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Elliptic Integral of the 2nd Kind

- By Dr. Minas E. Lemonis, PhD - Updated: October 30, 2019

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
k =
Result:

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About Elliptic Integrals of the Second Kind

Definitions

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

\textrm{E}(\varphi,k) = \int_{0}^{\varphi} \sqrt{1-k^2\sin^2\theta}\, d\theta

where k is the elliptic modulus, with -1 \le k \le 1 . Variable \varphi is the Jacobi's amplitude.

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

\textrm{E}(k) = \textrm{E}(\frac{\pi}{2},k) = \int_{0}^{\frac{\pi}{2}} \sqrt{1-k^2\sin^2\theta}\, d\theta

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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:

elliptic integral of the second kind xy plot/graph

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See also
Evaluate Elliptic Integral of the first kind
Evaluate Elliptic Integral of the third kind
Evaluate Carlson's form of Elliptic Integrals
All evaluation tools