======================================== Special functions (:mod:`scipy.special`) ======================================== .. module:: scipy.special Nearly all of the functions below are universal functions and follow broadcasting and automatic array-looping rules. Exceptions are noted. Error handling ============== Errors are handled by returning nans, or other appropriate values. Some of the special function routines will print an error message when an error occurs. By default this printing is disabled. To enable such messages use errprint(1) To disable such messages use errprint(0). Example: >>> print scipy.special.bdtr(-1,10,0.3) >>> scipy.special.errprint(1) >>> print scipy.special.bdtr(-1,10,0.3) .. autosummary:: :toctree: generated/ errprint errstate Available functions =================== Airy functions -------------- .. autosummary:: :toctree: generated/ airy airye ai_zeros bi_zeros Elliptic Functions and Integrals -------------------------------- .. autosummary:: :toctree: generated/ ellipj ellipk ellipkinc ellipe ellipeinc Bessel Functions ---------------- .. autosummary:: :toctree: generated/ jn jv jve yn yv yve kn kv kve iv ive hankel1 hankel1e hankel2 hankel2e The following is not an universal function: .. autosummary:: :toctree: generated/ lmbda Zeros of Bessel Functions ^^^^^^^^^^^^^^^^^^^^^^^^^ These are not universal functions: .. autosummary:: :toctree: generated/ jnjnp_zeros jnyn_zeros jn_zeros jnp_zeros yn_zeros ynp_zeros y0_zeros y1_zeros y1p_zeros Faster versions of common Bessel Functions ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autosummary:: :toctree: generated/ j0 j1 y0 y1 i0 i0e i1 i1e k0 k0e k1 k1e Integrals of Bessel Functions ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autosummary:: :toctree: generated/ itj0y0 it2j0y0 iti0k0 it2i0k0 besselpoly Derivatives of Bessel Functions ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autosummary:: :toctree: generated/ jvp yvp kvp ivp h1vp h2vp Spherical Bessel Functions ^^^^^^^^^^^^^^^^^^^^^^^^^^ These are not universal functions: .. autosummary:: :toctree: generated/ sph_jn sph_yn sph_jnyn sph_in sph_kn sph_inkn Riccati-Bessel Functions ^^^^^^^^^^^^^^^^^^^^^^^^ These are not universal functions: .. autosummary:: :toctree: generated/ riccati_jn riccati_yn Struve Functions ---------------- .. autosummary:: :toctree: generated/ struve modstruve itstruve0 it2struve0 itmodstruve0 Raw Statistical Functions ------------------------- .. seealso:: :mod:`scipy.stats`: Friendly versions of these functions. .. autosummary:: :toctree: generated/ bdtr bdtrc bdtri btdtr btdtri fdtr fdtrc fdtri gdtr gdtrc gdtria gdtrib gdtrix nbdtr nbdtrc nbdtri pdtr pdtrc pdtri stdtr stdtridf stdtrit chdtr chdtrc chdtri ndtr ndtri smirnov smirnovi kolmogorov kolmogi tklmbda Gamma and Related Functions --------------------------- .. autosummary:: :toctree: generated/ gamma gammaln gammainc gammaincinv gammaincc gammainccinv beta betaln betainc betaincinv psi rgamma polygamma multigammaln Error Function and Fresnel Integrals ------------------------------------ .. autosummary:: :toctree: generated/ erf erfc erfinv erfcinv erf_zeros fresnel fresnel_zeros modfresnelp modfresnelm These are not universal functions: .. autosummary:: :toctree: generated/ fresnelc_zeros fresnels_zeros Legendre Functions ------------------ .. autosummary:: :toctree: generated/ lpmv sph_harm These are not universal functions: .. autosummary:: :toctree: generated/ lpn lqn lpmn lqmn Orthogonal polynomials ---------------------- The following functions evaluate values of orthogonal polynomials: .. autosummary:: :toctree: generated/ eval_legendre eval_chebyt eval_chebyu eval_chebyc eval_chebys eval_jacobi eval_laguerre eval_genlaguerre eval_hermite eval_hermitenorm eval_gegenbauer eval_sh_legendre eval_sh_chebyt eval_sh_chebyu eval_sh_jacobi The functions below, in turn, return :ref:`orthopoly1d` objects, which functions similarly as :ref:`numpy.poly1d`. The :ref:`orthopoly1d` class also has an attribute ``weights`` which returns the roots, weights, and total weights for the appropriate form of Gaussian quadrature. These are returned in an ``n x 3`` array with roots in the first column, weights in the second column, and total weights in the final column. .. autosummary:: :toctree: generated/ legendre chebyt chebyu chebyc chebys jacobi laguerre genlaguerre hermite hermitenorm gegenbauer sh_legendre sh_chebyt sh_chebyu sh_jacobi .. warning:: Large-order polynomials obtained from these functions are numerically unstable. ``orthopoly1d`` objects are converted to ``poly1d``, when doing arithmetic. ``numpy.poly1d`` works in power basis and cannot represent high-order polynomials accurately, which can cause significant inaccuracy. Hypergeometric Functions ------------------------ .. autosummary:: :toctree: generated/ hyp2f1 hyp1f1 hyperu hyp0f1 hyp2f0 hyp1f2 hyp3f0 Parabolic Cylinder Functions ---------------------------- .. autosummary:: :toctree: generated/ pbdv pbvv pbwa These are not universal functions: .. autosummary:: :toctree: generated/ pbdv_seq pbvv_seq pbdn_seq Mathieu and Related Functions ----------------------------- .. autosummary:: :toctree: generated/ mathieu_a mathieu_b These are not universal functions: .. autosummary:: :toctree: generated/ mathieu_even_coef mathieu_odd_coef The following return both function and first derivative: .. autosummary:: :toctree: generated/ mathieu_cem mathieu_sem mathieu_modcem1 mathieu_modcem2 mathieu_modsem1 mathieu_modsem2 Spheroidal Wave Functions ------------------------- .. autosummary:: :toctree: generated/ pro_ang1 pro_rad1 pro_rad2 obl_ang1 obl_rad1 obl_rad2 pro_cv obl_cv pro_cv_seq obl_cv_seq The following functions require pre-computed characteristic value: .. autosummary:: :toctree: generated/ pro_ang1_cv pro_rad1_cv pro_rad2_cv obl_ang1_cv obl_rad1_cv obl_rad2_cv Kelvin Functions ---------------- .. autosummary:: :toctree: generated/ kelvin kelvin_zeros ber bei berp beip ker kei kerp keip These are not universal functions: .. autosummary:: :toctree: generated/ ber_zeros bei_zeros berp_zeros beip_zeros ker_zeros kei_zeros kerp_zeros keip_zeros Other Special Functions ----------------------- .. autosummary:: :toctree: generated/ expn exp1 expi wofz dawsn shichi sici spence lambertw zeta zetac Convenience Functions --------------------- .. autosummary:: :toctree: generated/ cbrt exp10 exp2 radian cosdg sindg tandg cotdg log1p expm1 cosm1 round