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About Elliptic Integrals of the First Kind
Definitions
The incomplete elliptic integral of the first 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 first kind are shown for a range of k values:
| k | K(k) |
|---|---|
| -1 | ∞ |
| -0.999 | 4.496 |
| -0.99 | 3.357 |
| -0.98 | 3.021 |
| -0.97 | 2.828 |
| -0.96 | 2.693 |
| -0.95 | 2.59 |
| -0.9 | 2.281 |
| -0.8 | 1.995 |
| -0.7 | 1.846 |
| -0.6 | 1.751 |
| -0.5 | 1.686 |
| -0.4 | 1.64 |
| -0.3 | 1.608 |
| -0.2 | 1.587 |
| -0.1 | 1.575 |
| 0 | 1.571 |
| 0.1 | 1.575 |
| 0.2 | 1.587 |
| 0.3 | 1.608 |
| 0.4 | 1.64 |
| 0.5 | 1.686 |
| 0.6 | 1.751 |
| 0.7 | 1.846 |
| 0.8 | 1.995 |
| 0.9 | 2.281 |
| 0.95 | 2.59 |
| 0.96 | 2.693 |
| 0.97 | 2.828 |
| 0.98 | 3.021 |
| 0.99 | 3.357 |
| 0.999 | 4.496 |
| 1 | ∞ |
In the following table the values of the incomplete elliptic integral of the first kind are shown for a range of k and φ values:
| k | F(30°,k) | F(45°,k) | F(60°,k) | F(90°,k) | F(135°,k) | F(225°,k) | F(270°,k) | F(315°,k) |
|---|---|---|---|---|---|---|---|---|
| -1 | 0.5493 | 0.8814 | 1.317 | ∞ | 0.8814 | -0.8814 | -∞ | -0.8814 |
| -0.999 | 0.5492 | 0.8811 | 1.316 | 4.496 | 0.8811 | -0.8811 | -4.496 | -0.8811 |
| -0.99 | 0.5487 | 0.8787 | 1.307 | 3.357 | 0.8787 | -0.8787 | -3.357 | -0.8787 |
| -0.98 | 0.5482 | 0.8762 | 1.297 | 3.021 | 0.8762 | -0.8762 | -3.021 | -0.8762 |
| -0.97 | 0.5476 | 0.8737 | 1.287 | 2.828 | 0.8737 | -0.8737 | -2.828 | -0.8737 |
| -0.96 | 0.547 | 0.8713 | 1.279 | 2.693 | 0.8713 | -0.8713 | -2.693 | -0.8713 |
| -0.95 | 0.5465 | 0.8689 | 1.27 | 2.59 | 0.8689 | -0.8689 | -2.59 | -0.8689 |
| -0.9 | 0.5439 | 0.8579 | 1.233 | 2.281 | 0.8579 | -0.8579 | -2.281 | -0.8579 |
| -0.8 | 0.5393 | 0.8396 | 1.179 | 1.995 | 0.8396 | -0.8396 | -1.995 | -0.8396 |
| -0.7 | 0.5354 | 0.8251 | 1.14 | 1.846 | 0.8251 | -0.8251 | -1.846 | -0.8251 |
| -0.6 | 0.5321 | 0.8135 | 1.111 | 1.751 | 0.8135 | -0.8135 | -1.751 | -0.8135 |
| -0.5 | 0.5294 | 0.8044 | 1.09 | 1.686 | 0.8044 | -0.8044 | -1.686 | -0.8044 |
| -0.4 | 0.5273 | 0.7973 | 1.073 | 1.64 | 0.7973 | -0.7973 | -1.64 | -0.7973 |
| -0.3 | 0.5257 | 0.792 | 1.061 | 1.608 | 0.792 | -0.792 | -1.608 | -0.792 |
| -0.2 | 0.5245 | 0.7883 | 1.053 | 1.587 | 0.7883 | -0.7883 | -1.587 | -0.7883 |
| -0.1 | 0.5238 | 0.7861 | 1.049 | 1.575 | 0.7861 | -0.7861 | -1.575 | -0.7861 |
| 0 | 0.5236 | 0.7854 | 1.047 | 1.571 | 0.7854 | -0.7854 | -1.571 | -0.7854 |
| 0.1 | 0.5238 | 0.7861 | 1.049 | 1.575 | 0.7861 | -0.7861 | -1.575 | -0.7861 |
| 0.2 | 0.5245 | 0.7883 | 1.053 | 1.587 | 0.7883 | -0.7883 | -1.587 | -0.7883 |
| 0.3 | 0.5257 | 0.792 | 1.061 | 1.608 | 0.792 | -0.792 | -1.608 | -0.792 |
| 0.4 | 0.5273 | 0.7973 | 1.073 | 1.64 | 0.7973 | -0.7973 | -1.64 | -0.7973 |
| 0.5 | 0.5294 | 0.8044 | 1.09 | 1.686 | 0.8044 | -0.8044 | -1.686 | -0.8044 |
| 0.6 | 0.5321 | 0.8135 | 1.111 | 1.751 | 0.8135 | -0.8135 | -1.751 | -0.8135 |
| 0.7 | 0.5354 | 0.8251 | 1.14 | 1.846 | 0.8251 | -0.8251 | -1.846 | -0.8251 |
| 0.8 | 0.5393 | 0.8396 | 1.179 | 1.995 | 0.8396 | -0.8396 | -1.995 | -0.8396 |
| 0.9 | 0.5439 | 0.8579 | 1.233 | 2.281 | 0.8579 | -0.8579 | -2.281 | -0.8579 |
| 0.95 | 0.5465 | 0.8689 | 1.27 | 2.59 | 0.8689 | -0.8689 | -2.59 | -0.8689 |
| 0.96 | 0.547 | 0.8713 | 1.279 | 2.693 | 0.8713 | -0.8713 | -2.693 | -0.8713 |
| 0.97 | 0.5476 | 0.8737 | 1.287 | 2.828 | 0.8737 | -0.8737 | -2.828 | -0.8737 |
| 0.98 | 0.5482 | 0.8762 | 1.297 | 3.021 | 0.8762 | -0.8762 | -3.021 | -0.8762 |
| 0.99 | 0.5487 | 0.8787 | 1.307 | 3.357 | 0.8787 | -0.8787 | -3.357 | -0.8787 |
| 0.999 | 0.5492 | 0.8811 | 1.316 | 4.496 | 0.8811 | -0.8811 | -4.496 | -0.8811 |
| 1 | 0.5493 | 0.8814 | 1.317 | ∞ | 0.8814 | -0.8814 | -∞ | -0.8814 |
The xy plots of the incomplete integral of the first kind for various values of amplitude φ are depicted in the following figure:
