## Elliptic Integral of the 2nd Kind

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

 Type Complete Incomplete k = Result:

## About Elliptic Integrals of the Second Kind

### Definitions

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

where k is the elliptic modulus, with . Variable is the Jacobi's amplitude.

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

### Values

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

kE(k)
-11
-0.9991.004
-0.991.028
-0.981.05
-0.971.069
-0.961.087
-0.951.103
-0.91.172
-0.81.276
-0.71.356
-0.61.418
-0.51.467
-0.41.506
-0.31.535
-0.21.555
-0.11.567
01.571
0.11.567
0.21.555
0.31.535
0.41.506
0.51.467
0.61.418
0.71.356
0.81.276
0.91.172
0.951.103
0.961.087
0.971.069
0.981.05
0.991.028
0.9991.004
11

In the following table the values of the incomplete elliptic integral of the second kind are shown for a range of k and φ values:

kE(30°,k)E(45°,k)E(60°,k)E(90°,k)E(135°,k)E(225°,k)E(270°,k)E(315°,k)
-10.50.70710.86610.7071-0.7071-1-0.7071
-0.9990.50.70730.86651.0040.7073-0.7073-1.004-0.7073
-0.990.50050.70880.87051.0280.7088-0.7088-1.028-0.7088
-0.980.5010.71050.87481.050.7105-0.7105-1.05-0.7105
-0.970.50150.71220.87911.0690.7122-0.7122-1.069-0.7122
-0.960.50190.71390.88331.0870.7139-0.7139-1.087-0.7139
-0.950.50240.71550.88731.1030.7155-0.7155-1.103-0.7155
-0.90.50460.72330.90651.1720.7233-0.7233-1.172-0.7233
-0.80.50870.73710.93951.2760.7371-0.7371-1.276-0.7371
-0.70.51230.7490.96671.3560.749-0.749-1.356-0.749
-0.60.51530.75890.98921.4180.7589-0.7589-1.418-0.7589
-0.50.51790.76721.0081.4670.7672-0.7672-1.467-0.7672
-0.40.520.77381.0221.5060.7738-0.7738-1.506-0.7738
-0.30.52160.77891.0331.5350.7789-0.7789-1.535-0.7789
-0.20.52270.78251.0411.5550.7825-0.7825-1.555-0.7825
-0.10.52340.78471.0461.5670.7847-0.7847-1.567-0.7847
00.52360.78541.0471.5710.7854-0.7854-1.571-0.7854
0.10.52340.78471.0461.5670.7847-0.7847-1.567-0.7847
0.20.52270.78251.0411.5550.7825-0.7825-1.555-0.7825
0.30.52160.77891.0331.5350.7789-0.7789-1.535-0.7789
0.40.520.77381.0221.5060.7738-0.7738-1.506-0.7738
0.50.51790.76721.0081.4670.7672-0.7672-1.467-0.7672
0.60.51530.75890.98921.4180.7589-0.7589-1.418-0.7589
0.70.51230.7490.96671.3560.749-0.749-1.356-0.749
0.80.50870.73710.93951.2760.7371-0.7371-1.276-0.7371
0.90.50460.72330.90651.1720.7233-0.7233-1.172-0.7233
0.950.50240.71550.88731.1030.7155-0.7155-1.103-0.7155
0.960.50190.71390.88331.0870.7139-0.7139-1.087-0.7139
0.970.50150.71220.87911.0690.7122-0.7122-1.069-0.7122
0.980.5010.71050.87481.050.7105-0.7105-1.05-0.7105
0.990.50050.70880.87051.0280.7088-0.7088-1.028-0.7088
0.9990.50.70730.86651.0040.7073-0.7073-1.004-0.7073
10.50.70710.86610.7071-0.7071-1-0.7071

The xy plots of the incomplete integral of the second kind for various values of amplitude φ are depicted in the following figure: