Algorythms/Elliptic-integrals/README.md

Elliptic Integrals Calculator

Description: Elliptic_Integrals::Elliptic(const double phi, const double k) computes elliptic integrals F(ϕ,k) and E(ϕ,k).

Usage:

std::vector<double> Elliptic_Integrals::Elliptic(const double phi, const double k)

Parameters:

• 'phi': The argument in degrees. It must be in the range [0,90]
• 'k': The modulus. It must be in the range [−1,1]

Returns: A vector containing the values of F(ϕ,k) (at the first position) and E(ϕ,k) (at the second position).

Exceptions:

• Throws an error if 'k' is not within the range [−1,1]
• Throws an error if 'phi' is not within the range [0,90]

Special Cases:

• If 'k' is equal to 1 and 'phi' is 90, the function returns ∞ for F(ϕ,k) and 1 for E(ϕ,k).
• If 'k' is equal to 1, the function applies specific formulas for computing F(ϕ,k) and E(ϕ,k).
• For other cases, the function iteratively computes the integrals until a desired level of precision is achieved.

Note: This implementation has a maximum iteration limit of 1000. Adjustments may be needed based on the desired precision and performance requirements.

**Related papers: ** Shanjie Zhang, Jian-Ming Jin, "Computation of Special Functions" Irene A. Stegun, Milton Abramowitz, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables"