椭圆积分

在积分学中，椭圆积分最初出现于椭圆的弧长有关的问题中。朱利奥·法尼亚诺（英语：Guilio Fagnano）和欧拉是最早的研究者。现代数学将椭圆积分定义为可以表达为如下形式的任何函数 ${\displaystyle f\,}$的积分

${\displaystyle f(x)=\int _{c}^{x}R[t,{\sqrt[{}]{P(t)}}]\ dt\,\!}$

其中${\displaystyle R\,}$是其两个参数的有理函数，${\displaystyle P\,}$是一个无重根的${\displaystyle 3\,}$或${\displaystyle 4\,}$阶多项式，而${\displaystyle c\,}$是一个常数。

通常，椭圆积分不能用基本函数表达。这个一般规则的例外出现在${\displaystyle P\,}$有重根的时候，或者是${\displaystyle R\,}$,${\displaystyle \left(x,y\right)\,}$没有${\displaystyle y\,}$的奇数幂时。但是，通过适当的简化公式，每个椭圆积分可以变为只涉及有理函数和三个经典形式的积分。（也即，第一，第二，和第三类的椭圆积分）。

除下面给出的形式之外，椭圆积分也可以表达为勒让德形式和Carlson对称形式。通过对施瓦茨-克里斯托费尔映射的研究可以加深对椭圆积分理论的理解。历史上，椭圆函数是作为椭圆积分的逆函数被发现的，特别是这一个：${\displaystyle F[{\textrm {sn}}\left(z;k\right);k]=z\,}$其中${\displaystyle {\textrm {sn}}\,}$是雅可比正弦椭圆函数。

椭圆积分通常表述为不同变量的函数。这些变量完全等价（它们给出同样的椭圆积分），但是它们看起来很不相同。很多文献使用单一一种标准命名规则。在定义积分之前，先来检视一下这些变量的命名常规：

• ${\displaystyle \alpha }$ 模角;
• ${\displaystyle k=\sin \alpha }$ 椭圆模;
• ${\displaystyle m=k^{2}=\sin ^{2}\alpha }$ 参数;

上述三种常规完全互相确定。规定其中一个和规定另外一个一样。椭圆积分也依赖于另一个变量，可以有如下几种不同的设定方法：

• ${\displaystyle \phi \,\!}$ 幅度
• ${\displaystyle x\,}$ 其中${\displaystyle x=\sin \phi ={\textrm {sn}}\;u\,\!}$
• ${\displaystyle u\,}$，其中${\displaystyle x={\textrm {sn}}\;u\,}$而${\displaystyle {\textrm {sn}}\,}$是雅可比椭圆函数之一

规定其中一个决定另外两个。这样，它们可以互换地使用。注意${\displaystyle u\,}$也依赖于${\displaystyle m\,}$。其它包含${\displaystyle u\,}$的关系有

${\displaystyle \cos \phi ={\textrm {cn}}\;u\,\!}$

和

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

后者有时称为δ幅度并写作${\displaystyle \Delta (\phi )={\textrm {dn}}\;u\,\!}$。有时文献也称之为补参数，补模或者补模角。这些在四分周期中有进一步的定义。

第一类不完全椭圆积分 ${\displaystyle F\,}$定义为

${\displaystyle F(\phi \setminus \alpha )=F(\phi |m)=\int _{0}^{\phi }{\frac {{\rm {d}}\theta }{\sqrt {1-(\sin \theta \sin \alpha )^{2}}}}.\,\!}$

与此等价，用雅可比的形式，可以设 ${\displaystyle x=\sin \phi ~,~t=\sin \theta \;\!}$；则

${\displaystyle F(\phi \setminus \alpha )=F(x;k)=\int _{0}^{x}{\frac {{\rm {d}}t}{\sqrt {(1-t^{2})(1-k^{2}t^{2})}}}\,\!}$

其中，假定任何有竖直条出现的地方，紧跟竖直条的变量是（如上定义的）参数；而且，当反斜杠出现的时候，跟着出现的是模角。 在这个意义下，${\displaystyle F(\sin \phi ;\sin \alpha )=F(\phi |\sin ^{2}\alpha )=F(\phi \setminus \alpha )~\,\!}$，这里的记法来自标准参考书Abramowitz and Stegun。

但是，还有许多不同的用于椭圆积分的记法。取值为椭圆积分的函数没有（像平方根，正弦和误差函数那样的）标准和唯一的名字。甚至关于该领域的文献也常常采用不同的记法。Gradstein, Ryzhik[1] （页面存档备份，存于互联网档案馆）, ${\displaystyle Eq\,}$.(8.111)]采用${\displaystyle F(\phi ,k)\,\!}$。该记法和这里的${\displaystyle F(\phi |k^{2})~\,\!}$；以及下面的${\displaystyle E(\phi ,k)=E(\phi |k^{2})~\,\!}$等价。

和上面的不同对应的是，如果从Mathematica语言翻译代码到Maple语言，必须将EllipticK函数的参数用它的平方根代替。反过来，如果从Maple翻到Mathematica，则参数应该用它的平方代替。Maple中的EllipticK(${\displaystyle x}$)几乎和Mathematica中的EllipticK[${\displaystyle x^{2}}$]相等；至少当${\displaystyle 0<x<1\,}$时是相等的。

注意

${\displaystyle F(x;k)=u\,\!}$

其中${\displaystyle u\,}$如上文所定义：由此可见，雅可比椭圆函数是椭圆积分的逆。

${\displaystyle \forall \varphi _{1},\varphi _{2}\in \left]-{\frac {\pi }{2}};{\frac {\pi }{2}}\right[,}$

${\displaystyle F\left(\varphi _{1},k\right)+F\left(\varphi _{2},k\right)=F\left(\arctan \left(\tan \varphi _{1}{\sqrt {1-k^{2}\sin ^{2}\varphi _{2}}}\right)+\arctan \left(\tan \varphi _{2}{\sqrt {1-k^{2}\sin ^{2}\varphi _{1}}}\right),k\right)}$

${\displaystyle \arctan \left(\tan \varphi _{1}{\sqrt {1-k^{2}\sin ^{2}\varphi _{2}}}\right)+\arctan \left(\tan \varphi _{2}{\sqrt {1-k^{2}\sin ^{2}\varphi _{1}}}\right)\in [-\pi /2;\pi /2]\Rightarrow }$

${\displaystyle F\left(\varphi _{1},k\right)+F\left(\varphi _{2},k\right)=F\left(\arcsin {\frac {\cos \varphi _{1}{\sqrt {1-k^{2}\sin ^{2}\varphi _{1}}}\sin \varphi _{2}+\cos \varphi _{2}{\sqrt {1-k^{2}\sin ^{2}\varphi _{2}}}\sin \varphi _{1}}{1-k^{2}\sin ^{2}\varphi _{1}\sin ^{2}\varphi _{2}}},k\right)}$

${\displaystyle \arctan \left(\tan \varphi _{1}{\sqrt {1-k^{2}\sin ^{2}\varphi _{2}}}\right)+\arctan \left(\tan \varphi _{2}{\sqrt {1-k^{2}\sin ^{2}\varphi _{1}}}\right)\in [0;\pi ]\Rightarrow }$

${\displaystyle F\left(\varphi _{1},k\right)+F\left(\varphi _{2},k\right)=F\left(\arccos {\frac {\cos \varphi _{1}\cos \varphi _{2}-\sin \varphi _{1}\sin \varphi _{2}{\sqrt {1-k^{2}\sin ^{2}\varphi _{1}}}{\sqrt {1-k^{2}\sin ^{2}\varphi _{1}}}{\sqrt {1-k^{2}\sin ^{2}\varphi _{2}}}}{1-k^{2}\sin ^{2}\varphi _{1}\sin ^{2}\varphi _{2}}},k\right)}$

${\displaystyle F(x+n\pi ;k)=F(x;k)+2nK(k)\,\!}$

${\displaystyle F(x+{\frac {n\pi }{2}};k)=nK(k)\,\!}$

${\displaystyle n\in \mathbb {Z} \,\!}$

${\displaystyle F(-x;k)=-F(x;k)\,\!}$

${\displaystyle F(x;0)=x\,\!}$

${\displaystyle F(0;k)=-F(x;k)\,\!}$

${\displaystyle F(x;1)={\rm {arctanh}}\sin x\,\!}$

${\displaystyle -{\frac {\pi }{2}}<\Re (x)<{\frac {\pi }{2}}\,\!}$

${\displaystyle {\frac {\rm {d}}{{\rm {d}}x}}F(x;k)={\frac {1}{\sqrt {1-k^{2}\sin ^{2}x}}}\,\!}$

${\displaystyle {\frac {\rm {d}}{{\rm {d}}k}}F(x;k)={\frac {E(x;k)}{2k(1-k)}}-{\frac {F(x;k)}{2k}}-{\frac {\sin 2x}{4(1-k){\sqrt {1-k\sin ^{2}x}}}}\,\!}$

第二类不完全椭圆积分 ${\displaystyle E\!}$是

${\displaystyle E(\phi \setminus \alpha )=E(\phi |m)=\int _{0}^{\phi }\!E'(\theta )\ {\rm {d}}\theta =\int _{0}^{\phi }{\sqrt {1-(\sin \theta \sin \alpha )^{2}}}\ {\rm {d}}\theta .\,\!}$

与此等价，采用另外一个记法（作变量替换${\displaystyle t=\sin \theta \,\!}$），

${\displaystyle E(x;k)=\int _{0}^{x}{\frac {\sqrt {1-k^{2}t^{2}}}{\sqrt {1-t^{2}}}}\ {\rm {d}}t.\,\!}$

其它关系包括

${\displaystyle E(\phi |m)=\int _{0}^{u}{\textrm {dn}}^{2}w\;{\rm {d}}w=u-m\int _{0}^{u}{\textrm {sn}}^{2}w\;{\rm {d}}w=(1-m)u+m\int _{0}^{u}{\textrm {cn}}^{2}w\;{\rm {d}}w.\,\!}$

${\displaystyle E(\phi |k^{2})=(1-k^{2})\int _{0}^{\phi }{\frac {{\rm {d}}\theta }{(1-k^{2}\sin ^{2}\theta ){\sqrt {1-k^{2}\sin ^{2}\theta }}}}+{\frac {k^{2}\sin \theta \cos \theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}}\,\!}$

${\displaystyle \forall \varphi _{1},\varphi _{2}\in \left]-{\frac {\pi }{2}};{\frac {\pi }{2}}\right[,}$

${\displaystyle \textstyle E\left(\varphi _{1},k\right)+E\left(\varphi _{2},k\right)=\left[{\begin{aligned}E\left(\arctan \left(\tan \varphi _{1}{\sqrt {1-k^{2}\sin ^{2}\varphi _{2}}}\right)+\arctan \left(\tan \varphi _{2}{\sqrt {1-k^{2}\sin ^{2}\varphi _{1}}}\right),k\right)\\+{\frac {k^{2}\sin \varphi _{1}\sin \varphi _{2}\left(\cos \varphi _{1}{\sqrt {1-k^{2}\sin ^{2}\varphi _{1}}}\sin \varphi _{2}+\cos \varphi _{2}{\sqrt {1-k^{2}\sin ^{2}\varphi _{2}}}\sin \varphi _{1}\right)}{1-k^{2}\sin \varphi _{1}\sin \varphi _{2}}}\end{aligned}}\right]}$

${\displaystyle E(\phi +n\pi ;k)=E(\phi ;k)+2nE(k)\,\!}$

${\displaystyle E(-\phi ;k)=-E(\phi ;k)\,\!}$

${\displaystyle {\frac {\rm {d}}{{\rm {d}}\phi }}E(\phi ;k)={\sqrt {1-k^{2}\sin ^{2}\phi }}\,\!}$

${\displaystyle {\frac {\rm {d}}{{\rm {d}}k}}E(\phi ;k)={\frac {E(\phi ;k)-F(\phi ;k)}{2k}}\,\!}$

${\displaystyle {\frac {{\rm {d}}^{n}}{{\rm {d}}k^{n}}}E(\phi ;k)={\frac {\pi }{2k^{n}}}{}_{2}F_{1}\left(-{\frac {1}{2}},{\frac {1}{2}};1-n;k\right)-{\frac {{\sqrt {\pi }}\cos \phi }{2k^{2n}}}F_{2\times 1\times 0}^{1\times 3\times 2}{\begin{bmatrix}{\frac {1}{2}};-{\frac {1}{2}},{\frac {1}{2}},1;{\frac {1}{2}},1;\\1,{\frac {3}{2}};1-n;;\\-k^{2}\cos \phi ,\cos ^{2}\phi \end{bmatrix}}+{\frac {\pi m^{1-n}\cos \phi }{8}}F_{3\times 1\times 1}^{2\times 1\times 1}{\begin{bmatrix}{\frac {1}{2}},{\frac {3}{2}},2;{\frac {1}{2}},1;\\2,2-n;1-n;{\frac {3}{2}};{\frac {3}{2}};\\-k^{2}\cos ^{2}\phi ,k^{2}\end{bmatrix}}\,\!}$

第三类不完全椭圆积分${\displaystyle \Pi \,\!}$是

${\displaystyle \Pi (n;\phi |m)=\int _{0}^{\phi }{\frac {{\rm {d}}\theta }{(1-n\sin ^{2}\theta ){\sqrt {1-(\sin \theta \sin o\!\varepsilon )^{2}}}}},\,\!}$

或者

${\displaystyle \Pi (n;\phi |m)=\int _{0}^{\sin \phi }{\frac {{\rm {d}}t}{(1-nt^{2}){\sqrt {(1-k^{2}t^{2})(1-t^{2})}}}},\,\!}$

或者

${\displaystyle \Pi (n;\phi |m)=\int _{0}^{F(\phi |m)}{\frac {{\rm {d}}w}{1-n{\textrm {sn}}^{2}(w|m)}}.\;\,\!}$

数字${\displaystyle n\,}$称为特征数，可以取任意值，和其它参数独立。但是要注意${\displaystyle \Pi (1;{\frac {\pi }{2}}|m)\,\!}$对于任意${\displaystyle m\,\!}$是无穷的。

