special_funcs
- special_funcs.Bernoulli(n, x)[source]
The Bernoulli polynomial B_n(x), n < 60.
- Parameters:
n (int) – Order of the Bernoulli polynomial (must be < 60).
x (float or array) – Argument of the polynomial.
- Returns:
Value of B_n(x).
- Return type:
float or array
Notes
See Wikipedia article on Bernoulli polynomials for definition.
- special_funcs.K0(z)[source]
Modified Bessel function of the second kind of order 0.
- Parameters:
z (float or array) – Argument of the Bessel function.
- Returns:
Value of K_0(z).
- Return type:
float or array
Notes
Uses different series approximations for z < 9 and z ≥ 9. Algorithm from Zhang and Jin.
- special_funcs.K1(z)[source]
Modified Bessel function of the second kind of order 1.
- Parameters:
z (float or array) – Argument of the Bessel function.
- Returns:
Value of K_1(z).
- Return type:
float or array
- special_funcs.K2(z)[source]
Modified Bessel function of the second kind of order 2.
- Parameters:
z (float or array) – Argument of the Bessel function.
- Returns:
Value of K_2(z).
- Return type:
float or array
Notes
Computed using recurrence relation: K_2(z) = K_0(z) + 2/z K_1(z).
- special_funcs.Li(n, z)[source]
Polylogarithm of order n and argument z.
- Parameters:
n (int) – Order of the polylogarithm (static argument).
z (float or complex) – Argument of the polylogarithm.
- Returns:
Value of Li_n(z).
- Return type:
float
Notes
Uses different series expansions depending on |z|: - |z| ≤ 0.5: direct series - 0.5 < |z| < 2: intermediate series with harmonic terms - |z| ≥ 2: reciprocal series with Bernoulli polynomials
- special_funcs.Riemann_zeta(n)[source]
Riemann zeta function with extended domain.
- Parameters:
n (int or array) – Argument of the zeta function (must be > -60).
- Returns:
Value of ζ(n).
- Return type:
float or array
Notes
Uses Bernoulli numbers for negative arguments. Returns 0 for negative even integers.