Special case of elliptic integral (M = arithmetic geometric mean)

lanxiyu · 2023-11-10 22:00:26

$equation?tex=%5Cint_0%5Ex%5Cfrac%7B%5Cmathrm%7Bd%7Dt%7D%7B%5Csqrt%7B1%2Bat%5E2%2Bbt%5E4%7D%7D%3D%5Cbegin%7Bcases%7Dx%5C%2CR_F%5C!%5Cleft(%5Cleft(1%2B%5Csqrt%7Bb%7D%5C%2Cx%5E2%5Cright)%5E2%2C%5Cleft(1-%5Csqrt%7Bb%7D%5C%2Cx%5E2%5Cright)%5E2%2C1%2Bax%5E2%2Bbx%5E4%5Cright)%2C%26%5Ctext%7Bif%20%7Db%5Cge0%5C%5Cx%5C%2CR_F%5C!%5Cleft(1%2B%5Cfrac12%5Cleft(a%2B%5Csqrt%7Ba%5E2-4b%7D%5Cright)x%5E2%2C1%2B%5Cfrac12%5Cleft(a-%5Csqrt%7Ba%5E2-4b%7D%5Cright)x%5E2%2C1%5Cright)%2C%26%5Ctext%7Bif%20%7Db%3C0%5C%5C%5Cend%7Bcases%7D$

Duplication formulas:

Derivatives:

k-derivatives:

Lemniscatic cases:

Perimeter of ellipse with semiaxes x, y:

Surface area of ellipsoid with semiaxes x, y, z:

Definition of Legendre’s elliptic Integrals
Definition of Jacobi zeta function
Notation for the theta functions

Last edited by lanxiyu (2025-03-25 23:30:29)

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#1 2023-11-10 22:00:26

Special case of elliptic integral (M = arithmetic geometric mean)

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