${\displaystyle \Pi (n;\phi _{1},k)+\Pi (n;\phi _{2},k)=\Pi \left[n;\arccos {\frac {\cos \phi _{1}\cos \phi _{2}-\sin \phi _{1}\sin \phi _{2}{\sqrt {(1-k^{2}\sin ^{2}\phi _{1})(1-k^{2}\sin ^{2}\phi _{2})}}}{1-k^{2}\sin ^{2}\phi _{1}\sin ^{2}\phi _{2}}},k\right]-{\sqrt {\frac {n}{(1-n)(n-k^{2})}}}\arctan {\frac {{\sqrt {(1-n)n(n-k^{2})}}\sin \arccos {\frac {\cos \phi _{1}\cos \phi _{2}-\sin \phi _{1}\sin \phi _{2}{\sqrt {(1-k^{2}\sin ^{2}\phi _{1})(1-k^{2}\sin ^{2}\phi _{2})}}}{1-k^{2}\sin ^{2}\phi _{1}\sin ^{2}\phi _{2}}}\sin \phi _{1}\sin \phi _{2}}{{\frac {n\cos \phi _{1}\cos \phi _{2}-n\sin \phi _{1}\sin \phi _{2}{\sqrt {(1-k^{2}\sin ^{2}\phi _{1})(1-k^{2}\sin ^{2}\phi _{2})}}}{1-k^{2}\sin ^{2}\phi _{1}\sin ^{2}\phi _{2}}}{\sqrt {1-k^{2}\sin ^{2}\arccos {\frac {\cos \phi _{1}\cos \phi _{2}-\sin \phi _{1}\sin \phi _{2}{\sqrt {(1-k^{2}\sin ^{2}\phi _{1})(1-k^{2}\sin ^{2}\phi _{2})}}}{1-k^{2}\sin ^{2}\phi _{1}\sin ^{2}\phi _{2}}}}}\sin \phi _{1}\sin \phi _{2}+1-n\sin ^{2}\arccos {\frac {\cos \phi _{1}\cos \phi _{2}-\sin \phi _{1}\sin \phi _{2}{\sqrt {(1-k^{2}\sin ^{2}\phi _{1})(1-k^{2}\sin ^{2}\phi _{2})}}}{1-k^{2}\sin ^{2}\phi _{1}\sin ^{2}\phi _{2}}}}}}$

${\displaystyle {\frac {\partial }{\partial n}}\Pi (n;\phi ,k)={\frac {1}{2(k^{2}-n)(n-1)}}\left[E(\phi ;k)+{\frac {(k^{2}-n)F(\phi ;k)}{n}}+{\frac {(n^{2}-k^{2})\Pi (n;\phi ,k)}{n}}-{\frac {n{\sqrt {1-k^{2}\sin \phi }}\sin 2\phi }{2(1-n\sin ^{2}\phi )}}\right]}$

${\displaystyle {\frac {{\partial }^{m}}{\partial n^{m}}}\Pi (n;\phi ,k)={\frac {\sin \phi }{n^{m}}}\sum _{q=0}^{\infty }{\frac {q!(n\sin ^{2}\phi )^{q}}{(2q+1)\Gamma (q-m+1)}}F_{1}\left(q+{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}};q+{\frac {3}{2}};\sin ^{2}\phi ,k^{2}\sin ^{2}\phi \right)}$

${\displaystyle {\frac {\partial }{\partial \phi }}\Pi (n;\phi ,k)={\frac {1}{(1-k^{2}\sin ^{2}\phi )}}\!}$

${\displaystyle {\frac {\partial }{\partial k}}\Pi (n;\phi ,k)={\frac {k}{n-k^{2}}}\left[{\frac {E(\phi ;k)}{k^{2}-1}}+\Pi (n;\phi ,k)-{\frac {k^{2}\sin 2\phi }{2(k^{2}-1){\sqrt {1-k^{2}\sin ^{2}\phi }}}}\right]\!}$

${\displaystyle \Pi (n;\phi ,1)={\frac {1}{2n-2}}\left[{\sqrt {n}}\ln {\frac {1+{\sqrt {n}}\sin \phi }{1-{\sqrt {n}}\sin \phi }}-2\ln(\sec \phi +\tan \phi )\right]\!}$

${\displaystyle -{\frac {\pi }{2}}\leq \Re (\phi )\leq {\frac {\pi }{2}}\!}$

${\displaystyle \Pi (0;\phi ,k)=F(\phi ,k)\!}$

${\displaystyle \Pi (n;\phi ,0)={\frac {{\rm {arctanh}}({\sqrt {n-1}}\tan \phi )}{\sqrt {n-1}}}\!}$

${\displaystyle -{\frac {\pi }{2}}\leq \Re (\phi )\leq {\frac {\pi }{2}}\!}$

${\displaystyle \Pi (n;\phi ,{\sqrt {n}})={\frac {1}{1-n}}\left[E(\phi ,{\sqrt {n}})-{\frac {n\sin 2\phi }{2{\sqrt {1-n\sin ^{2}\phi }}}}\right]\!}$

${\displaystyle \Pi \left(n;{\frac {1}{k}},k\right)={\frac {1}{k}}\Pi \left({\frac {n}{k^{2}}},{\frac {1}{k}}\right)\!}$

${\displaystyle \Pi \left(1;\phi ,k\right)={\frac {{\sqrt {1-k^{2}\sin ^{2}\phi }}\tan \phi -E(\phi ,k)}{1-k^{2}}}+F(\phi ,k)\!}$

如果幅度为${\displaystyle {\frac {\pi }{2}}\,}$或者${\displaystyle x=1\,}$，则称椭圆积分为完全的。 第一类完全椭圆积分${\displaystyle K\,}$可以定義为

${\displaystyle K(k)=\int _{0}^{\frac {\pi }{2}}{\frac {{\rm {d}}\theta }{\sqrt {1-k^{2}\sin ^{2}\theta }}}}$

或者

${\displaystyle K(k)=\int _{0}^{1}{\frac {{\rm {d}}t}{\sqrt {(1-t^{2})(1-k^{2}t^{2})}}}.\!}$

它是第一类不完全椭圆积分的特例：

${\displaystyle K(k)=F(1;\,k)=F\left({\frac {\pi }{2}}\,|\,k^{2}\right)\!}$

这个特例可以表达为幂级数

${\displaystyle K(k)={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left[{\frac {(2n)!}{2^{2n}n!^{2}}}\right]^{2}k^{2n}\!}$

它等价于

${\displaystyle K(k)={\frac {\pi }{2}}\left\{1+\left({\frac {1}{2}}\right)^{2}k^{2}+\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}k^{4}+\cdots +\left[{\frac {(2n-1)!!}{(2n)!!}}\right]^{2}k^{2n}+\cdots \right\}.\!}$

其中${\displaystyle n!!\,}$表示双阶乘。利用高斯的超几何函数，第一类完全椭圆积分可以表达为

${\displaystyle K(k)={\frac {\pi }{2}}\,_{2}F_{1}\left({\frac {1}{2}},{\frac {1}{2}};1;k^{2}\right).\,\!}$

第一类完全椭圆积分有时称为四分周期。它可以利用算术几何平均值來快速计算。

${\displaystyle K(k)={\frac {\frac {\pi }{2}}{\mathrm {agm} (1,{\sqrt {1-k^{2}}})}}.}$

${\displaystyle \Re \left[K(x+y{\rm {i}})\right]={\frac {\pi }{2}}F_{2\times 1\times 1}^{4\times 0\times 0}{\begin{bmatrix}{\frac {3}{4}},{\frac {3}{4}},{\frac {5}{4}},{\frac {5}{4}},;;;\\1,{\frac {3}{2}};{\frac {1}{2}};{\frac {3}{2}};\\-y^{2},x^{2}\end{bmatrix}}+{\frac {\pi }{8}}xF_{2\times 1\times 1}^{4\times 0\times 0}{\begin{bmatrix}{\frac {1}{4}},{\frac {1}{4}},{\frac {3}{4}},{\frac {3}{4}},;;;\\1,{\frac {1}{2}};{\frac {1}{2}};{\frac {1}{2}};\\-y^{2},x^{2}\end{bmatrix}}\,\!}$

${\displaystyle \Im \left[K(x+y{\rm {i}})\right]={\frac {\pi }{8}}yF_{2\times 1\times 1}^{4\times 0\times 0}{\begin{bmatrix}{\frac {3}{4}},{\frac {5}{4}},{\frac {3}{4}},{\frac {5}{4}},;;;\\1,{\frac {3}{2}};{\frac {3}{2}};{\frac {1}{2}};\\-y^{2},x^{2}\end{bmatrix}}+{\frac {9}{64}}\pi xyF_{2\times 1\times 1}^{4\times 0\times 0}{\begin{bmatrix}{\frac {5}{4}},{\frac {7}{4}},{\frac {7}{4}},{\frac {5}{4}},;;;\\2,{\frac {3}{2}};{\frac {3}{2}};{\frac {3}{2}};\\-y^{2},x^{2}\end{bmatrix}}\,\!}$

${\displaystyle K(\pm \infty )=0\,}$

${\displaystyle K(\pm {\rm {i}}\infty )=0\,}$

${\displaystyle K(0)={\frac {\pi }{2}}\!}$

${\displaystyle K(1)=\infty \!}$

${\displaystyle K({\frac {\sqrt {2}}{2}})={\frac {8\pi }{\Gamma ^{2}\left(-{\frac {1}{4}}\right)}}{\sqrt {\pi }}\,}$

${\displaystyle K\left({\sqrt {17-12{\sqrt {2}}}}\right)={\frac {(4+2{\sqrt {2}})\pi }{\Gamma ^{2}\left(-{\frac {1}{4}}\right)}}{\sqrt {\pi }}\,}$

${\displaystyle K\left({\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}\right)={\frac {{\sqrt[{3}]{4}}\cdot {\sqrt[{4}]{3}}}{8\pi }}\Gamma ^{3}\left({\frac {1}{3}}\right)\,}$

${\displaystyle K\left({\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}\right)={\frac {{\sqrt[{3}]{4}}\cdot {\sqrt[{4}]{27}}}{8\pi }}\Gamma ^{3}\left({\frac {1}{3}}\right)\,}$

${\displaystyle K(i)={\frac {\sqrt {2\pi }}{8\pi }}\Gamma ^{2}\left({\frac {1}{4}}\right)\,}$

${\displaystyle K({\sqrt {2}})={\frac {4{\sqrt {2\pi }}\pi }{\Gamma ^{2}\left({\frac {1}{4}}\right)}}+{\frac {4{\sqrt {2\pi }}\pi }{\Gamma ^{2}\left({\frac {1}{4}}\right)}}{\rm {i}}\,}$

${\displaystyle K({\rm {i}}k)={\frac {1}{\sqrt {k^{2}+1}}}K\left({\sqrt {\frac {k^{2}}{k^{2}+1}}}\right)\,}$

其中

${\displaystyle \Gamma \left({\frac {1}{4}}\right)\approx 3.62561\,}$

${\displaystyle \Gamma \left({\frac {1}{3}}\right)\approx 2.67893\,}$

第一类完全椭圆积分满足

${\displaystyle E(k)K'(k)+E'(k)K(k)-K(k)K'(k)={\frac {\pi }{2}}\,}$

${\displaystyle {\frac {\rm {d}}{{\rm {d}}k}}K^{n}(k)={\frac {nK^{n-1}(k)E(k)}{2k(1-k)}}-{\frac {nK^{n}(k)}{2k}}}$

${\displaystyle K(k^{2})\approx {\frac {\pi }{2}}+{\frac {\pi }{8}}{\frac {k^{2}}{1-k^{2}}}-{\frac {\pi }{16}}{\frac {k^{4}}{1-k^{2}}}}$

這個近似在k<1/2時相對誤差小於3×10⁻⁴，若只保留前兩項則誤差在k<1/2時小於0.01

此函數滿足以下微分方程

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} k}}\left[k(1-k^{2}){\frac {\mathrm {d} K(k)}{\mathrm {d} k}}\right]=kK(k)}$

此微分方程之另一解為${\displaystyle K({\sqrt {1-k^{2}}})}$，此解滿足以下關係。

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} k}}K({\sqrt {1-k^{2}}})={\frac {E(k)}{k(1-k^{2})}}-{\frac {K(k)}{k}}}$.

第二类完全椭圆积分 ${\displaystyle E\,}$可以定义为

${\displaystyle E(k)=\int _{0}^{\frac {\pi }{2}}{\sqrt {1-k^{2}\sin ^{2}\theta }}\ {\rm {d}}\theta \!}$

或者

${\displaystyle E(k)=\int _{0}^{1}{\frac {\sqrt {1-k^{2}t^{2}}}{\sqrt {1-t^{2}}}}\ {\rm {d}}t.\!}$

它是第二类不完全椭圆积分的特殊情况：

${\displaystyle E(k)=E(1;\,k)=E({\frac {\pi }{2}}\,|\,k^{2})\!}$

它可以用幂级数表达

${\displaystyle E(k)={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left[{\frac {(2n)!}{2^{2n}n!^{2}}}\right]^{2}{\frac {k^{2n}}{1-2n}}\!}$

也就是

${\displaystyle E(k)={\frac {\pi }{2}}\left\{1-\left({\frac {1}{2}}\right)^{2}{\frac {k^{2}}{1}}-\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}{\frac {k^{4}}{3}}-\cdots -\left[{\frac {\left(2n-1\right)!!}{\left(2n\right)!!}}\right]^{2}{\frac {k^{2n}}{2n-1}}-\cdots \right\}.\!}$

用高斯超几何函数表示的话，第二类完全椭圆积分可以写作

${\displaystyle E(k)={\frac {\pi }{2}}\,_{2}F_{1}\left(-{\frac {1}{2}},{\frac {1}{2}};1;k^{2}\right).\,\!}$

有如下性质

${\displaystyle E({\frac {n\pi }{2}};k)=nE(k)\,\!}$

${\displaystyle n\in \mathbb {Z} \,\!}$

${\displaystyle E(x+y{\rm {i}})=\left\{{\frac {\pi }{2}}F_{2\times 1\times 1}^{4\times 0\times 0}{\begin{bmatrix}{\frac {3}{4}},{\frac {5}{4}},{\frac {1}{4}},{\frac {3}{4}},;-;-;\\1,{\frac {3}{2}};{\frac {1}{2}};{\frac {3}{2}};\\-y^{2},x^{2}\end{bmatrix}}-{\frac {\pi }{8}}xF_{2\times 1\times 1}^{4\times 0\times 0}{\begin{bmatrix}{\frac {1}{4}},{\frac {3}{4}},-{\frac {1}{4}},{\frac {1}{4}},;-;-;\\1,{\frac {1}{2}};{\frac {1}{2}};{\frac {1}{2}};\\-y^{2},x^{2}\end{bmatrix}}\right\}+{\rm {i}}\left\{-{\frac {\pi }{8}}yF_{2\times 1\times 1}^{4\times 0\times 0}{\begin{bmatrix}{\frac {3}{4}},{\frac {5}{4}},{\frac {1}{4}},{\frac {3}{4}},;-;-;\\1,{\frac {3}{2}};{\frac {1}{2}};{\frac {3}{2}};\\-y^{2},x^{2}\end{bmatrix}}-{\frac {3}{64}}\pi xyF_{2\times 1\times 1}^{4\times 0\times 0}{\begin{bmatrix}{\frac {5}{4}},{\frac {7}{4}},{\frac {3}{4}},{\frac {5}{4}},;-;-;\\2,{\frac {3}{2}};{\frac {3}{2}};{\frac {3}{2}};\\-y^{2},x^{2}\end{bmatrix}}\right\}\,\!}$

${\displaystyle E(0)={\frac {\pi }{2}}\!}$

${\displaystyle E(1)=1\!}$

${\displaystyle E(\infty )={\rm {i}}\infty \,}$

${\displaystyle E(-\infty )=\infty \,}$

${\displaystyle E({\rm {i}}\infty )=({\frac {\sqrt {2}}{2}}-{\frac {\sqrt {2}}{2}}{\rm {i}})\infty \,}$

${\displaystyle E({\rm {i}})={\frac {\sqrt {2\pi }}{2\pi }}\Gamma ^{2}\left({\frac {3}{4}}\right)+{\frac {{\sqrt {2\pi }}{\pi }^{2}}{4\pi \Gamma ^{2}\left({\frac {3}{4}}\right)}}={\frac {\pi {\sqrt {2\pi }}}{\Gamma ^{2}\left({\frac {1}{4}}\right)}}+{\frac {\sqrt {2\pi }}{8\pi }}\Gamma ^{2}\left({\frac {1}{4}}\right)\,}$

${\displaystyle E(-{\rm {i}}\infty )=({\frac {\sqrt {2}}{2}}+{\frac {\sqrt {2}}{2}}{\rm {i}})\infty \,}$

${\displaystyle E\left({\tfrac {\sqrt {2}}{2}}\right)=\pi ^{\frac {3}{2}}\Gamma \left({\tfrac {1}{4}}\right)^{-2}+{\tfrac {1}{8{\sqrt {\pi }}}}\Gamma \left({\tfrac {1}{4}}\right)^{2}}$

${\displaystyle E\left({\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}\right)={\frac {{\sqrt[{3}]{2}}\cdot \ {\sqrt[{4}]{3}}}{3\Gamma ^{3}\left({\frac {1}{3}}\right)}}{\pi }^{2}+{\frac {{\sqrt[{3}]{4}}\left(3{\sqrt[{4}]{3}}+{\sqrt[{4}]{27}}\right)}{48{\pi }}}\Gamma ^{3}\left({\frac {1}{3}}\right)\!}$

${\displaystyle E\left({\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}\right)={\frac {{\sqrt[{3}]{2}}\cdot \ {\sqrt[{4}]{27}}}{3\Gamma ^{3}\left({\frac {1}{3}}\right)}}{\pi }^{2}+{\frac {{\sqrt[{3}]{4}}\left({\sqrt[{4}]{27}}-{\sqrt[{4}]{3}}\right)}{16{\pi }}}\Gamma ^{3}\left({\frac {1}{3}}\right)\!}$

${\displaystyle E({\sqrt {2}}-1)={\frac {\sqrt {\pi }}{8}}\left[{\frac {\Gamma ({\frac {1}{8}})}{\Gamma ({\frac {5}{8}})}}+{\frac {\Gamma ({\frac {5}{8}})}{\Gamma ({\frac {9}{8}})}}\right]\!}$

${\displaystyle E({\sqrt {2}})={\sqrt {\frac {1}{2\pi }}}\Gamma ^{2}\left({\frac {3}{4}}\right)+{\sqrt {\frac {1}{2\pi }}}\Gamma ^{2}\left({\frac {3}{4}}\right){\rm {i}}}$

其中

${\displaystyle \Gamma \left({\frac {1}{8}}\right)\approx 7.53394\,}$

${\displaystyle \Gamma \left({\frac {5}{8}}\right)\approx 1.43452\,}$

${\displaystyle \Gamma \left({\frac {9}{8}}\right)\approx 0.94174\,}$

${\displaystyle \Gamma \left({\frac {3}{4}}\right)\approx 1.22541\,}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} k}}E(k)={\frac {E(k)-K(k)}{k}}}$

${\displaystyle \int E(k){\rm {d}}k={\frac {2}{3}}\left[kK(k)-K(k)+kE(k)+E(k)\right]}$

