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Elliptic Integral of the 3rd Kind

- By Dr. Minas E. Lemonis, PhD - Updated: October 30, 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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Result:

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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

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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
-0.9990.77191.1771.9482.644.496-3.242-0.8537-0.384
-0.990.66660.98491.5642.0663.357-2.066-0.5619-0.2545
-0.980.63470.92661.4491.8943.021-1.701-0.472-0.2147
-0.970.61590.89251.3821.7952.828-1.484-0.4186-0.191
-0.960.60250.86821.3341.7252.693-1.329-0.3802-0.1741
-0.950.59210.84941.2981.6712.59-1.208-0.3502-0.1608
-0.90.55970.79131.1861.5072.281-0.8352-0.2554-0.1187
-0.80.52760.73461.0781.3521.995-0.4841-0.16-0.07571
-0.70.50960.70311.0191.2681.846-0.3036-0.1061-0.05094
-0.60.49770.68240.98131.2151.751-0.1933-0.07043-0.03421
-0.50.48920.66790.95481.1771.686-0.1207-0.04539-0.02225
-0.40.48310.65740.93591.1511.64-0.07137-0.02748-0.01357
-0.30.47880.650.92261.1321.608-0.0379-0.01484-0.007372
-0.20.47580.64510.91371.121.587-0.01619-0.006418-0.0032
-0.10.47420.64220.90861.1131.575-0.003957-0.001579-0.000789
00.47360.64130.90691.1111.571000
0.10.47420.64220.90861.1131.575-0.003957-0.001579-0.000789
0.20.47580.64510.91371.121.587-0.01619-0.006418-0.0032
0.30.47880.650.92261.1321.608-0.0379-0.01484-0.007372
0.40.48310.65740.93591.1511.64-0.07137-0.02748-0.01357
0.50.48920.66790.95481.1771.686-0.1207-0.04539-0.02225
0.60.49770.68240.98131.2151.751-0.1933-0.07043-0.03421
0.70.50960.70311.0191.2681.846-0.3036-0.1061-0.05094
0.80.52760.73461.0781.3521.995-0.4841-0.16-0.07571
0.90.55970.79131.1861.5072.281-0.8352-0.2554-0.1187
0.950.59210.84941.2981.6712.59-1.208-0.3502-0.1608
0.960.60250.86821.3341.7252.693-1.329-0.3802-0.1741
0.970.61590.89251.3821.7952.828-1.484-0.4186-0.191
0.980.63470.92661.4491.8943.021-1.701-0.472-0.2147
0.990.66660.98491.5642.0663.357-2.066-0.5619-0.2545
0.9990.77191.1771.9482.644.496-3.242-0.8537-0.384
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)
-10.54930.88141.3170.8814-0.8814-∞-0.8814
-0.9990.54920.88111.3164.4960.8811-0.8811-4.496-0.8811
-0.990.54870.87871.3073.3570.8787-0.8787-3.357-0.8787
-0.90.54390.85791.2332.2810.8579-0.8579-2.281-0.8579
-0.80.53930.83961.1791.9950.8396-0.8396-1.995-0.8396
-0.60.53210.81351.1111.7510.8135-0.8135-1.751-0.8135
-0.40.52730.79731.0731.640.7973-0.7973-1.64-0.7973
-0.20.52450.78831.0531.5870.7883-0.7883-1.587-0.7883
00.52360.78541.0471.5710.7854-0.7854-1.571-0.7854
0.20.52450.78831.0531.5870.7883-0.7883-1.587-0.7883
0.40.52730.79731.0731.640.7973-0.7973-1.64-0.7973
0.60.53210.81351.1111.7510.8135-0.8135-1.751-0.8135
0.80.53930.83961.1791.9950.8396-0.8396-1.995-0.8396
0.90.54390.85791.2332.2810.8579-0.8579-2.281-0.8579
0.990.54870.87871.3073.3570.8787-0.8787-3.357-0.8787
0.9990.54920.88111.3164.4960.8811-0.8811-4.496-0.8811
10.54930.88141.3170.8814-0.8814-∞-0.8814

