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Elliptic Integral of the 3rd Kind

- By Dr. Minas E. Lemonis, PhD - Updated: March 3, 2019

This tool evaluates the complete or incomplete elliptic integral of the third kind: Π(k,n) or Π(φ,k,n) respectively. Select the desired type of the calculation and enter the appropriate arguments below.

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About Elliptic Integrals of the Third Kind

Definitions

The incomplete elliptic integral of the third kind is defined as:

\[ \Pi(\varphi,k,n) = \int_{0}^{\varphi}\frac{d\theta}{\left(1-n\sin^2\theta \right)\sqrt{1-k^2\sin^2\theta}} \]

where k is the elliptic modulus, with \(-1 \le k \le 1 \), and n is a parameter called elliptic characteristic, that can take any real value. Variable \(\varphi \) is the Jacobi's amplitude.

For n=0, the elliptic integral of the third kind is identical to the respective integral of the first kind:

\[ \Pi(\varphi,k,0) = \textrm{F}(\varphi,k) \]

The complete elliptic integral of the third kind is defined as:

\[ \begin{split} \Pi(k,n) & = \Pi(\frac{\pi}{2},k,n) \\ & = \int_{0}^{\frac{\pi}{2}}\frac{d\theta}{\left(1-n\sin^2\theta \right)\sqrt{1-k^2\sin^2\theta}} \end{split} \]

For n=1, the complete elliptic integral of the third kind becomes infinite for any k:

\[ \Pi(k,1) = \infty \]

Values

In the following table the values of the complete elliptic integral of the third kind are shown for a range of k and n values:

kΠ(k,-10)Π(k,-5)Π(k,-2)Π(k,-1)Π(k,0)Π(k,1)Π(k,2)Π(k,5)Π(k,10)
-1
-0.999
-0.99
-0.98
-0.97
-0.96
-0.95
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
0.96
0.97
0.98
0.99
0.999
1

The xy plots of the complete integrals of the third kind for various values of characteristic n are depicted in the following figure:

elliptic integral of the third kind xy plot/graph

The curve for n=1 is not drawn because, the integral becomes infinite. Also, for n>1 it is \(\Pi(0,n)=0\).

In the following table the values of the incomplete elliptic integral of the third kind are shown for n=0 and a range of k and φ values. These values are identical to the respective ones of the incomplete elliptic integral of the first kind because \(\Pi(\varphi,k,0) = \textrm{F}(\varphi,k)\).

kΠ(30°,k,0)Π(45°,k,0)Π(60°,k,0)Π(90°,k,0)Π(135°,k,0)Π(225°,k,0)Π(270°,k,0)Π(315°,k,0)
-1
-0.999
-0.99
-0.9
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0.9
0.99
0.999
1

The xy plots of the incomplete integrals of the third kind for characteristic n=0 are depicted in the following figure. The curves are identical to the respective ones of the first elliptic integral.

elliptic integral of the third kind xy plot/graph

In the following table the values of the incomplete elliptic integral of the third kind are shown for n=1 and a range of k and φ values:

kΠ(30°,k,1)Π(45°,k,1)Π(60°,k,1)Π(90°,k,1)Π(135°,k,1)Π(225°,k,1)Π(270°,k,1)Π(315°,k,1)
-1
-0.999
-0.99
-0.9
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0.9
0.99
0.999
1

The xy plots of the incomplete integrals of the third kind for characteristic n=1 are depicted in the following figure.

elliptic integral of the third kind xy plot/graph

In the following table the values of the incomplete elliptic integral of the third kind are shown for n=-1 and a range of k and φ values:

kΠ(30°,k,-1)Π(45°,k,-1)Π(60°,k,-1)Π(90°,k,-1)Π(135°,k,-1)Π(225°,k,-1)Π(270°,k,-1)Π(315°,k,-1)
-1
-0.999
-0.99
-0.9
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0.9
0.99
0.999
1

The xy plots of the incomplete integrals of the third kind for characteristic n=-1 are depicted in the following figure.

elliptic integral of the third kind xy plot/graph

In the following table the values of the incomplete elliptic integral of the third kind are shown for n=5 and a range of k and φ values:

kΠ(30°,k,5)Π(45°,k,5)Π(60°,k,5)Π(90°,k,5)Π(135°,k,5)Π(225°,k,5)Π(270°,k,5)Π(315°,k,5)
-1
-0.999
-0.99
-0.9
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0.9
0.99
0.999
1

The xy plots of the incomplete integrals of the third kind for characteristic n=5 are depicted in the following figure.

elliptic integral of the third kind xy plot/graph

In the following table the values of the incomplete elliptic integral of the third kind are shown for n=-5 and a range of k and φ values:

kΠ(30°,k,-5)Π(45°,k,-5)Π(60°,k,-5)Π(90°,k,-5)Π(135°,k,-5)Π(225°,k,-5)Π(270°,k,-5)Π(315°,k,-5)
-1
-0.999
-0.99
-0.9
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0.9
0.99
0.999
1

The xy plots of the incomplete integrals of the third kind for characteristic n=-5 are depicted in the following figure.

elliptic integral of the third kind xy plot/graph

In the following table the values of the incomplete elliptic integral of the third kind are shown for n=10 and a range of k and φ values:

kΠ(30°,k,10)Π(45°,k,10)Π(60°,k,10)Π(90°,k,10)Π(135°,k,10)Π(225°,k,10)Π(270°,k,10)Π(315°,k,10)
-1
-0.999
-0.99
-0.9
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0.9
0.99
0.999
1

The xy plots of the incomplete integrals of the third kind for characteristic n=10 are depicted in the following figure.

elliptic integral of the third kind xy plot/graph

In the following table the values of the incomplete elliptic integral of the third kind are shown for n=-10 and a range of k and φ values:

kΠ(30°,k,-10)Π(45°,k,-10)Π(60°,k,-10)Π(90°,k,-10)Π(135°,k,-10)Π(225°,k,-10)Π(270°,k,-10)Π(315°,k,-10)
-1
-0.999
-0.99
-0.9
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0.9
0.99
0.999
1

The xy plots of the incomplete integrals of the third kind for characteristic n=-10 are depicted in the following figure.

elliptic integral of the third kind xy plot/graph
See also
Evaluate Elliptic Integral of the first kind
Evaluate Elliptic Integral of the second kind
Evaluate Carlson's form of Elliptic Integrals
All evaluation tools