${\displaystyle (k^{2}-1){\frac {\mathrm {d} }{\mathrm {d} k}}\left[k\;{\frac {\mathrm {d} E(k)}{\mathrm {d} k}}\right]=kE(k)}$

此微分方程之另解為${\displaystyle E({\sqrt {1-k^{2}}})-K({\sqrt {1-k^{2}}})}$。

第三类完全椭圆积分${\displaystyle \Pi \,}$可以定义为

${\displaystyle \Pi (n,k)=\int _{0}^{\frac {\pi }{2}}{\frac {\ {\rm {d}}\theta }{(1-n\sin ^{2}\theta ){\sqrt {1-k^{2}\sin ^{2}\theta }}}}}$

注意有时第三类椭圆积分被定义为带相反符号的${\displaystyle n\,}$，也即

${\displaystyle \Pi '(n,k)=\int _{0}^{\frac {\pi }{2}}{\frac {\ {\rm {d}}\theta }{(1+n\sin ^{2}\theta ){\sqrt {1-k'^{2}\sin ^{2}\theta }}}}.}$

用阿佩尔函数可表示为

${\displaystyle \Pi (m,n)={\frac {\pi }{2}}F_{1}\left({\frac {1}{2}};1,{\frac {1}{2}};1;m,n\right)\,}$

第三类完全椭圆积分和第一类椭圆积分之间的关系

${\displaystyle \Pi \left[{\frac {(1+x)(1-3x)}{(1-x)(1+3x)}},{\frac {(1+x)^{3}(1-3x)}{(1-x)^{3}(1+3x)}}\right]-{\frac {1+3x}{6x}}K\left[{\frac {(1+x)^{3}(1-3x)}{(1-x)^{3}(1+3x)}}\right]=\,}$

${\displaystyle {\begin{cases}0&{\mbox{for }}0<x<1\!\,\\-{\frac {\pi (x-1){\sqrt {(x-1)(1+3x)}}}{12x}}&{\mbox{for }}x<0,x>1\!\,\\\end{cases}}}$

如

${\displaystyle K\left({\frac {\sqrt {2}}{2}}\right)={\frac {\sqrt {\pi }}{4\pi }}\Gamma ^{2}\left({\frac {1}{4}}\right)={\frac {3-{\sqrt {6{\sqrt {3}}-9}}}{2}}\Pi \left({\frac {1-{\sqrt {2{\sqrt {3}}-3}}}{2}},{\frac {1}{2}}\right)\,}$

${\displaystyle ={\frac {3+{\sqrt {6{\sqrt {3}}-9}}}{2}}\Pi \left({\frac {1+{\sqrt {2{\sqrt {3}}-3}}}{2}},{\frac {1}{2}}\right)-\pi {\sqrt {2+{\sqrt {3}}+{\sqrt {7+{\frac {38}{9}}{\sqrt {3}}}}}}\,}$

${\displaystyle {\frac {\partial }{\partial n}}\Pi (n,k)={\frac {1}{2(k^{2}-n)(n-1)}}\left[E(k)+{\frac {(k^{2}-n)K(k)}{n}}+{\frac {(n^{2}-k^{2})\Pi (n,k)}{n}}\right]}$

${\displaystyle {\frac {\partial }{\partial k}}\Pi (n,k)={\frac {k}{n-k^{2}}}\left[{\frac {E(k)}{k^{2}-1}}+\Pi (n,k)\right]}$

${\displaystyle \Pi (0,0)={\frac {\pi }{2}}\,}$

${\displaystyle \Pi (n,0)={\frac {\pi }{2{\sqrt {1-n}}}}\,}$

${\displaystyle \Pi (n,1)=-{\frac {\infty }{\operatorname {sgn} {n-1}}}\,}$

${\displaystyle \Pi (n,{\sqrt {n}})={\frac {E(n)}{1-n}}\,}$

${\displaystyle \Pi (0,{\sqrt {n}})=K(n)\,}$

${\displaystyle \Pi (\pm \infty ,{\sqrt {n}})=0\,}$

${\displaystyle \Pi (n,\pm \infty )=0\,}$

勒讓得關係指出了第一类和第二类完全椭圆积分之间的联系:

${\displaystyle K(k)E\left({\sqrt {1-k^{2}}}\right)+E(k)K\left({\sqrt {1-k^{2}}}\right)-K(k)K\left({\sqrt {1-k^{2}}}\right)={\frac {\pi }{2}}.}$

• 椭圆曲线
• 施瓦茨-克里斯托费尔映射
• 雅可比橢圓函數
• 魏爾斯特拉斯橢圓函數
• Θ函數
• 算术-几何平均数

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 0-486-61272-4. (See chapter 17).
• Harris Hancock Lectures on the theory of Elliptic functions (New York, J. Wiley & sons, 1910)
• Alfred George Greenhill The applications of elliptic functions (New York, Macmillan, 1892)
• Louis V. King On The Direct Numerical Calculation Of Elliptic Functions And Integrals (Cambridge University Press, 1924)