Special Functions (26)

Compute the modified Bessel function of the first kind I_n(x). The order n must be a non-negative integer.

Compute the Bessel function of the first kind J_n(x). The order n must be a non-negative integer.

Compute the modified Bessel function of the second kind K_n(x). Only valid for x > 0. Returns Infinity for x = 0.

Compute the Bessel function of the second kind Y_n(x). Only valid for x > 0. Returns -Infinity for x = 0.

Compute the Beta function B(a, b) = Gamma(a) * Gamma(b) / Gamma(a + b). Both parameters must be positive.

Compute the regularized incomplete beta function I_x(a, b). x must be in [0, 1], a and b must be positive. Returns values in [0, 1].

Evaluate the Chebyshev polynomial of the first kind T_n(x) using the three-term recurrence: T_0=1, T_1=x, T_{n+1}=2*x*T_n - T_{n-1}. n must be a non-negative integer.

Compute the cosine integral Ci(x) = gamma + ln(x) + integral from 0 to x of (cos(t)-1)/t dt, where gamma is the Euler-Mascheroni constant. Defined for x > 0.

Compute the digamma (psi) function, the logarithmic derivative of the gamma function: psi(x) = d/dx ln(Gamma(x)). Has poles at non-positive integers.

Compute the complete elliptic integral of the second kind E(m), where m is the parameter (m = k^2). Uses the Arithmetic-Geometric Mean algorithm.

Compute the complete elliptic integral of the first kind K(m), where m is the parameter (m = k^2). Uses the Arithmetic-Geometric Mean algorithm.

Compute the erf function of a value using a rational Chebyshev approximations for different intervals of x

Compute the complementary error function erfc(x) = 1 - erf(x). Computed directly to avoid cancellation for large x.

Compute the imaginary error function erfi(x) = -i * erf(ix) = (2/sqrt(pi)) * integral from 0 to x of exp(t^2) dt. Unlike erf, erfi is unbounded.

Compute the exponential integral Ei(x) = -P.V. integral from -x to infinity of e^(-t)/t dt.

Compute the Fresnel cosine integral C(x) = integral from 0 to x of cos(pi * t^2 / 2) dt. Approaches 0.5 as x -> Infinity.

Compute the Fresnel sine integral S(x) = integral from 0 to x of sin(pi * t^2 / 2) dt. Approaches 0.5 as x -> Infinity.

Compute the regularized lower incomplete gamma function P(a, x) = gamma(a, x) / Gamma(a). a must be positive, x must be non-negative.

Compute the regularized upper incomplete gamma function Q(a, x) = 1 - P(a, x) = Gamma(a, x) / Gamma(a). a must be positive, x must be non-negative.

Evaluate the Laguerre polynomial L_n(x) using the recurrence: L_0=1, L_1=1-x, (n+1)*L_{n+1}=(2n+1-x)*L_n - n*L_{n-1}. n must be a non-negative integer.

Compute the Lambert W function W(x), the principal branch W_0. W(x) satisfies W(x)*exp(W(x)) = x. Defined for x >= -1/e.

Evaluate the Legendre polynomial P_n(x) using Bonnet\

Compute the logarithmic integral li(x) = integral from 0 to x of 1/ln(t) dt. Computed via li(x) = Ei(ln(x)). Returns -Infinity at x=1. Defined for x > 0.

Compute the sine integral Si(x) = integral from 0 to x of sin(t)/t dt. Approaches pi/2 as x -> Infinity.

Compute the Riemann Zeta Function using an infinite series and Riemann\