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：）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。

记法
--

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
----------

**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 （页面存档备份，存于）, {\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。

### 加法公式

{\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 {1-\cos \varphi \_{1}\sin \varphi \_{1}\cos \varphi \_{2}\sin \varphi \_{2}{\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)}
### 性质

{\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}}\,\!}
### 第一类不完全椭圆积分的导数

{\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
----------

**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 }}}\,\!}
### 加法公式

{\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]}
### 性质

{\displaystyle E(\phi +n\pi ;k)=E(\phi ;k)+2nE(k)\,\!}
{\displaystyle E(-\phi ;k)=-E(\phi ;k)\,\!}
### 第二类不完全椭圆积分的导数

{\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
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**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。

### 加法公式

{\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}}}}}}
### 第三类不完全椭圆积分的导数

{\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]\!}
### 特殊值

{\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)\!}

第一类完全椭圆积分
---------

第一类完全椭圆积分{\displaystyle K(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}}})}}.}
### 复数值

{\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}}\,\!}

  

### 特殊值

{\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}}\,}
### 导数

{\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}}}
### 漸近表示

{\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

### 微分方程

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
---------

第二类完全椭圆积分{\displaystyle E(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} \,\!}

  

### 复数值

{\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\}\,\!}
### 特殊值

{\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\,}
### 导数、積分及微分方程

{\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
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不同{\displaystyle n}值的第三类完全椭圆积分{\displaystyle \Pi (n,k)}
**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}}}}}}\,}
### 偏导数

{\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]}
### 特殊值

{\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\,}

函數關係
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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}}.}

参看
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* 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