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

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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)
-10.6081.1482.3911.148-1.148-∞-1.148
-0.9990.60791.1472.3881.147-1.147-∞-1.147
-0.990.60731.1442.3631.144-1.144-∞-1.144
-0.90.60151.1112.1741.111-1.111-∞-1.111
-0.80.5961.0832.0391.083-1.083-∞-1.083
-0.60.58751.0431.8781.043-1.043-∞-1.043
-0.40.58171.0181.7911.018-1.018-∞-1.018
-0.20.57841.0041.7461.004-1.004-∞-1.004
00.577411.7321-1-∞-1
0.20.57841.0041.7461.004-1.004-∞-1.004
0.40.58171.0181.7911.018-1.018-∞-1.018
0.60.58751.0431.8781.043-1.043-∞-1.043
0.80.5961.0832.0391.083-1.083-∞-1.083
0.90.60151.1112.1741.111-1.111-∞-1.111
0.990.60731.1442.3631.144-1.144-∞-1.144
0.9990.60791.1472.3881.147-1.147-∞-1.147
10.6081.1482.3911.148-1.148-∞-1.148

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)
-10.50650.74841.0150.7484-0.7484-∞-0.7484
-0.9990.50640.74821.0152.640.7482-0.7482-2.64-0.7482
-0.990.5060.74651.0092.0660.7465-0.7465-2.066-0.7465
-0.90.50180.73080.96131.5070.7308-0.7308-1.507-0.7308
-0.80.49780.71690.92541.3520.7169-0.7169-1.352-0.7169
-0.60.49160.69710.88021.2150.6971-0.6971-1.215-0.6971
-0.40.48740.68460.85451.1510.6846-0.6846-1.151-0.6846
-0.20.4850.67770.84091.120.6777-0.6777-1.12-0.6777
00.48420.67550.83661.1110.6755-0.6755-1.111-0.6755
0.20.4850.67770.84091.120.6777-0.6777-1.12-0.6777
0.40.48740.68460.85451.1510.6846-0.6846-1.151-0.6846
0.60.49160.69710.88021.2150.6971-0.6971-1.215-0.6971
0.80.49780.71690.92541.3520.7169-0.7169-1.352-0.7169
0.90.50180.73080.96131.5070.7308-0.7308-1.507-0.7308
0.990.5060.74651.0092.0660.7465-0.7465-2.066-0.7465
0.9990.50640.74821.0152.640.7482-0.7482-2.64-0.7482
10.50650.74841.0150.7484-0.7484-∞-0.7484

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)
-10.66970.1964-0.0098090.1964-0.1964-∞-0.1964
-0.9990.66970.1966-0.00921-0.85370.1966-0.19660.8537-0.1966
-0.990.66950.1987-0.003963-0.56190.1987-0.19870.5619-0.1987
-0.90.66810.2170.038-0.25540.217-0.2170.2554-0.217
-0.80.66630.23250.07019-0.160.2325-0.23250.16-0.2325
-0.60.66310.25360.1106-0.070430.2536-0.25360.07043-0.2536
-0.40.66060.2660.1332-0.027480.266-0.2660.02748-0.266
-0.20.6590.27260.1449-0.0064180.2726-0.27260.006418-0.2726
00.65850.27470.148600.2747-0.27470-0.2747
0.20.6590.27260.1449-0.0064180.2726-0.27260.006418-0.2726
0.40.66060.2660.1332-0.027480.266-0.2660.02748-0.266
0.60.66310.25360.1106-0.070430.2536-0.25360.07043-0.2536
0.80.66630.23250.07019-0.160.2325-0.23250.16-0.2325
0.90.66810.2170.038-0.25540.217-0.2170.2554-0.217
0.990.66950.1987-0.003963-0.56190.1987-0.19870.5619-0.1987
0.9990.66970.1966-0.00921-0.85370.1966-0.19660.8537-0.1966
10.66970.1964-0.0098090.1964-0.1964-∞-0.1964

