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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:
| k | E(k) |
|---|---|
| -1 | 1 |
| -0.999 | 1.004 |
| -0.99 | 1.028 |
| -0.98 | 1.05 |
| -0.97 | 1.069 |
| -0.96 | 1.087 |
| -0.95 | 1.103 |
| -0.9 | 1.172 |
| -0.8 | 1.276 |
| -0.7 | 1.356 |
| -0.6 | 1.418 |
| -0.5 | 1.467 |
| -0.4 | 1.506 |
| -0.3 | 1.535 |
| -0.2 | 1.555 |
| -0.1 | 1.567 |
| 0 | 1.571 |
| 0.1 | 1.567 |
| 0.2 | 1.555 |
| 0.3 | 1.535 |
| 0.4 | 1.506 |
| 0.5 | 1.467 |
| 0.6 | 1.418 |
| 0.7 | 1.356 |
| 0.8 | 1.276 |
| 0.9 | 1.172 |
| 0.95 | 1.103 |
| 0.96 | 1.087 |
| 0.97 | 1.069 |
| 0.98 | 1.05 |
| 0.99 | 1.028 |
| 0.999 | 1.004 |
| 1 | 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:
| k | E(30°,k) | E(45°,k) | E(60°,k) | E(90°,k) | E(135°,k) | E(225°,k) | E(270°,k) | E(315°,k) |
|---|---|---|---|---|---|---|---|---|
| -1 | 0.5 | 0.7071 | 0.866 | 1 | 0.7071 | -0.7071 | -1 | -0.7071 |
| -0.999 | 0.5 | 0.7073 | 0.8665 | 1.004 | 0.7073 | -0.7073 | -1.004 | -0.7073 |
| -0.99 | 0.5005 | 0.7088 | 0.8705 | 1.028 | 0.7088 | -0.7088 | -1.028 | -0.7088 |
| -0.98 | 0.501 | 0.7105 | 0.8748 | 1.05 | 0.7105 | -0.7105 | -1.05 | -0.7105 |
| -0.97 | 0.5015 | 0.7122 | 0.8791 | 1.069 | 0.7122 | -0.7122 | -1.069 | -0.7122 |
| -0.96 | 0.5019 | 0.7139 | 0.8833 | 1.087 | 0.7139 | -0.7139 | -1.087 | -0.7139 |
| -0.95 | 0.5024 | 0.7155 | 0.8873 | 1.103 | 0.7155 | -0.7155 | -1.103 | -0.7155 |
| -0.9 | 0.5046 | 0.7233 | 0.9065 | 1.172 | 0.7233 | -0.7233 | -1.172 | -0.7233 |
| -0.8 | 0.5087 | 0.7371 | 0.9395 | 1.276 | 0.7371 | -0.7371 | -1.276 | -0.7371 |
| -0.7 | 0.5123 | 0.749 | 0.9667 | 1.356 | 0.749 | -0.749 | -1.356 | -0.749 |
| -0.6 | 0.5153 | 0.7589 | 0.9892 | 1.418 | 0.7589 | -0.7589 | -1.418 | -0.7589 |
| -0.5 | 0.5179 | 0.7672 | 1.008 | 1.467 | 0.7672 | -0.7672 | -1.467 | -0.7672 |
| -0.4 | 0.52 | 0.7738 | 1.022 | 1.506 | 0.7738 | -0.7738 | -1.506 | -0.7738 |
| -0.3 | 0.5216 | 0.7789 | 1.033 | 1.535 | 0.7789 | -0.7789 | -1.535 | -0.7789 |
| -0.2 | 0.5227 | 0.7825 | 1.041 | 1.555 | 0.7825 | -0.7825 | -1.555 | -0.7825 |
| -0.1 | 0.5234 | 0.7847 | 1.046 | 1.567 | 0.7847 | -0.7847 | -1.567 | -0.7847 |
| 0 | 0.5236 | 0.7854 | 1.047 | 1.571 | 0.7854 | -0.7854 | -1.571 | -0.7854 |
| 0.1 | 0.5234 | 0.7847 | 1.046 | 1.567 | 0.7847 | -0.7847 | -1.567 | -0.7847 |
| 0.2 | 0.5227 | 0.7825 | 1.041 | 1.555 | 0.7825 | -0.7825 | -1.555 | -0.7825 |
| 0.3 | 0.5216 | 0.7789 | 1.033 | 1.535 | 0.7789 | -0.7789 | -1.535 | -0.7789 |
| 0.4 | 0.52 | 0.7738 | 1.022 | 1.506 | 0.7738 | -0.7738 | -1.506 | -0.7738 |
| 0.5 | 0.5179 | 0.7672 | 1.008 | 1.467 | 0.7672 | -0.7672 | -1.467 | -0.7672 |
| 0.6 | 0.5153 | 0.7589 | 0.9892 | 1.418 | 0.7589 | -0.7589 | -1.418 | -0.7589 |
| 0.7 | 0.5123 | 0.749 | 0.9667 | 1.356 | 0.749 | -0.749 | -1.356 | -0.749 |
| 0.8 | 0.5087 | 0.7371 | 0.9395 | 1.276 | 0.7371 | -0.7371 | -1.276 | -0.7371 |
| 0.9 | 0.5046 | 0.7233 | 0.9065 | 1.172 | 0.7233 | -0.7233 | -1.172 | -0.7233 |
| 0.95 | 0.5024 | 0.7155 | 0.8873 | 1.103 | 0.7155 | -0.7155 | -1.103 | -0.7155 |
| 0.96 | 0.5019 | 0.7139 | 0.8833 | 1.087 | 0.7139 | -0.7139 | -1.087 | -0.7139 |
| 0.97 | 0.5015 | 0.7122 | 0.8791 | 1.069 | 0.7122 | -0.7122 | -1.069 | -0.7122 |
| 0.98 | 0.501 | 0.7105 | 0.8748 | 1.05 | 0.7105 | -0.7105 | -1.05 | -0.7105 |
| 0.99 | 0.5005 | 0.7088 | 0.8705 | 1.028 | 0.7088 | -0.7088 | -1.028 | -0.7088 |
| 0.999 | 0.5 | 0.7073 | 0.8665 | 1.004 | 0.7073 | -0.7073 | -1.004 | -0.7073 |
| 1 | 0.5 | 0.7071 | 0.866 | 1 | 0.7071 | -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:
