行四邊形代替上述矩形，以考慮更一般的週期。

以上定義略顯抽象，更具體的定義是將之表為某類橢圓積分(第一類不完全橢圓積分)之逆。設

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

橢圓正弦函數 sn u 定義為

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

而橢圓余弦函數 cn u 定義為

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

同理，椭圆德尔塔函数有

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

這裡的 ${\displaystyle m\in \mathbb {R} }$ 是自由變元，通常取 ${\displaystyle 0\leq m\leq 1}$。

剩下的九種橢圓函數能由這三種構造。

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函數来定义。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：

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

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

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

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

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

由此可見 (cn,sn,dn) 描出射影空間 ${\displaystyle \mathbb {P} ^{3}(\mathbb {C} )}$ 中兩個二次曲面之交，這同構於一條橢圓曲線。曲線上的群運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. . New York: Dover. 1972. ISBN 0-486-61272-4. 引文使用过时参数coauthors (帮助); 使用|accessdate=需要含有|url= (帮助) 見 第16章 （页面存档备份，存于）

• 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)
• H. Hancock Lectures on the theory of elliptic functions (New York, J. Wiley & sons, 1910)
• A. C. Dixon The elementary properties of the elliptic functions, with examples (Macmillan, 1894)