1""" 2======================================== 3Special functions (:mod:`scipy.special`) 4======================================== 5 6.. currentmodule:: scipy.special 7 8Nearly all of the functions below are universal functions and follow 9broadcasting and automatic array-looping rules. 10 11.. seealso:: 12 13 `scipy.special.cython_special` -- Typed Cython versions of special functions 14 15 16Error handling 17============== 18 19Errors are handled by returning NaNs or other appropriate values. 20Some of the special function routines can emit warnings or raise 21exceptions when an error occurs. By default this is disabled; to 22query and control the current error handling state the following 23functions are provided. 24 25.. autosummary:: 26 :toctree: generated/ 27 28 geterr -- Get the current way of handling special-function errors. 29 seterr -- Set how special-function errors are handled. 30 errstate -- Context manager for special-function error handling. 31 SpecialFunctionWarning -- Warning that can be emitted by special functions. 32 SpecialFunctionError -- Exception that can be raised by special functions. 33 34Available functions 35=================== 36 37Airy functions 38-------------- 39 40.. autosummary:: 41 :toctree: generated/ 42 43 airy -- Airy functions and their derivatives. 44 airye -- Exponentially scaled Airy functions and their derivatives. 45 ai_zeros -- Compute `nt` zeros and values of the Airy function Ai and its derivative. 46 bi_zeros -- Compute `nt` zeros and values of the Airy function Bi and its derivative. 47 itairy -- Integrals of Airy functions 48 49 50Elliptic functions and integrals 51-------------------------------- 52 53.. autosummary:: 54 :toctree: generated/ 55 56 ellipj -- Jacobian elliptic functions. 57 ellipk -- Complete elliptic integral of the first kind. 58 ellipkm1 -- Complete elliptic integral of the first kind around `m` = 1. 59 ellipkinc -- Incomplete elliptic integral of the first kind. 60 ellipe -- Complete elliptic integral of the second kind. 61 ellipeinc -- Incomplete elliptic integral of the second kind. 62 63Bessel functions 64---------------- 65 66.. autosummary:: 67 :toctree: generated/ 68 69 jv -- Bessel function of the first kind of real order and \ 70 complex argument. 71 jve -- Exponentially scaled Bessel function of order `v`. 72 yn -- Bessel function of the second kind of integer order and \ 73 real argument. 74 yv -- Bessel function of the second kind of real order and \ 75 complex argument. 76 yve -- Exponentially scaled Bessel function of the second kind \ 77 of real order. 78 kn -- Modified Bessel function of the second kind of integer \ 79 order `n` 80 kv -- Modified Bessel function of the second kind of real order \ 81 `v` 82 kve -- Exponentially scaled modified Bessel function of the \ 83 second kind. 84 iv -- Modified Bessel function of the first kind of real order. 85 ive -- Exponentially scaled modified Bessel function of the \ 86 first kind. 87 hankel1 -- Hankel function of the first kind. 88 hankel1e -- Exponentially scaled Hankel function of the first kind. 89 hankel2 -- Hankel function of the second kind. 90 hankel2e -- Exponentially scaled Hankel function of the second kind. 91 wright_bessel -- Wright's generalized Bessel function. 92 93The following is not a universal function: 94 95.. autosummary:: 96 :toctree: generated/ 97 98 lmbda -- Jahnke-Emden Lambda function, Lambdav(x). 99 100Zeros of Bessel functions 101^^^^^^^^^^^^^^^^^^^^^^^^^ 102 103These are not universal functions: 104 105.. autosummary:: 106 :toctree: generated/ 107 108 jnjnp_zeros -- Compute zeros of integer-order Bessel functions Jn and Jn'. 109 jnyn_zeros -- Compute nt zeros of Bessel functions Jn(x), Jn'(x), Yn(x), and Yn'(x). 110 jn_zeros -- Compute zeros of integer-order Bessel function Jn(x). 111 jnp_zeros -- Compute zeros of integer-order Bessel function derivative Jn'(x). 112 yn_zeros -- Compute zeros of integer-order Bessel function Yn(x). 113 ynp_zeros -- Compute zeros of integer-order Bessel function derivative Yn'(x). 114 y0_zeros -- Compute nt zeros of Bessel function Y0(z), and derivative at each zero. 115 y1_zeros -- Compute nt zeros of Bessel function Y1(z), and derivative at each zero. 116 y1p_zeros -- Compute nt zeros of Bessel derivative Y1'(z), and value at each zero. 117 118Faster versions of common Bessel functions 119^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 120 121.. autosummary:: 122 :toctree: generated/ 123 124 j0 -- Bessel function of the first kind of order 0. 125 j1 -- Bessel function of the first kind of order 1. 126 y0 -- Bessel function of the second kind of order 0. 127 y1 -- Bessel function of the second kind of order 1. 128 i0 -- Modified Bessel function of order 0. 129 i0e -- Exponentially scaled modified Bessel function of order 0. 130 i1 -- Modified Bessel function of order 1. 131 i1e -- Exponentially scaled modified Bessel function of order 1. 132 k0 -- Modified Bessel function of the second kind of order 0, :math:`K_0`. 133 k0e -- Exponentially scaled modified Bessel function K of order 0 134 k1 -- Modified Bessel function of the second kind of order 1, :math:`K_1(x)`. 135 k1e -- Exponentially scaled modified Bessel function K of order 1. 136 137Integrals of Bessel functions 138^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 139 140.. autosummary:: 141 :toctree: generated/ 142 143 itj0y0 -- Integrals of Bessel functions of order 0. 144 it2j0y0 -- Integrals related to Bessel functions of order 0. 145 iti0k0 -- Integrals of modified Bessel functions of order 0. 146 it2i0k0 -- Integrals related to modified Bessel functions of order 0. 147 besselpoly -- Weighted integral of a Bessel function. 148 149Derivatives of Bessel functions 150^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 151 152.. autosummary:: 153 :toctree: generated/ 154 155 jvp -- Compute nth derivative of Bessel function Jv(z) with respect to `z`. 156 yvp -- Compute nth derivative of Bessel function Yv(z) with respect to `z`. 157 kvp -- Compute nth derivative of real-order modified Bessel function Kv(z) 158 ivp -- Compute nth derivative of modified Bessel function Iv(z) with respect to `z`. 159 h1vp -- Compute nth derivative of Hankel function H1v(z) with respect to `z`. 160 h2vp -- Compute nth derivative of Hankel function H2v(z) with respect to `z`. 161 162Spherical Bessel functions 163^^^^^^^^^^^^^^^^^^^^^^^^^^ 164 165.. autosummary:: 166 :toctree: generated/ 167 168 spherical_jn -- Spherical Bessel function of the first kind or its derivative. 169 spherical_yn -- Spherical Bessel function of the second kind or its derivative. 170 spherical_in -- Modified spherical Bessel function of the first kind or its derivative. 171 spherical_kn -- Modified spherical Bessel function of the second kind or its derivative. 172 173Riccati-Bessel functions 174^^^^^^^^^^^^^^^^^^^^^^^^ 175 176These are not universal functions: 177 178.. autosummary:: 179 :toctree: generated/ 180 181 riccati_jn -- Compute Ricatti-Bessel function of the first kind and its derivative. 182 riccati_yn -- Compute Ricatti-Bessel function of the second kind and its derivative. 183 184Struve functions 185---------------- 186 187.. autosummary:: 188 :toctree: generated/ 189 190 struve -- Struve function. 191 modstruve -- Modified Struve function. 192 itstruve0 -- Integral of the Struve function of order 0. 193 it2struve0 -- Integral related to the Struve function of order 0. 194 itmodstruve0 -- Integral of the modified Struve function of order 0. 195 196 197Raw statistical functions 198------------------------- 199 200.. seealso:: :mod:`scipy.stats`: Friendly versions of these functions. 201 202.. autosummary:: 203 :toctree: generated/ 204 205 bdtr -- Binomial distribution cumulative distribution function. 206 bdtrc -- Binomial distribution survival function. 207 bdtri -- Inverse function to `bdtr` with respect to `p`. 208 bdtrik -- Inverse function to `bdtr` with respect to `k`. 209 bdtrin -- Inverse function to `bdtr` with respect to `n`. 210 btdtr -- Cumulative distribution function of the beta distribution. 211 btdtri -- The `p`-th quantile of the beta distribution. 212 btdtria -- Inverse of `btdtr` with respect to `a`. 213 btdtrib -- btdtria(a, p, x). 214 fdtr -- F cumulative distribution function. 215 fdtrc -- F survival function. 216 fdtri -- The `p`-th quantile of the F-distribution. 217 fdtridfd -- Inverse to `fdtr` vs dfd. 218 gdtr -- Gamma distribution cumulative distribution function. 219 gdtrc -- Gamma distribution survival function. 220 gdtria -- Inverse of `gdtr` vs a. 221 gdtrib -- Inverse of `gdtr` vs b. 222 gdtrix -- Inverse of `gdtr` vs x. 223 nbdtr -- Negative binomial cumulative distribution function. 224 nbdtrc -- Negative binomial survival function. 225 nbdtri -- Inverse of `nbdtr` vs `p`. 226 nbdtrik -- Inverse of `nbdtr` vs `k`. 227 nbdtrin -- Inverse of `nbdtr` vs `n`. 228 ncfdtr -- Cumulative distribution function of the non-central F distribution. 229 ncfdtridfd -- Calculate degrees of freedom (denominator) for the noncentral F-distribution. 230 ncfdtridfn -- Calculate degrees of freedom (numerator) for the noncentral F-distribution. 231 ncfdtri -- Inverse cumulative distribution function of the non-central F distribution. 232 ncfdtrinc -- Calculate non-centrality parameter for non-central F distribution. 233 nctdtr -- Cumulative distribution function of the non-central `t` distribution. 234 nctdtridf -- Calculate degrees of freedom for non-central t distribution. 235 nctdtrit -- Inverse cumulative distribution function of the non-central t distribution. 236 nctdtrinc -- Calculate non-centrality parameter for non-central t distribution. 237 nrdtrimn -- Calculate mean of normal distribution given other params. 238 nrdtrisd -- Calculate standard deviation of normal distribution given other params. 239 pdtr -- Poisson cumulative distribution function. 240 pdtrc -- Poisson survival function. 241 pdtri -- Inverse to `pdtr` vs m. 242 pdtrik -- Inverse to `pdtr` vs k. 243 stdtr -- Student t distribution cumulative distribution function. 244 stdtridf -- Inverse of `stdtr` vs df. 245 stdtrit -- Inverse of `stdtr` vs `t`. 246 chdtr -- Chi square cumulative distribution function. 247 chdtrc -- Chi square survival function. 248 chdtri -- Inverse to `chdtrc`. 249 chdtriv -- Inverse to `chdtr` vs `v`. 250 ndtr -- Gaussian cumulative distribution function. 251 log_ndtr -- Logarithm of Gaussian cumulative distribution function. 252 ndtri -- Inverse of `ndtr` vs x. 253 ndtri_exp -- Inverse of `log_ndtr` vs x. 254 chndtr -- Non-central chi square cumulative distribution function. 255 chndtridf -- Inverse to `chndtr` vs `df`. 256 chndtrinc -- Inverse to `chndtr` vs `nc`. 257 chndtrix -- Inverse to `chndtr` vs `x`. 258 smirnov -- Kolmogorov-Smirnov complementary cumulative distribution function. 259 smirnovi -- Inverse to `smirnov`. 260 kolmogorov -- Complementary cumulative distribution function of Kolmogorov distribution. 261 kolmogi -- Inverse function to `kolmogorov`. 262 tklmbda -- Tukey-Lambda cumulative distribution function. 263 logit -- Logit ufunc for ndarrays. 264 expit -- Expit ufunc for ndarrays. 265 boxcox -- Compute the Box-Cox transformation. 266 boxcox1p -- Compute the Box-Cox transformation of 1 + `x`. 267 inv_boxcox -- Compute the inverse of the Box-Cox transformation. 268 inv_boxcox1p -- Compute the inverse of the Box-Cox transformation. 269 owens_t -- Owen's T Function. 270 271 272Information Theory functions 273---------------------------- 274 275.. autosummary:: 276 :toctree: generated/ 277 278 entr -- Elementwise function for computing entropy. 279 rel_entr -- Elementwise function for computing relative entropy. 280 kl_div -- Elementwise function for computing Kullback-Leibler divergence. 281 huber -- Huber loss function. 282 pseudo_huber -- Pseudo-Huber loss function. 283 284 285Gamma and related functions 286--------------------------- 287 288.. autosummary:: 289 :toctree: generated/ 290 291 gamma -- Gamma function. 292 gammaln -- Logarithm of the absolute value of the Gamma function for real inputs. 293 loggamma -- Principal branch of the logarithm of the Gamma function. 294 gammasgn -- Sign of the gamma function. 295 gammainc -- Regularized lower incomplete gamma function. 296 gammaincinv -- Inverse to `gammainc`. 297 gammaincc -- Regularized upper incomplete gamma function. 298 gammainccinv -- Inverse to `gammaincc`. 299 beta -- Beta function. 300 betaln -- Natural logarithm of absolute value of beta function. 301 betainc -- Incomplete beta integral. 302 betaincinv -- Inverse function to beta integral. 303 psi -- The digamma function. 304 rgamma -- Gamma function inverted. 305 polygamma -- Polygamma function n. 306 multigammaln -- Returns the log of multivariate gamma, also sometimes called the generalized gamma. 307 digamma -- psi(x[, out]). 308 poch -- Rising factorial (z)_m. 309 310 311Error function and Fresnel integrals 312------------------------------------ 313 314.. autosummary:: 315 :toctree: generated/ 316 317 erf -- Returns the error function of complex argument. 318 erfc -- Complementary error function, ``1 - erf(x)``. 319 erfcx -- Scaled complementary error function, ``exp(x**2) * erfc(x)``. 320 erfi -- Imaginary error function, ``-i erf(i z)``. 321 erfinv -- Inverse function for erf. 322 erfcinv -- Inverse function for erfc. 323 wofz -- Faddeeva function. 324 dawsn -- Dawson's integral. 325 fresnel -- Fresnel sin and cos integrals. 326 fresnel_zeros -- Compute nt complex zeros of sine and cosine Fresnel integrals S(z) and C(z). 327 modfresnelp -- Modified Fresnel positive integrals. 328 modfresnelm -- Modified Fresnel negative integrals. 329 voigt_profile -- Voigt profile. 330 331These are not universal functions: 332 333.. autosummary:: 334 :toctree: generated/ 335 336 erf_zeros -- Compute nt complex zeros of error function erf(z). 337 fresnelc_zeros -- Compute nt complex zeros of cosine Fresnel integral C(z). 338 fresnels_zeros -- Compute nt complex zeros of sine Fresnel integral S(z). 339 340Legendre functions 341------------------ 342 343.. autosummary:: 344 :toctree: generated/ 345 346 lpmv -- Associated Legendre function of integer order and real degree. 347 sph_harm -- Compute spherical harmonics. 348 349These are not universal functions: 350 351.. autosummary:: 352 :toctree: generated/ 353 354 clpmn -- Associated Legendre function of the first kind for complex arguments. 355 lpn -- Legendre function of the first kind. 356 lqn -- Legendre function of the second kind. 357 lpmn -- Sequence of associated Legendre functions of the first kind. 358 lqmn -- Sequence of associated Legendre functions of the second kind. 359 360Ellipsoidal harmonics 361--------------------- 362 363.. autosummary:: 364 :toctree: generated/ 365 366 ellip_harm -- Ellipsoidal harmonic functions E^p_n(l). 367 ellip_harm_2 -- Ellipsoidal harmonic functions F^p_n(l). 368 ellip_normal -- Ellipsoidal harmonic normalization constants gamma^p_n. 369 370Orthogonal polynomials 371---------------------- 372 373The following functions evaluate values of orthogonal polynomials: 374 375.. autosummary:: 376 :toctree: generated/ 377 378 assoc_laguerre -- Compute the generalized (associated) Laguerre polynomial of degree n and order k. 379 eval_legendre -- Evaluate Legendre polynomial at a point. 380 eval_chebyt -- Evaluate Chebyshev polynomial of the first kind at a point. 381 eval_chebyu -- Evaluate Chebyshev polynomial of the second kind at a point. 382 eval_chebyc -- Evaluate Chebyshev polynomial of the first kind on [-2, 2] at a point. 383 eval_chebys -- Evaluate Chebyshev polynomial of the second kind on [-2, 2] at a point. 384 eval_jacobi -- Evaluate Jacobi polynomial at a point. 385 eval_laguerre -- Evaluate Laguerre polynomial at a point. 386 eval_genlaguerre -- Evaluate generalized Laguerre polynomial at a point. 387 eval_hermite -- Evaluate physicist's Hermite polynomial at a point. 388 eval_hermitenorm -- Evaluate probabilist's (normalized) Hermite polynomial at a point. 389 eval_gegenbauer -- Evaluate Gegenbauer polynomial at a point. 390 eval_sh_legendre -- Evaluate shifted Legendre polynomial at a point. 391 eval_sh_chebyt -- Evaluate shifted Chebyshev polynomial of the first kind at a point. 392 eval_sh_chebyu -- Evaluate shifted Chebyshev polynomial of the second kind at a point. 393 eval_sh_jacobi -- Evaluate shifted Jacobi polynomial at a point. 394 395The following functions compute roots and quadrature weights for 396orthogonal polynomials: 397 398.. autosummary:: 399 :toctree: generated/ 400 401 roots_legendre -- Gauss-Legendre quadrature. 402 roots_chebyt -- Gauss-Chebyshev (first kind) quadrature. 403 roots_chebyu -- Gauss-Chebyshev (second kind) quadrature. 404 roots_chebyc -- Gauss-Chebyshev (first kind) quadrature. 405 roots_chebys -- Gauss-Chebyshev (second kind) quadrature. 406 roots_jacobi -- Gauss-Jacobi quadrature. 407 roots_laguerre -- Gauss-Laguerre quadrature. 408 roots_genlaguerre -- Gauss-generalized Laguerre quadrature. 409 roots_hermite -- Gauss-Hermite (physicst's) quadrature. 410 roots_hermitenorm -- Gauss-Hermite (statistician's) quadrature. 411 roots_gegenbauer -- Gauss-Gegenbauer quadrature. 412 roots_sh_legendre -- Gauss-Legendre (shifted) quadrature. 413 roots_sh_chebyt -- Gauss-Chebyshev (first kind, shifted) quadrature. 414 roots_sh_chebyu -- Gauss-Chebyshev (second kind, shifted) quadrature. 415 roots_sh_jacobi -- Gauss-Jacobi (shifted) quadrature. 416 417The functions below, in turn, return the polynomial coefficients in 418``orthopoly1d`` objects, which function similarly as `numpy.poly1d`. 419The ``orthopoly1d`` class also has an attribute ``weights``, which returns 420the roots, weights, and total weights for the appropriate form of Gaussian 421quadrature. These are returned in an ``n x 3`` array with roots in the first 422column, weights in the second column, and total weights in the final column. 423Note that ``orthopoly1d`` objects are converted to `~numpy.poly1d` when doing 424arithmetic, and lose information of the original orthogonal polynomial. 425 426.. autosummary:: 427 :toctree: generated/ 428 429 legendre -- Legendre polynomial. 430 chebyt -- Chebyshev polynomial of the first kind. 431 chebyu -- Chebyshev polynomial of the second kind. 432 chebyc -- Chebyshev polynomial of the first kind on :math:`[-2, 2]`. 433 chebys -- Chebyshev polynomial of the second kind on :math:`[-2, 2]`. 434 jacobi -- Jacobi polynomial. 435 laguerre -- Laguerre polynomial. 436 genlaguerre -- Generalized (associated) Laguerre polynomial. 437 hermite -- Physicist's Hermite polynomial. 438 hermitenorm -- Normalized (probabilist's) Hermite polynomial. 439 gegenbauer -- Gegenbauer (ultraspherical) polynomial. 440 sh_legendre -- Shifted Legendre polynomial. 441 sh_chebyt -- Shifted Chebyshev polynomial of the first kind. 442 sh_chebyu -- Shifted Chebyshev polynomial of the second kind. 443 sh_jacobi -- Shifted Jacobi polynomial. 444 445.. warning:: 446 447 Computing values of high-order polynomials (around ``order > 20``) using 448 polynomial coefficients is numerically unstable. To evaluate polynomial 449 values, the ``eval_*`` functions should be used instead. 450 451 452Hypergeometric functions 453------------------------ 454 455.. autosummary:: 456 :toctree: generated/ 457 458 hyp2f1 -- Gauss hypergeometric function 2F1(a, b; c; z). 459 hyp1f1 -- Confluent hypergeometric function 1F1(a, b; x). 460 hyperu -- Confluent hypergeometric function U(a, b, x) of the second kind. 461 hyp0f1 -- Confluent hypergeometric limit function 0F1. 462 463 464Parabolic cylinder functions 465---------------------------- 466 467.. autosummary:: 468 :toctree: generated/ 469 470 pbdv -- Parabolic cylinder function D. 471 pbvv -- Parabolic cylinder function V. 472 pbwa -- Parabolic cylinder function W. 473 474These are not universal functions: 475 476.. autosummary:: 477 :toctree: generated/ 478 479 pbdv_seq -- Parabolic cylinder functions Dv(x) and derivatives. 480 pbvv_seq -- Parabolic cylinder functions Vv(x) and derivatives. 481 pbdn_seq -- Parabolic cylinder functions Dn(z) and derivatives. 482 483Mathieu and related functions 484----------------------------- 485 486.. autosummary:: 487 :toctree: generated/ 488 489 mathieu_a -- Characteristic value of even Mathieu functions. 490 mathieu_b -- Characteristic value of odd Mathieu functions. 491 492These are not universal functions: 493 494.. autosummary:: 495 :toctree: generated/ 496 497 mathieu_even_coef -- Fourier coefficients for even Mathieu and modified Mathieu functions. 498 mathieu_odd_coef -- Fourier coefficients for even Mathieu and modified Mathieu functions. 499 500The following return both function and first derivative: 501 502.. autosummary:: 503 :toctree: generated/ 504 505 mathieu_cem -- Even Mathieu function and its derivative. 506 mathieu_sem -- Odd Mathieu function and its derivative. 507 mathieu_modcem1 -- Even modified Mathieu function of the first kind and its derivative. 508 mathieu_modcem2 -- Even modified Mathieu function of the second kind and its derivative. 509 mathieu_modsem1 -- Odd modified Mathieu function of the first kind and its derivative. 510 mathieu_modsem2 -- Odd modified Mathieu function of the second kind and its derivative. 511 512Spheroidal wave functions 513------------------------- 514 515.. autosummary:: 516 :toctree: generated/ 517 518 pro_ang1 -- Prolate spheroidal angular function of the first kind and its derivative. 519 pro_rad1 -- Prolate spheroidal radial function of the first kind and its derivative. 520 pro_rad2 -- Prolate spheroidal radial function of the secon kind and its derivative. 521 obl_ang1 -- Oblate spheroidal angular function of the first kind and its derivative. 522 obl_rad1 -- Oblate spheroidal radial function of the first kind and its derivative. 523 obl_rad2 -- Oblate spheroidal radial function of the second kind and its derivative. 524 pro_cv -- Characteristic value of prolate spheroidal function. 525 obl_cv -- Characteristic value of oblate spheroidal function. 526 pro_cv_seq -- Characteristic values for prolate spheroidal wave functions. 527 obl_cv_seq -- Characteristic values for oblate spheroidal wave functions. 528 529The following functions require pre-computed characteristic value: 530 531.. autosummary:: 532 :toctree: generated/ 533 534 pro_ang1_cv -- Prolate spheroidal angular function pro_ang1 for precomputed characteristic value. 535 pro_rad1_cv -- Prolate spheroidal radial function pro_rad1 for precomputed characteristic value. 536 pro_rad2_cv -- Prolate spheroidal radial function pro_rad2 for precomputed characteristic value. 537 obl_ang1_cv -- Oblate spheroidal angular function obl_ang1 for precomputed characteristic value. 538 obl_rad1_cv -- Oblate spheroidal radial function obl_rad1 for precomputed characteristic value. 539 obl_rad2_cv -- Oblate spheroidal radial function obl_rad2 for precomputed characteristic value. 540 541Kelvin functions 542---------------- 543 544.. autosummary:: 545 :toctree: generated/ 546 547 kelvin -- Kelvin functions as complex numbers. 548 kelvin_zeros -- Compute nt zeros of all Kelvin functions. 549 ber -- Kelvin function ber. 550 bei -- Kelvin function bei 551 berp -- Derivative of the Kelvin function `ber`. 552 beip -- Derivative of the Kelvin function `bei`. 553 ker -- Kelvin function ker. 554 kei -- Kelvin function ker. 555 kerp -- Derivative of the Kelvin function ker. 556 keip -- Derivative of the Kelvin function kei. 557 558These are not universal functions: 559 560.. autosummary:: 561 :toctree: generated/ 562 563 ber_zeros -- Compute nt zeros of the Kelvin function ber(x). 564 bei_zeros -- Compute nt zeros of the Kelvin function bei(x). 565 berp_zeros -- Compute nt zeros of the Kelvin function ber'(x). 566 beip_zeros -- Compute nt zeros of the Kelvin function bei'(x). 567 ker_zeros -- Compute nt zeros of the Kelvin function ker(x). 568 kei_zeros -- Compute nt zeros of the Kelvin function kei(x). 569 kerp_zeros -- Compute nt zeros of the Kelvin function ker'(x). 570 keip_zeros -- Compute nt zeros of the Kelvin function kei'(x). 571 572Combinatorics 573------------- 574 575.. autosummary:: 576 :toctree: generated/ 577 578 comb -- The number of combinations of N things taken k at a time. 579 perm -- Permutations of N things taken k at a time, i.e., k-permutations of N. 580 581Lambert W and related functions 582------------------------------- 583 584.. autosummary:: 585 :toctree: generated/ 586 587 lambertw -- Lambert W function. 588 wrightomega -- Wright Omega function. 589 590Other special functions 591----------------------- 592 593.. autosummary:: 594 :toctree: generated/ 595 596 agm -- Arithmetic, Geometric Mean. 597 bernoulli -- Bernoulli numbers B0..Bn (inclusive). 598 binom -- Binomial coefficient 599 diric -- Periodic sinc function, also called the Dirichlet function. 600 euler -- Euler numbers E0..En (inclusive). 601 expn -- Exponential integral E_n. 602 exp1 -- Exponential integral E_1 of complex argument z. 603 expi -- Exponential integral Ei. 604 factorial -- The factorial of a number or array of numbers. 605 factorial2 -- Double factorial. 606 factorialk -- Multifactorial of n of order k, n(!!...!). 607 shichi -- Hyperbolic sine and cosine integrals. 608 sici -- Sine and cosine integrals. 609 softmax -- Softmax function. 610 log_softmax -- Logarithm of softmax function. 611 spence -- Spence's function, also known as the dilogarithm. 612 zeta -- Riemann zeta function. 613 zetac -- Riemann zeta function minus 1. 614 615Convenience functions 616--------------------- 617 618.. autosummary:: 619 :toctree: generated/ 620 621 cbrt -- Cube root of `x`. 622 exp10 -- 10**x. 623 exp2 -- 2**x. 624 radian -- Convert from degrees to radians. 625 cosdg -- Cosine of the angle `x` given in degrees. 626 sindg -- Sine of angle given in degrees. 627 tandg -- Tangent of angle x given in degrees. 628 cotdg -- Cotangent of the angle `x` given in degrees. 629 log1p -- Calculates log(1+x) for use when `x` is near zero. 630 expm1 -- exp(x) - 1 for use when `x` is near zero. 631 cosm1 -- cos(x) - 1 for use when `x` is near zero. 632 round -- Round to nearest integer. 633 xlogy -- Compute ``x*log(y)`` so that the result is 0 if ``x = 0``. 634 xlog1py -- Compute ``x*log1p(y)`` so that the result is 0 if ``x = 0``. 635 logsumexp -- Compute the log of the sum of exponentials of input elements. 636 exprel -- Relative error exponential, (exp(x)-1)/x, for use when `x` is near zero. 637 sinc -- Return the sinc function. 638 639""" 640 641from .sf_error import SpecialFunctionWarning, SpecialFunctionError 642 643from . import _ufuncs 644from ._ufuncs import * 645 646from . import _basic 647from ._basic import * 648 649from ._logsumexp import logsumexp, softmax, log_softmax 650 651from . import orthogonal 652from .orthogonal import * 653 654from .spfun_stats import multigammaln 655from ._ellip_harm import ( 656 ellip_harm, 657 ellip_harm_2, 658 ellip_normal 659) 660from ._lambertw import lambertw 661from ._spherical_bessel import ( 662 spherical_jn, 663 spherical_yn, 664 spherical_in, 665 spherical_kn 666) 667 668__all__ = _ufuncs.__all__ + _basic.__all__ + orthogonal.__all__ + [ 669 'SpecialFunctionWarning', 670 'SpecialFunctionError', 671 'orthogonal', # Not public, but kept in __all__ for back-compat 672 'logsumexp', 673 'softmax', 674 'log_softmax', 675 'multigammaln', 676 'ellip_harm', 677 'ellip_harm_2', 678 'ellip_normal', 679 'lambertw', 680 'spherical_jn', 681 'spherical_yn', 682 'spherical_in', 683 'spherical_kn', 684] 685 686from scipy._lib._testutils import PytestTester 687test = PytestTester(__name__) 688del PytestTester 689