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

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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)
-10.4050.52210.62720.5221-0.5221-∞-0.5221
-0.9990.4050.5220.6271.1770.522-0.522-1.177-0.522
-0.990.40470.52110.62440.98490.5211-0.5211-0.9849-0.5211
-0.90.40190.51290.60380.79130.5129-0.5129-0.7913-0.5129
-0.80.39920.50550.58780.73460.5055-0.5055-0.7346-0.5055
-0.60.3950.49480.56720.68240.4948-0.4948-0.6824-0.4948
-0.40.39220.4880.55520.65740.488-0.488-0.6574-0.488
-0.20.39050.48430.54880.64510.4843-0.4843-0.6451-0.4843
00.390.4830.54680.64130.483-0.483-0.6413-0.483
0.20.39050.48430.54880.64510.4843-0.4843-0.6451-0.4843
0.40.39220.4880.55520.65740.488-0.488-0.6574-0.488
0.60.3950.49480.56720.68240.4948-0.4948-0.6824-0.4948
0.80.39920.50550.58780.73460.5055-0.5055-0.7346-0.5055
0.90.40190.51290.60380.79130.5129-0.5129-0.7913-0.5129
0.990.40470.52110.62440.98490.5211-0.5211-0.9849-0.5211
0.9990.4050.5220.6271.1770.522-0.522-1.177-0.522
10.4050.52210.62720.5221-0.5221-∞-0.5221

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)
-10.20090.07115-0.011820.07115-0.07115-∞-0.07115
-0.9990.2010.07127-0.01156-0.3840.07127-0.071270.384-0.07127
-0.990.20140.07233-0.009216-0.25450.07233-0.072330.2545-0.07233
-0.90.2050.081770.009852-0.11870.08177-0.081770.1187-0.08177
-0.80.20850.09020.02502-0.075710.0902-0.09020.07571-0.0902
-0.60.21360.10230.04497-0.034210.1023-0.10230.03421-0.1023
-0.40.2170.110.05669-0.013570.11-0.110.01357-0.11
-0.20.21890.11420.06297-0.00320.1142-0.11420.0032-0.1142
00.21950.11550.0649600.1155-0.11550-0.1155
0.20.21890.11420.06297-0.00320.1142-0.11420.0032-0.1142
0.40.2170.110.05669-0.013570.11-0.110.01357-0.11
0.60.21360.10230.04497-0.034210.1023-0.10230.03421-0.1023
0.80.20850.09020.02502-0.075710.0902-0.09020.07571-0.0902
0.90.2050.081770.009852-0.11870.08177-0.081770.1187-0.08177
0.990.20140.07233-0.009216-0.25450.07233-0.072330.2545-0.07233
0.9990.2010.07127-0.01156-0.3840.07127-0.071270.384-0.07127
10.20090.07115-0.011820.07115-0.07115-∞-0.07115

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)
-10.33940.41080.47060.4108-0.4108-∞-0.4108
-0.9990.33940.41070.47050.77190.4107-0.4107-0.7719-0.4107
-0.990.33910.41010.46890.66660.4101-0.4101-0.6666-0.4101
-0.90.33710.40480.45660.55970.4048-0.4048-0.5597-0.4048
-0.80.33520.40010.4470.52760.4001-0.4001-0.5276-0.4001
-0.60.33210.39310.43430.49770.3931-0.3931-0.4977-0.3931
-0.40.33010.38860.42690.48310.3886-0.3886-0.4831-0.3886
-0.20.32890.38610.42290.47580.3861-0.3861-0.4758-0.3861
00.32850.38530.42160.47360.3853-0.3853-0.4736-0.3853
0.20.32890.38610.42290.47580.3861-0.3861-0.4758-0.3861
0.40.33010.38860.42690.48310.3886-0.3886-0.4831-0.3886
0.60.33210.39310.43430.49770.3931-0.3931-0.4977-0.3931
0.80.33520.40010.4470.52760.4001-0.4001-0.5276-0.4001
0.90.33710.40480.45660.55970.4048-0.4048-0.5597-0.4048
0.990.33910.41010.46890.66660.4101-0.4101-0.6666-0.4101
0.9990.33940.41070.47050.77190.4107-0.4107-0.7719-0.4107
10.33940.41080.47060.4108-0.4108-∞-0.4108

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

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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