## Elliptic Integral of the 3rd Kind

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.

 Type Complete Incomplete k = n = Result:

## About Elliptic Integrals of the Third Kind

### Definitions

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

where k is the elliptic modulus, with , and n is a parameter called elliptic characteristic, that can take any real value. Variable 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:

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

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

### 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:

The curve for n=1 is not drawn because, the integral becomes infinite. Also, for n>1 it is .

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 .

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.

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.

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.

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.

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.

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.

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.