## 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:

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

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:

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

### 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)
-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

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)
-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 incomplete integral of the second kind for various values of amplitude φ are depicted in the following figure: