Module bessel 

Bessel functions of integer order: J_n, Y_n (first/second kind) and the modified I_n, K_n. The order-0 functions come from their convergent power series (with the log + harmonic-number terms for the second kinds); order 1 from the Wronskian relations; higher orders from the standard upward recurrences. Accurate for moderate |x| (the series converge for all x but lose digits for large argument — documented).

Functions

bessel_i

Modified Bessel function of the first kind I_n(x), integer order. I_{-n} = I_n.

bessel_j

Bessel function of the first kind J_n(x), integer order (any sign). Defined for all real x. J_{-n} = (−1)^n J_n.

bessel_k

Modified Bessel function of the second kind K_n(x), integer order n ≥ 0. Requires x > 0 → None otherwise. Order 1 via the Wronskian I_0 K_1 + I_1 K_0 = 1/x; higher via K_{n+1} = (2n/x)K_n + K_{n-1}.

bessel_y

Bessel function of the second kind Y_n(x), integer order n ≥ 0. Requires x > 0 (singular at the origin) → None otherwise. Order 1 via the Wronskian J_1 Y_0 − J_0 Y_1 = 2/(πx); higher via Y_{n+1} = (2n/x)Y_n − Y_{n-1}.