Incomplete Elliptic Integral of the Third Kind Pi Calculator
Enter the amplitude phi, characteristic n, and modulus k to compute the incomplete elliptic integral of the third kind Pi(phi,n,k) using Carlson symmetric forms.
Input
Enter the amplitude phi in radians, the characteristic n, and the modulus k to compute the incomplete elliptic integral of the third kind Pi(phi,n,k).
Upper limit of the integral. Example: pi/4 is about 0.785398
A real number. Values near n sin²phi = 1 cannot be computed.
A value with |k| less than or equal to 1. Here m = k² is the parameter m.
Result
Incomplete elliptic integral of the third kind Pi(phi,n,k)
0.8930657289
Characteristic n
0.5
Modulus k
0.5
Amplitude phi
0.785398
Parameter m = k²
0.25
Pi(t,n,k) from t = 0 to phi
How it works
- The incomplete elliptic integral of the third kind is defined as Pi(phi,n,k)=integral from 0 to phi of dtheta/((1−n sin²theta)√(1−k²sin²theta)).
- This tool uses the modulus k convention, where m=k² is called the parameter m.
- Enter a modulus k with |k|≤1; if k²sin²theta exceeds 1 the integrand becomes non-real.
- The characteristic n is real, and values near n sin²phi=1 give a pole, so that neighborhood is excluded from computation.
- It is evaluated with Carlson symmetric integrals R_F and R_J as Pi=sin(phi)·R_F(cos²phi, 1−k²sin²phi, 1)+(n/3)sin³phi·R_J(cos²phi, 1−k²sin²phi, 1, 1−n sin²phi).
- The amplitude phi is given in radians; for example pi/4 is about 0.7853981634, and R_F and R_J converge to roughly 10 significant digits.
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Incomplete Elliptic Integral of the Third Kind Pi Calculator