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## About Elliptic Integrals of the Third Kind

### Definitions

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

where k is the elliptic modulus, with , and n is a parameter called elliptic characteristic, that can take any real value. Variable is the Jacobi's amplitude.

For n=0, the elliptic integral of the third kind is identical to the respective integral of the first kind:

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

For n=1, the complete elliptic integral of the third kind becomes infinite for any k:

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### Values

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

k | Π(k,-10) | Π(k,-5) | Π(k,-2) | Π(k,-1) | Π(k,0) | Π(k,1) | Π(k,2) | Π(k,5) | Π(k,10) |
---|---|---|---|---|---|---|---|---|---|

-1 | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ |

-0.999 | 0.7719 | 1.177 | 1.948 | 2.64 | 4.496 | ∞ | -3.242 | -0.8537 | -0.384 |

-0.99 | 0.6666 | 0.9849 | 1.564 | 2.066 | 3.357 | ∞ | -2.066 | -0.5619 | -0.2545 |

-0.98 | 0.6347 | 0.9266 | 1.449 | 1.894 | 3.021 | ∞ | -1.701 | -0.472 | -0.2147 |

-0.97 | 0.6159 | 0.8925 | 1.382 | 1.795 | 2.828 | ∞ | -1.484 | -0.4186 | -0.191 |

-0.96 | 0.6025 | 0.8682 | 1.334 | 1.725 | 2.693 | ∞ | -1.329 | -0.3802 | -0.1741 |

-0.95 | 0.5921 | 0.8494 | 1.298 | 1.671 | 2.59 | ∞ | -1.208 | -0.3502 | -0.1608 |

-0.9 | 0.5597 | 0.7913 | 1.186 | 1.507 | 2.281 | ∞ | -0.8352 | -0.2554 | -0.1187 |

-0.8 | 0.5276 | 0.7346 | 1.078 | 1.352 | 1.995 | ∞ | -0.4841 | -0.16 | -0.07571 |

-0.7 | 0.5096 | 0.7031 | 1.019 | 1.268 | 1.846 | ∞ | -0.3036 | -0.1061 | -0.05094 |

-0.6 | 0.4977 | 0.6824 | 0.9813 | 1.215 | 1.751 | ∞ | -0.1933 | -0.07043 | -0.03421 |

-0.5 | 0.4892 | 0.6679 | 0.9548 | 1.177 | 1.686 | ∞ | -0.1207 | -0.04539 | -0.02225 |

-0.4 | 0.4831 | 0.6574 | 0.9359 | 1.151 | 1.64 | ∞ | -0.07137 | -0.02748 | -0.01357 |

-0.3 | 0.4788 | 0.65 | 0.9226 | 1.132 | 1.608 | ∞ | -0.0379 | -0.01484 | -0.007372 |

-0.2 | 0.4758 | 0.6451 | 0.9137 | 1.12 | 1.587 | ∞ | -0.01619 | -0.006418 | -0.0032 |

-0.1 | 0.4742 | 0.6422 | 0.9086 | 1.113 | 1.575 | ∞ | -0.003957 | -0.001579 | -0.000789 |

0 | 0.4736 | 0.6413 | 0.9069 | 1.111 | 1.571 | ∞ | 0 | 0 | 0 |

0.1 | 0.4742 | 0.6422 | 0.9086 | 1.113 | 1.575 | ∞ | -0.003957 | -0.001579 | -0.000789 |

0.2 | 0.4758 | 0.6451 | 0.9137 | 1.12 | 1.587 | ∞ | -0.01619 | -0.006418 | -0.0032 |

0.3 | 0.4788 | 0.65 | 0.9226 | 1.132 | 1.608 | ∞ | -0.0379 | -0.01484 | -0.007372 |

0.4 | 0.4831 | 0.6574 | 0.9359 | 1.151 | 1.64 | ∞ | -0.07137 | -0.02748 | -0.01357 |

0.5 | 0.4892 | 0.6679 | 0.9548 | 1.177 | 1.686 | ∞ | -0.1207 | -0.04539 | -0.02225 |

0.6 | 0.4977 | 0.6824 | 0.9813 | 1.215 | 1.751 | ∞ | -0.1933 | -0.07043 | -0.03421 |

0.7 | 0.5096 | 0.7031 | 1.019 | 1.268 | 1.846 | ∞ | -0.3036 | -0.1061 | -0.05094 |

0.8 | 0.5276 | 0.7346 | 1.078 | 1.352 | 1.995 | ∞ | -0.4841 | -0.16 | -0.07571 |

0.9 | 0.5597 | 0.7913 | 1.186 | 1.507 | 2.281 | ∞ | -0.8352 | -0.2554 | -0.1187 |

0.95 | 0.5921 | 0.8494 | 1.298 | 1.671 | 2.59 | ∞ | -1.208 | -0.3502 | -0.1608 |

0.96 | 0.6025 | 0.8682 | 1.334 | 1.725 | 2.693 | ∞ | -1.329 | -0.3802 | -0.1741 |

0.97 | 0.6159 | 0.8925 | 1.382 | 1.795 | 2.828 | ∞ | -1.484 | -0.4186 | -0.191 |

0.98 | 0.6347 | 0.9266 | 1.449 | 1.894 | 3.021 | ∞ | -1.701 | -0.472 | -0.2147 |

0.99 | 0.6666 | 0.9849 | 1.564 | 2.066 | 3.357 | ∞ | -2.066 | -0.5619 | -0.2545 |

0.999 | 0.7719 | 1.177 | 1.948 | 2.64 | 4.496 | ∞ | -3.242 | -0.8537 | -0.384 |

1 | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ |

The xy plots of the complete integrals of the third kind for various values of characteristic n are depicted in the following figure:

The curve for n=1 is not drawn because, the integral becomes infinite. Also, for n>1 it is .

In the following table the values of the incomplete elliptic integral of the third kind are shown for n=0 and a range of k and φ values. These values are identical to the respective ones of the incomplete elliptic integral of the first kind because .

k | Π(30°,k,0) | Π(45°,k,0) | Π(60°,k,0) | Π(90°,k,0) | Π(135°,k,0) | Π(225°,k,0) | Π(270°,k,0) | Π(315°,k,0) |
---|---|---|---|---|---|---|---|---|

-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 |

-0.9 | 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 |

-0.6 | 0.5321 | 0.8135 | 1.111 | 1.751 | 0.8135 | -0.8135 | -1.751 | -0.8135 |

-0.4 | 0.5273 | 0.7973 | 1.073 | 1.64 | 0.7973 | -0.7973 | -1.64 | -0.7973 |

-0.2 | 0.5245 | 0.7883 | 1.053 | 1.587 | 0.7883 | -0.7883 | -1.587 | -0.7883 |

0 | 0.5236 | 0.7854 | 1.047 | 1.571 | 0.7854 | -0.7854 | -1.571 | -0.7854 |

0.2 | 0.5245 | 0.7883 | 1.053 | 1.587 | 0.7883 | -0.7883 | -1.587 | -0.7883 |

0.4 | 0.5273 | 0.7973 | 1.073 | 1.64 | 0.7973 | -0.7973 | -1.64 | -0.7973 |

0.6 | 0.5321 | 0.8135 | 1.111 | 1.751 | 0.8135 | -0.8135 | -1.751 | -0.8135 |

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 |

0.99 | 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 integrals of the third kind for characteristic n=0 are depicted in the following figure. The curves are identical to the respective ones of the first elliptic integral.

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In the following table the values of the incomplete elliptic integral of the third kind are shown for n=1 and a range of k and φ values:

k | Π(30°,k,1) | Π(45°,k,1) | Π(60°,k,1) | Π(90°,k,1) | Π(135°,k,1) | Π(225°,k,1) | Π(270°,k,1) | Π(315°,k,1) |
---|---|---|---|---|---|---|---|---|

-1 | 0.608 | 1.148 | 2.391 | ∞ | 1.148 | -1.148 | -∞ | -1.148 |

-0.999 | 0.6079 | 1.147 | 2.388 | ∞ | 1.147 | -1.147 | -∞ | -1.147 |

-0.99 | 0.6073 | 1.144 | 2.363 | ∞ | 1.144 | -1.144 | -∞ | -1.144 |

-0.9 | 0.6015 | 1.111 | 2.174 | ∞ | 1.111 | -1.111 | -∞ | -1.111 |

-0.8 | 0.596 | 1.083 | 2.039 | ∞ | 1.083 | -1.083 | -∞ | -1.083 |

-0.6 | 0.5875 | 1.043 | 1.878 | ∞ | 1.043 | -1.043 | -∞ | -1.043 |

-0.4 | 0.5817 | 1.018 | 1.791 | ∞ | 1.018 | -1.018 | -∞ | -1.018 |

-0.2 | 0.5784 | 1.004 | 1.746 | ∞ | 1.004 | -1.004 | -∞ | -1.004 |

0 | 0.5774 | 1 | 1.732 | ∞ | 1 | -1 | -∞ | -1 |

0.2 | 0.5784 | 1.004 | 1.746 | ∞ | 1.004 | -1.004 | -∞ | -1.004 |

0.4 | 0.5817 | 1.018 | 1.791 | ∞ | 1.018 | -1.018 | -∞ | -1.018 |

0.6 | 0.5875 | 1.043 | 1.878 | ∞ | 1.043 | -1.043 | -∞ | -1.043 |

0.8 | 0.596 | 1.083 | 2.039 | ∞ | 1.083 | -1.083 | -∞ | -1.083 |

0.9 | 0.6015 | 1.111 | 2.174 | ∞ | 1.111 | -1.111 | -∞ | -1.111 |

0.99 | 0.6073 | 1.144 | 2.363 | ∞ | 1.144 | -1.144 | -∞ | -1.144 |

0.999 | 0.6079 | 1.147 | 2.388 | ∞ | 1.147 | -1.147 | -∞ | -1.147 |

1 | 0.608 | 1.148 | 2.391 | ∞ | 1.148 | -1.148 | -∞ | -1.148 |

The xy plots of the incomplete integrals of the third kind for characteristic n=1 are depicted in the following figure.

In the following table the values of the incomplete elliptic integral of the third kind are shown for n=-1 and a range of k and φ values:

k | Π(30°,k,-1) | Π(45°,k,-1) | Π(60°,k,-1) | Π(90°,k,-1) | Π(135°,k,-1) | Π(225°,k,-1) | Π(270°,k,-1) | Π(315°,k,-1) |
---|---|---|---|---|---|---|---|---|

-1 | 0.5065 | 0.7484 | 1.015 | ∞ | 0.7484 | -0.7484 | -∞ | -0.7484 |

-0.999 | 0.5064 | 0.7482 | 1.015 | 2.64 | 0.7482 | -0.7482 | -2.64 | -0.7482 |

-0.99 | 0.506 | 0.7465 | 1.009 | 2.066 | 0.7465 | -0.7465 | -2.066 | -0.7465 |

-0.9 | 0.5018 | 0.7308 | 0.9613 | 1.507 | 0.7308 | -0.7308 | -1.507 | -0.7308 |

-0.8 | 0.4978 | 0.7169 | 0.9254 | 1.352 | 0.7169 | -0.7169 | -1.352 | -0.7169 |

-0.6 | 0.4916 | 0.6971 | 0.8802 | 1.215 | 0.6971 | -0.6971 | -1.215 | -0.6971 |

-0.4 | 0.4874 | 0.6846 | 0.8545 | 1.151 | 0.6846 | -0.6846 | -1.151 | -0.6846 |

-0.2 | 0.485 | 0.6777 | 0.8409 | 1.12 | 0.6777 | -0.6777 | -1.12 | -0.6777 |

0 | 0.4842 | 0.6755 | 0.8366 | 1.111 | 0.6755 | -0.6755 | -1.111 | -0.6755 |

0.2 | 0.485 | 0.6777 | 0.8409 | 1.12 | 0.6777 | -0.6777 | -1.12 | -0.6777 |

0.4 | 0.4874 | 0.6846 | 0.8545 | 1.151 | 0.6846 | -0.6846 | -1.151 | -0.6846 |

0.6 | 0.4916 | 0.6971 | 0.8802 | 1.215 | 0.6971 | -0.6971 | -1.215 | -0.6971 |

0.8 | 0.4978 | 0.7169 | 0.9254 | 1.352 | 0.7169 | -0.7169 | -1.352 | -0.7169 |

0.9 | 0.5018 | 0.7308 | 0.9613 | 1.507 | 0.7308 | -0.7308 | -1.507 | -0.7308 |

0.99 | 0.506 | 0.7465 | 1.009 | 2.066 | 0.7465 | -0.7465 | -2.066 | -0.7465 |

0.999 | 0.5064 | 0.7482 | 1.015 | 2.64 | 0.7482 | -0.7482 | -2.64 | -0.7482 |

1 | 0.5065 | 0.7484 | 1.015 | ∞ | 0.7484 | -0.7484 | -∞ | -0.7484 |

The xy plots of the incomplete integrals of the third kind for characteristic n=-1 are depicted in the following figure.

In the following table the values of the incomplete elliptic integral of the third kind are shown for n=5 and a range of k and φ values:

k | Π(30°,k,5) | Π(45°,k,5) | Π(60°,k,5) | Π(90°,k,5) | Π(135°,k,5) | Π(225°,k,5) | Π(270°,k,5) | Π(315°,k,5) |
---|---|---|---|---|---|---|---|---|

-1 | 0.6697 | 0.1964 | -0.009809 | ∞ | 0.1964 | -0.1964 | -∞ | -0.1964 |

-0.999 | 0.6697 | 0.1966 | -0.00921 | -0.8537 | 0.1966 | -0.1966 | 0.8537 | -0.1966 |

-0.99 | 0.6695 | 0.1987 | -0.003963 | -0.5619 | 0.1987 | -0.1987 | 0.5619 | -0.1987 |

-0.9 | 0.6681 | 0.217 | 0.038 | -0.2554 | 0.217 | -0.217 | 0.2554 | -0.217 |

-0.8 | 0.6663 | 0.2325 | 0.07019 | -0.16 | 0.2325 | -0.2325 | 0.16 | -0.2325 |

-0.6 | 0.6631 | 0.2536 | 0.1106 | -0.07043 | 0.2536 | -0.2536 | 0.07043 | -0.2536 |

-0.4 | 0.6606 | 0.266 | 0.1332 | -0.02748 | 0.266 | -0.266 | 0.02748 | -0.266 |

-0.2 | 0.659 | 0.2726 | 0.1449 | -0.006418 | 0.2726 | -0.2726 | 0.006418 | -0.2726 |

0 | 0.6585 | 0.2747 | 0.1486 | 0 | 0.2747 | -0.2747 | 0 | -0.2747 |

0.2 | 0.659 | 0.2726 | 0.1449 | -0.006418 | 0.2726 | -0.2726 | 0.006418 | -0.2726 |

0.4 | 0.6606 | 0.266 | 0.1332 | -0.02748 | 0.266 | -0.266 | 0.02748 | -0.266 |

0.6 | 0.6631 | 0.2536 | 0.1106 | -0.07043 | 0.2536 | -0.2536 | 0.07043 | -0.2536 |

0.8 | 0.6663 | 0.2325 | 0.07019 | -0.16 | 0.2325 | -0.2325 | 0.16 | -0.2325 |

0.9 | 0.6681 | 0.217 | 0.038 | -0.2554 | 0.217 | -0.217 | 0.2554 | -0.217 |

0.99 | 0.6695 | 0.1987 | -0.003963 | -0.5619 | 0.1987 | -0.1987 | 0.5619 | -0.1987 |

0.999 | 0.6697 | 0.1966 | -0.00921 | -0.8537 | 0.1966 | -0.1966 | 0.8537 | -0.1966 |

1 | 0.6697 | 0.1964 | -0.009809 | ∞ | 0.1964 | -0.1964 | -∞ | -0.1964 |

The xy plots of the incomplete integrals of the third kind for characteristic n=5 are depicted in the following figure.

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In the following table the values of the incomplete elliptic integral of the third kind are shown for n=-5 and a range of k and φ values:

k | Π(30°,k,-5) | Π(45°,k,-5) | Π(60°,k,-5) | Π(90°,k,-5) | Π(135°,k,-5) | Π(225°,k,-5) | Π(270°,k,-5) | Π(315°,k,-5) |
---|---|---|---|---|---|---|---|---|

-1 | 0.405 | 0.5221 | 0.6272 | ∞ | 0.5221 | -0.5221 | -∞ | -0.5221 |

-0.999 | 0.405 | 0.522 | 0.627 | 1.177 | 0.522 | -0.522 | -1.177 | -0.522 |

-0.99 | 0.4047 | 0.5211 | 0.6244 | 0.9849 | 0.5211 | -0.5211 | -0.9849 | -0.5211 |

-0.9 | 0.4019 | 0.5129 | 0.6038 | 0.7913 | 0.5129 | -0.5129 | -0.7913 | -0.5129 |

-0.8 | 0.3992 | 0.5055 | 0.5878 | 0.7346 | 0.5055 | -0.5055 | -0.7346 | -0.5055 |

-0.6 | 0.395 | 0.4948 | 0.5672 | 0.6824 | 0.4948 | -0.4948 | -0.6824 | -0.4948 |

-0.4 | 0.3922 | 0.488 | 0.5552 | 0.6574 | 0.488 | -0.488 | -0.6574 | -0.488 |

-0.2 | 0.3905 | 0.4843 | 0.5488 | 0.6451 | 0.4843 | -0.4843 | -0.6451 | -0.4843 |

0 | 0.39 | 0.483 | 0.5468 | 0.6413 | 0.483 | -0.483 | -0.6413 | -0.483 |

0.2 | 0.3905 | 0.4843 | 0.5488 | 0.6451 | 0.4843 | -0.4843 | -0.6451 | -0.4843 |

0.4 | 0.3922 | 0.488 | 0.5552 | 0.6574 | 0.488 | -0.488 | -0.6574 | -0.488 |

0.6 | 0.395 | 0.4948 | 0.5672 | 0.6824 | 0.4948 | -0.4948 | -0.6824 | -0.4948 |

0.8 | 0.3992 | 0.5055 | 0.5878 | 0.7346 | 0.5055 | -0.5055 | -0.7346 | -0.5055 |

0.9 | 0.4019 | 0.5129 | 0.6038 | 0.7913 | 0.5129 | -0.5129 | -0.7913 | -0.5129 |

0.99 | 0.4047 | 0.5211 | 0.6244 | 0.9849 | 0.5211 | -0.5211 | -0.9849 | -0.5211 |

0.999 | 0.405 | 0.522 | 0.627 | 1.177 | 0.522 | -0.522 | -1.177 | -0.522 |

1 | 0.405 | 0.5221 | 0.6272 | ∞ | 0.5221 | -0.5221 | -∞ | -0.5221 |

The xy plots of the incomplete integrals of the third kind for characteristic n=-5 are depicted in the following figure.

In the following table the values of the incomplete elliptic integral of the third kind are shown for n=10 and a range of k and φ values:

k | Π(30°,k,10) | Π(45°,k,10) | Π(60°,k,10) | Π(90°,k,10) | Π(135°,k,10) | Π(225°,k,10) | Π(270°,k,10) | Π(315°,k,10) |
---|---|---|---|---|---|---|---|---|

-1 | 0.2009 | 0.07115 | -0.01182 | ∞ | 0.07115 | -0.07115 | -∞ | -0.07115 |

-0.999 | 0.201 | 0.07127 | -0.01156 | -0.384 | 0.07127 | -0.07127 | 0.384 | -0.07127 |

-0.99 | 0.2014 | 0.07233 | -0.009216 | -0.2545 | 0.07233 | -0.07233 | 0.2545 | -0.07233 |

-0.9 | 0.205 | 0.08177 | 0.009852 | -0.1187 | 0.08177 | -0.08177 | 0.1187 | -0.08177 |

-0.8 | 0.2085 | 0.0902 | 0.02502 | -0.07571 | 0.0902 | -0.0902 | 0.07571 | -0.0902 |

-0.6 | 0.2136 | 0.1023 | 0.04497 | -0.03421 | 0.1023 | -0.1023 | 0.03421 | -0.1023 |

-0.4 | 0.217 | 0.11 | 0.05669 | -0.01357 | 0.11 | -0.11 | 0.01357 | -0.11 |

-0.2 | 0.2189 | 0.1142 | 0.06297 | -0.0032 | 0.1142 | -0.1142 | 0.0032 | -0.1142 |

0 | 0.2195 | 0.1155 | 0.06496 | 0 | 0.1155 | -0.1155 | 0 | -0.1155 |

0.2 | 0.2189 | 0.1142 | 0.06297 | -0.0032 | 0.1142 | -0.1142 | 0.0032 | -0.1142 |

0.4 | 0.217 | 0.11 | 0.05669 | -0.01357 | 0.11 | -0.11 | 0.01357 | -0.11 |

0.6 | 0.2136 | 0.1023 | 0.04497 | -0.03421 | 0.1023 | -0.1023 | 0.03421 | -0.1023 |

0.8 | 0.2085 | 0.0902 | 0.02502 | -0.07571 | 0.0902 | -0.0902 | 0.07571 | -0.0902 |

0.9 | 0.205 | 0.08177 | 0.009852 | -0.1187 | 0.08177 | -0.08177 | 0.1187 | -0.08177 |

0.99 | 0.2014 | 0.07233 | -0.009216 | -0.2545 | 0.07233 | -0.07233 | 0.2545 | -0.07233 |

0.999 | 0.201 | 0.07127 | -0.01156 | -0.384 | 0.07127 | -0.07127 | 0.384 | -0.07127 |

1 | 0.2009 | 0.07115 | -0.01182 | ∞ | 0.07115 | -0.07115 | -∞ | -0.07115 |

The xy plots of the incomplete integrals of the third kind for characteristic n=10 are depicted in the following figure.

In the following table the values of the incomplete elliptic integral of the third kind are shown for n=-10 and a range of k and φ values:

k | Π(30°,k,-10) | Π(45°,k,-10) | Π(60°,k,-10) | Π(90°,k,-10) | Π(135°,k,-10) | Π(225°,k,-10) | Π(270°,k,-10) | Π(315°,k,-10) |
---|---|---|---|---|---|---|---|---|

-1 | 0.3394 | 0.4108 | 0.4706 | ∞ | 0.4108 | -0.4108 | -∞ | -0.4108 |

-0.999 | 0.3394 | 0.4107 | 0.4705 | 0.7719 | 0.4107 | -0.4107 | -0.7719 | -0.4107 |

-0.99 | 0.3391 | 0.4101 | 0.4689 | 0.6666 | 0.4101 | -0.4101 | -0.6666 | -0.4101 |

-0.9 | 0.3371 | 0.4048 | 0.4566 | 0.5597 | 0.4048 | -0.4048 | -0.5597 | -0.4048 |

-0.8 | 0.3352 | 0.4001 | 0.447 | 0.5276 | 0.4001 | -0.4001 | -0.5276 | -0.4001 |

-0.6 | 0.3321 | 0.3931 | 0.4343 | 0.4977 | 0.3931 | -0.3931 | -0.4977 | -0.3931 |

-0.4 | 0.3301 | 0.3886 | 0.4269 | 0.4831 | 0.3886 | -0.3886 | -0.4831 | -0.3886 |

-0.2 | 0.3289 | 0.3861 | 0.4229 | 0.4758 | 0.3861 | -0.3861 | -0.4758 | -0.3861 |

0 | 0.3285 | 0.3853 | 0.4216 | 0.4736 | 0.3853 | -0.3853 | -0.4736 | -0.3853 |

0.2 | 0.3289 | 0.3861 | 0.4229 | 0.4758 | 0.3861 | -0.3861 | -0.4758 | -0.3861 |

0.4 | 0.3301 | 0.3886 | 0.4269 | 0.4831 | 0.3886 | -0.3886 | -0.4831 | -0.3886 |

0.6 | 0.3321 | 0.3931 | 0.4343 | 0.4977 | 0.3931 | -0.3931 | -0.4977 | -0.3931 |

0.8 | 0.3352 | 0.4001 | 0.447 | 0.5276 | 0.4001 | -0.4001 | -0.5276 | -0.4001 |

0.9 | 0.3371 | 0.4048 | 0.4566 | 0.5597 | 0.4048 | -0.4048 | -0.5597 | -0.4048 |

0.99 | 0.3391 | 0.4101 | 0.4689 | 0.6666 | 0.4101 | -0.4101 | -0.6666 | -0.4101 |

0.999 | 0.3394 | 0.4107 | 0.4705 | 0.7719 | 0.4107 | -0.4107 | -0.7719 | -0.4107 |

1 | 0.3394 | 0.4108 | 0.4706 | ∞ | 0.4108 | -0.4108 | -∞ | -0.4108 |

The xy plots of the incomplete integrals of the third kind for characteristic n=-10 are depicted in the following figure.

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