### Table Of Contents

- Special functions (`scipy.special`)
  - Error handling
  - Available functions
    - Airy functions
    - Elliptic Functions and Integrals
    - Bessel Functions
      - Zeros of Bessel Functions
      - Faster versions of common Bessel Functions
      - Integrals of Bessel Functions
      - Derivatives of Bessel Functions
      - Spherical Bessel Functions
      - Riccati-Bessel Functions
    - Struve Functions
    - Raw Statistical Functions
    - Information Theory Functions
    - Gamma and Related Functions
    - Error Function and Fresnel Integrals
    - Legendre Functions
    - Ellipsoidal Harmonics
    - Orthogonal polynomials
    - Hypergeometric Functions
    - Parabolic Cylinder Functions
    - Mathieu and Related Functions
    - Spheroidal Wave Functions
    - Kelvin Functions
    - Combinatorics
    - Other Special Functions
    - Convenience Functions

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# Special functions (`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 emit warnings when an error occurs. By default this is disabled. To enable such messages use `errprint(1)`, and 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)

`errprint`([inflag]) | Sets or returns the error printing flag for special functions.
`SpecialFunctionWarning` | Warning that can be issued with `errprint(True)`

## Available functions

### Airy functions

`airy`(z) | Airy functions and their derivatives.
`airye`(z) | Exponentially scaled Airy functions and their derivatives.
`ai_zeros`(nt) | Compute nt zeros and values of the Airy function Ai and its derivative.
`bi_zeros`(nt) | Compute nt zeros and values of the Airy function Bi and its derivative.
`itairy`(x) | Integrals of Airy functions

### Elliptic Functions and Integrals

`ellipj`(u, m) | Jacobian elliptic functions
`ellipk`(m) | Complete elliptic integral of the first kind.
`ellipkm1`(p) | Complete elliptic integral of the first kind around m = 1
`ellipkinc`(phi, m) | Incomplete elliptic integral of the first kind
`ellipe`(m) | Complete elliptic integral of the second kind
`ellipeinc`(phi, m) | Incomplete elliptic integral of the second kind

### Bessel Functions

`jv`(v, z) | Bessel function of the first kind of real order v
`jn`(v, z) | Bessel function of the first kind of real order v
`jve`(v, z) | Exponentially scaled Bessel function of order v
`yn`(n, x) | Bessel function of the second kind of integer order
`yv`(v, z) | Bessel function of the second kind of real order
`yve`(v, z) | Exponentially scaled Bessel function of the second kind of real order
`kn`(n, x) | Modified Bessel function of the second kind of integer order n
`kv`(v, z) | Modified Bessel function of the second kind of real order v
`kve`(v, z) | Exponentially scaled modified Bessel function of the second kind.
`iv`(v, z) | Modified Bessel function of the first kind of real order
`ive`(v, z) | Exponentially scaled modified Bessel function of the first kind
`hankel1`(v, z) | Hankel function of the first kind
`hankel1e`(v, z) | Exponentially scaled Hankel function of the first kind
`hankel2`(v, z) | Hankel function of the second kind
`hankel2e`(v, z) | Exponentially scaled Hankel function of the second kind

The following is not an universal function:

`lmbda`(v, x) | Jahnke-Emden Lambda function, Lambdav(x).

#### Zeros of Bessel Functions

These are not universal functions:

`jnjnp_zeros`(nt) | Compute zeros of integer-order Bessel functions Jn and Jn’.
`jnyn_zeros`(n, nt) | Compute nt zeros of Bessel functions Jn(x), Jn’(x), Yn(x), and Yn’(x).
`jn_zeros`(n, nt) | Compute zeros of integer-order Bessel function Jn(x).
`jnp_zeros`(n, nt) | Compute zeros of integer-order Bessel function derivative Jn’(x).
`yn_zeros`(n, nt) | Compute zeros of integer-order Bessel function Yn(x).
`ynp_zeros`(n, nt) | Compute zeros of integer-order Bessel function derivative Yn’(x).
`y0_zeros`(nt[, complex]) | Compute nt zeros of Bessel function Y0(z), and derivative at each zero.
`y1_zeros`(nt[, complex]) | Compute nt zeros of Bessel function Y1(z), and derivative at each zero.
`y1p_zeros`(nt[, complex]) | Compute nt zeros of Bessel derivative Y1’(z), and value at each zero.

#### Faster versions of common Bessel Functions

`j0`(x) | Bessel function the first kind of order 0
`j1`(x) | Bessel function of the first kind of order 1
`y0`(x) | Bessel function of the second kind of order 0
`y1`(x) | Bessel function of the second kind of order 1
`i0`(x) | Modified Bessel function of order 0
`i0e`(x) | Exponentially scaled modified Bessel function of order 0.
`i1`(x) | Modified Bessel function of order 1
`i1e`(x) | Exponentially scaled modified Bessel function of order 1.
`k0`(x) | Modified Bessel function K of order 0
`k0e`(x) | Exponentially scaled modified Bessel function K of order 0
`k1`(x) | Modified Bessel function of the first kind of order 1
`k1e`(x) | Exponentially scaled modified Bessel function K of order 1

#### Integrals of Bessel Functions

`itj0y0`(x) | Integrals of Bessel functions of order 0
`it2j0y0`(x) | Integrals related to Bessel functions of order 0
`iti0k0`(x) | Integrals of modified Bessel functions of order 0
`it2i0k0`(x) | Integrals related to modified Bessel functions of order 0
`besselpoly`(a, lmb, nu) | Weighted integral of a Bessel function.

#### Derivatives of Bessel Functions

`jvp`(v, z[, n]) | Compute nth derivative of Bessel function Jv(z) with respect to z.
`yvp`(v, z[, n]) | Compute nth derivative of Bessel function Yv(z) with respect to z.
`kvp`(v, z[, n]) | Compute nth derivative of real-order modified Bessel function Kv(z)
`ivp`(v, z[, n]) | Compute nth derivative of modified Bessel function Iv(z) with respect to z.
`h1vp`(v, z[, n]) | Compute nth derivative of Hankel function H1v(z) with respect to z.
`h2vp`(v, z[, n]) | Compute nth derivative of Hankel function H2v(z) with respect to z.

#### Spherical Bessel Functions

These are not universal functions:

`sph_jn`(n, z) | Compute spherical Bessel function jn(z) and derivative.
`sph_yn`(n, z) | Compute spherical Bessel function yn(z) and derivative.
`sph_jnyn`(n, z) | Compute spherical Bessel functions jn(z) and yn(z) and derivatives.
`sph_in`(n, z) | Compute spherical Bessel function in(z) and derivative.
`sph_kn`(n, z) | Compute spherical Bessel function kn(z) and derivative.
`sph_inkn`(n, z) | Compute spherical Bessel functions in(z), kn(z), and derivatives.

#### Riccati-Bessel Functions

These are not universal functions:

`riccati_jn`(n, x) | Compute Ricatti-Bessel function of the first kind and derivative.
`riccati_yn`(n, x) | Compute Ricatti-Bessel function of the second kind and derivative.

### Struve Functions

`struve`(v, x) | Struve function
`modstruve`(v, x) | Modified Struve function
`itstruve0`(x) | Integral of the Struve function of order 0
`it2struve0`(x) | Integral related to Struve function of order 0
`itmodstruve0`(x) | Integral of the modified Struve function of order 0

### Raw Statistical Functions

See also

`scipy.stats`: Friendly versions of these functions.

`bdtr`(k, n, p) | Binomial distribution cumulative distribution function.
`bdtrc`(k, n, p) | Binomial distribution survival function.
`bdtri`(k, n, y) | Inverse function to `bdtr` vs.
`bdtrik`(y, n, p) | Inverse function to `bdtr` vs k
`bdtrin`(k, y, p) | Inverse function to `bdtr` vs n
`btdtr`(a, b, x) | Cumulative beta distribution.
`btdtri`(a, b, p) | p-th quantile of the beta distribution.
`btdtria`(p, b, x) | Inverse of `btdtr` vs a
`btdtrib`(a, p, x) | Inverse of `btdtr` vs b
`fdtr`(dfn, dfd, x) | F cumulative distribution function
`fdtrc`(dfn, dfd, x) | F survival function
`fdtri`(dfn, dfd, p) | Inverse to `fdtr` vs x
`fdtridfd`(dfn, p, x) | Inverse to `fdtr` vs dfd
`gdtr`(a, b, x) | Gamma distribution cumulative density function.
`gdtrc`(a, b, x) | Gamma distribution survival function.
`gdtria`(p, b, x[, out]) | Inverse of `gdtr` vs a.
`gdtrib`(a, p, x[, out]) | Inverse of `gdtr` vs b.
`gdtrix`(a, b, p[, out]) | Inverse of `gdtr` vs x.
`nbdtr`(k, n, p) | Negative binomial cumulative distribution function
`nbdtrc`(k, n, p) | Negative binomial survival function
`nbdtri`(k, n, y) | Inverse of `nbdtr` vs p
`nbdtrik`(y, n, p) | Inverse of `nbdtr` vs k
`nbdtrin`(k, y, p) | Inverse of `nbdtr` vs n
`ncfdtr`(dfn, dfd, nc, f) | Cumulative distribution function of the non-central F distribution.
`ncfdtridfd`(p, f, dfn, nc) | Calculate degrees of freedom (denominator) for the noncentral F-distribution.
`ncfdtridfn`(p, f, dfd, nc) | Calculate degrees of freedom (numerator) for the noncentral F-distribution.
`ncfdtri`(p, dfn, dfd, nc) | Inverse cumulative distribution function of the non-central F distribution.
`ncfdtrinc`(p, f, dfn, dfd) | Calculate non-centrality parameter for non-central F distribution.
`nctdtr`(df, nc, t) | Cumulative distribution function of the non-central t distribution.
`nctdtridf`(p, nc, t) | Calculate degrees of freedom for non-central t distribution.
`nctdtrit`(df, nc, p) | Inverse cumulative distribution function of the non-central t distribution.
`nctdtrinc`(df, p, t) | Calculate non-centrality parameter for non-central t distribution.
`nrdtrimn`(p, x, std) | Calculate mean of normal distribution given other params.
`nrdtrisd`(p, x, mn) | Calculate standard deviation of normal distribution given other params.
`pdtr`(k, m) | Poisson cumulative distribution function
`pdtrc`(k, m) | Poisson survival function
`pdtri`(k, y) | Inverse to `pdtr` vs m
`pdtrik`(p, m) | Inverse to `pdtr` vs k
`stdtr`(df, t) | Student t distribution cumulative density function
`stdtridf`(p, t) | Inverse of `stdtr` vs df
`stdtrit`(df, p) | Inverse of `stdtr` vs t
`chdtr`(v, x) | Chi square cumulative distribution function
`chdtrc`(v, x) | Chi square survival function
`chdtri`(v, p) | Inverse to `chdtrc`
`chdtriv`(p, x) | Inverse to `chdtr` vs v
`ndtr`(x) | Gaussian cumulative distribution function
`log_ndtr`(x) | Logarithm of Gaussian cumulative distribution function
`ndtri`(y) | Inverse of `ndtr` vs x
`chndtr`(x, df, nc) | Non-central chi square cumulative distribution function
`chndtridf`(x, p, nc) | Inverse to `chndtr` vs df
`chndtrinc`(x, df, p) | Inverse to `chndtr` vs nc
`chndtrix`(p, df, nc) | Inverse to `chndtr` vs x
`smirnov`(n, e) | Kolmogorov-Smirnov complementary cumulative distribution function
`smirnovi`(n, y) | Inverse to `smirnov`
`kolmogorov`(y) | Complementary cumulative distribution function of Kolmogorov distribution
`kolmogi`(p) | Inverse function to kolmogorov
`tklmbda`(x, lmbda) | Tukey-Lambda cumulative distribution function
`logit`(x) | Logit ufunc for ndarrays.
`expit`(x) | Expit ufunc for ndarrays.
`boxcox`(x, lmbda) | Compute the Box-Cox transformation.
`boxcox1p`(x, lmbda) | Compute the Box-Cox transformation of 1 + x.
`inv_boxcox`(y, lmbda) | Compute the inverse of the Box-Cox transformation.
`inv_boxcox1p`(y, lmbda) | Compute the inverse of the Box-Cox transformation.

### Information Theory Functions

`entr`(x) | Elementwise function for computing entropy.
`rel_entr`(x, y) | Elementwise function for computing relative entropy.
`kl_div`(x, y) | Elementwise function for computing Kullback-Leibler divergence.
`huber`(delta, r) | Huber loss function.
`pseudo_huber`(delta, r) | Pseudo-Huber loss function.

### Gamma and Related Functions

`gamma`(z) | Gamma function
`gammaln`(z) | Logarithm of absolute value of gamma function
`gammasgn`(x) | Sign of the gamma function.
`gammainc`(a, x) | Incomplete gamma function
`gammaincinv`(a, y) | Inverse to `gammainc`
`gammaincc`(a, x) | Complemented incomplete gamma integral
`gammainccinv`(a, y) | Inverse to `gammaincc`
`beta`(a, b) | Beta function.
`betaln`(a, b) | Natural logarithm of absolute value of beta function.
`betainc`(a, b, x) | Incomplete beta integral.
`betaincinv`(a, b, y) | Inverse function to beta integral.
`psi`(z) | Digamma function
`rgamma`(z) | Gamma function inverted
`polygamma`(n, x) | Polygamma function n.
`multigammaln`(a, d) | Returns the log of multivariate gamma, also sometimes called the generalized gamma.
`digamma`(z) | Digamma function
`poch`(z, m) | Rising factorial (z)_m

### Error Function and Fresnel Integrals

`erf`(z) | Returns the error function of complex argument.
`erfc`(x) | Complementary error function, 1 - erf(x).
`erfcx`(x) | Scaled complementary error function, exp(x^2) erfc(x).
`erfi`(z) | Imaginary error function, -i erf(i z).
`erfinv`(y) | Inverse function for erf.
`erfcinv`(y) | Inverse function for erfc.
`wofz`(z) | Faddeeva function
`dawsn`(x) | Dawson’s integral.
`fresnel`(z) | Fresnel sin and cos integrals
`fresnel_zeros`(nt) | Compute nt complex zeros of sine and cosine Fresnel integrals S(z) and C(z).
`modfresnelp`(x) | Modified Fresnel positive integrals
`modfresnelm`(x) | Modified Fresnel negative integrals

These are not universal functions:

`erf_zeros`(nt) | Compute nt complex zeros of error function erf(z).
`fresnelc_zeros`(nt) | Compute nt complex zeros of cosine Fresnel integral C(z).
`fresnels_zeros`(nt) | Compute nt complex zeros of sine Fresnel integral S(z).

### Legendre Functions

`lpmv`(m, v, x) | Associated legendre function of integer order.
`sph_harm`(m, n, theta, phi) | Compute spherical harmonics.

These are not universal functions:

`clpmn`(m, n, z[, type]) | Associated Legendre function of the first kind, Pmn(z).
`lpn`(n, z) | Legendre functions of the first kind, Pn(z).
`lqn`(n, z) | Legendre functions of the second kind, Qn(z).
`lpmn`(m, n, z) | Associated Legendre function of the first kind, Pmn(z).
`lqmn`(m, n, z) | Associated Legendre function of the second kind, Qmn(z).

### Ellipsoidal Harmonics

`ellip_harm`(h2, k2, n, p, s[, signm, signn]) | Ellipsoidal harmonic functions E^p_n(l)
`ellip_harm_2`(h2, k2, n, p, s) | Ellipsoidal harmonic functions F^p_n(l)
`ellip_normal`(h2, k2, n, p) | Ellipsoidal harmonic normalization constants gamma^p_n

### Orthogonal polynomials

The following functions evaluate values of orthogonal polynomials:

`assoc_laguerre`(x, n[, k]) | Compute nth-order generalized (associated) Laguerre polynomial.
`eval_legendre`(n, x[, out]) | Evaluate Legendre polynomial at a point.
`eval_chebyt`(n, x[, out]) | Evaluate Chebyshev T polynomial at a point.
`eval_chebyu`(n, x[, out]) | Evaluate Chebyshev U polynomial at a point.
`eval_chebyc`(n, x[, out]) | Evaluate Chebyshev C polynomial at a point.
`eval_chebys`(n, x[, out]) | Evaluate Chebyshev S polynomial at a point.
`eval_jacobi`(n, alpha, beta, x[, out]) | Evaluate Jacobi polynomial at a point.
`eval_laguerre`(n, x[, out]) | Evaluate Laguerre polynomial at a point.
`eval_genlaguerre`(n, alpha, x[, out]) | Evaluate generalized Laguerre polynomial at a point.
`eval_hermite`(n, x[, out]) | Evaluate Hermite polynomial at a point.
`eval_hermitenorm`(n, x[, out]) | Evaluate normalized Hermite polynomial at a point.
`eval_gegenbauer`(n, alpha, x[, out]) | Evaluate Gegenbauer polynomial at a point.
`eval_sh_legendre`(n, x[, out]) | Evaluate shifted Legendre polynomial at a point.
`eval_sh_chebyt`(n, x[, out]) | Evaluate shifted Chebyshev T polynomial at a point.
`eval_sh_chebyu`(n, x[, out]) | Evaluate shifted Chebyshev U polynomial at a point.
`eval_sh_jacobi`(n, p, q, x[, out]) | Evaluate shifted Jacobi polynomial at a point.

The functions below, in turn, return the polynomial coefficients in `orthopoly1d` objects, which function similarly as numpy.poly1d. The `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. Note that `orthopoly1d` objects are converted to `poly1d` when doing arithmetic, and lose information of the original orthogonal polynomial.

`legendre`(n[, monic]) | Legendre polynomial coefficients
`chebyt`(n[, monic]) | Return nth order Chebyshev polynomial of first kind, Tn(x).
`chebyu`(n[, monic]) | Return nth order Chebyshev polynomial of second kind, Un(x).
`chebyc`(n[, monic]) | Return n-th order Chebyshev polynomial of first kind, \(C_n(x)\).
`chebys`(n[, monic]) | Return nth order Chebyshev polynomial of second kind, \(S_n(x)\).
`jacobi`(n, alpha, beta[, monic]) | Returns the nth order Jacobi polynomial, P^(alpha,beta)_n(x) orthogonal over [-1,1] with weighting function (1-x)**alpha (1+x)**beta with alpha,beta > -1.
`laguerre`(n[, monic]) | Return the nth order Laguerre polynoimal, L_n(x), orthogonal over
`genlaguerre`(n, alpha[, monic]) | Returns the nth order generalized (associated) Laguerre polynomial,
`hermite`(n[, monic]) | Return the nth order Hermite polynomial, H_n(x), orthogonal over
`hermitenorm`(n[, monic]) | Return the nth order normalized Hermite polynomial, He_n(x), orthogonal
`gegenbauer`(n, alpha[, monic]) | Return the nth order Gegenbauer (ultraspherical) polynomial,
`sh_legendre`(n[, monic]) | Returns the nth order shifted Legendre polynomial, P^*_n(x), orthogonal over [0,1] with weighting function 1.
`sh_chebyt`(n[, monic]) | Return nth order shifted Chebyshev polynomial of first kind, Tn(x).
`sh_chebyu`(n[, monic]) | Return nth order shifted Chebyshev polynomial of second kind, Un(x).
`sh_jacobi`(n, p, q[, monic]) | Returns the nth order Jacobi polynomial, G_n(p,q,x) orthogonal over [0,1] with weighting function (1-x)**(p-q) (x)**(q-1) with p>q-1 and q > 0.

Warning

Computing values of high-order polynomials (around `order > 20`) using polynomial coefficients is numerically unstable. To evaluate polynomial values, the `eval_*` functions should be used instead.

Roots and weights for orthogonal polynomials

`c_roots`(n[, mu]) | Gauss-Chebyshev (first kind) quadrature.
`cg_roots`(n, alpha[, mu]) | Gauss-Gegenbauer quadrature.
`h_roots`(n[, mu]) | Gauss-Hermite (physicst’s) quadrature.
`he_roots`(n[, mu]) | Gauss-Hermite (statistician’s) quadrature.
`j_roots`(n, alpha, beta[, mu]) | Gauss-Jacobi quadrature.
`js_roots`(n, p1, q1[, mu]) | Gauss-Jacobi (shifted) quadrature.
`l_roots`(n[, mu]) | Gauss-Laguerre quadrature.
`la_roots`(n, alpha[, mu]) | Gauss-generalized Laguerre quadrature.
`p_roots`(n[, mu]) | Gauss-Legendre quadrature.
`ps_roots`(n[, mu]) | Gauss-Legendre (shifted) quadrature.
`s_roots`(n[, mu]) | Gauss-Chebyshev (second kind) quadrature.
`t_roots`(n[, mu]) | Gauss-Chebyshev (first kind) quadrature.
`ts_roots`(n[, mu]) | Gauss-Chebyshev (first kind, shifted) quadrature.
`u_roots`(n[, mu]) | Gauss-Chebyshev (second kind) quadrature.
`us_roots`(n[, mu]) | Gauss-Chebyshev (second kind, shifted) quadrature.

### Hypergeometric Functions

`hyp2f1`(a, b, c, z) | Gauss hypergeometric function 2F1(a, b; c; z).
`hyp1f1`(a, b, x) | Confluent hypergeometric function 1F1(a, b; x)
`hyperu`(a, b, x) | Confluent hypergeometric function U(a, b, x) of the second kind
`hyp0f1`(v, z) | Confluent hypergeometric limit function 0F1.
`hyp2f0`(a, b, x, type) | Hypergeometric function 2F0 in y and an error estimate
`hyp1f2`(a, b, c, x) | Hypergeometric function 1F2 and error estimate
`hyp3f0`(a, b, c, x) | Hypergeometric function 3F0 in y and an error estimate

### Parabolic Cylinder Functions

`pbdv`(v, x) | Parabolic cylinder function D
`pbvv`(v, x) | Parabolic cylinder function V
`pbwa`(a, x) | Parabolic cylinder function W

These are not universal functions:

`pbdv_seq`(v, x) | Parabolic cylinder functions Dv(x) and derivatives.
`pbvv_seq`(v, x) | Parabolic cylinder functions Vv(x) and derivatives.
`pbdn_seq`(n, z) | Parabolic cylinder functions Dn(z) and derivatives.

### Mathieu and Related Functions

`mathieu_a`(m, q) | Characteristic value of even Mathieu functions
`mathieu_b`(m, q) | Characteristic value of odd Mathieu functions

These are not universal functions:

`mathieu_even_coef`(m, q) | Fourier coefficients for even Mathieu and modified Mathieu functions.
`mathieu_odd_coef`(m, q) | Fourier coefficients for even Mathieu and modified Mathieu functions.

The following return both function and first derivative:

`mathieu_cem`(m, q, x) | Even Mathieu function and its derivative
`mathieu_sem`(m, q, x) | Odd Mathieu function and its derivative
`mathieu_modcem1`(m, q, x) | Even modified Mathieu function of the first kind and its derivative
`mathieu_modcem2`(m, q, x) | Even modified Mathieu function of the second kind and its derivative
`mathieu_modsem1`(m, q, x) | Odd modified Mathieu function of the first kind and its derivative
`mathieu_modsem2`(m, q, x) | Odd modified Mathieu function of the second kind and its derivative

### Spheroidal Wave Functions

`pro_ang1`(m, n, c, x) | Prolate spheroidal angular function of the first kind and its derivative
`pro_rad1`(m, n, c, x) | Prolate spheroidal radial function of the first kind and its derivative
`pro_rad2`(m, n, c, x) | Prolate spheroidal radial function of the secon kind and its derivative
`obl_ang1`(m, n, c, x) | Oblate spheroidal angular function of the first kind and its derivative
`obl_rad1`(m, n, c, x) | Oblate spheroidal radial function of the first kind and its derivative
`obl_rad2`(m, n, c, x) | Oblate spheroidal radial function of the second kind and its derivative.
`pro_cv`(m, n, c) | Characteristic value of prolate spheroidal function
`obl_cv`(m, n, c) | Characteristic value of oblate spheroidal function
`pro_cv_seq`(m, n, c) | Characteristic values for prolate spheroidal wave functions.
`obl_cv_seq`(m, n, c) | Characteristic values for oblate spheroidal wave functions.

The following functions require pre-computed characteristic value:

`pro_ang1_cv`(m, n, c, cv, x) | Prolate spheroidal angular function pro_ang1 for precomputed characteristic value
`pro_rad1_cv`(m, n, c, cv, x) | Prolate spheroidal radial function pro_rad1 for precomputed characteristic value
`pro_rad2_cv`(m, n, c, cv, x) | Prolate spheroidal radial function pro_rad2 for precomputed characteristic value
`obl_ang1_cv`(m, n, c, cv, x) | Oblate spheroidal angular function obl_ang1 for precomputed characteristic value
`obl_rad1_cv`(m, n, c, cv, x) | Oblate spheroidal radial function obl_rad1 for precomputed characteristic value
`obl_rad2_cv`(m, n, c, cv, x) | Oblate spheroidal radial function obl_rad2 for precomputed characteristic value

### Kelvin Functions

`kelvin`(x) | Kelvin functions as complex numbers
`kelvin_zeros`(nt) | Compute nt zeros of all Kelvin functions.
`ber`(x) | Kelvin function ber.
`bei`(x) | Kelvin function bei
`berp`(x) | Derivative of the Kelvin function `ber`
`beip`(x) | Derivative of the Kelvin function `bei`
`ker`(x) | Kelvin function ker
`kei`(x) | Kelvin function ker
`kerp`(x) | Derivative of the Kelvin function ker
`keip`(x) | Derivative of the Kelvin function kei

These are not universal functions:

`ber_zeros`(nt) | Compute nt zeros of the Kelvin function ber(x).
`bei_zeros`(nt) | Compute nt zeros of the Kelvin function bei(x).
`berp_zeros`(nt) | Compute nt zeros of the Kelvin function ber’(x).
`beip_zeros`(nt) | Compute nt zeros of the Kelvin function bei’(x).
`ker_zeros`(nt) | Compute nt zeros of the Kelvin function ker(x).
`kei_zeros`(nt) | Compute nt zeros of the Kelvin function kei(x).
`kerp_zeros`(nt) | Compute nt zeros of the Kelvin function ker’(x).
`keip_zeros`(nt) | Compute nt zeros of the Kelvin function kei’(x).

### Combinatorics

`comb`(N, k[, exact, repetition]) | The number of combinations of N things taken k at a time.
`perm`(N, k[, exact]) | Permutations of N things taken k at a time, i.e., k-permutations of N.

### Other Special Functions

`agm`(a, b) | Arithmetic, Geometric Mean.
`bernoulli`(n) | Bernoulli numbers B0..Bn (inclusive).
`binom`(n, k) | Binomial coefficient
`diric`(x, n) | Periodic sinc function, also called the Dirichlet function.
`euler`(n) | Euler numbers E0..En (inclusive).
`expn`(n, x) | Exponential integral E_n
`exp1`(z) | Exponential integral E_1 of complex argument z
`expi`(x) | Exponential integral Ei
`factorial`(n[, exact]) | The factorial function, n! = special.gamma(n+1).
`factorial2`(n[, exact]) | Double factorial.
`factorialk`(n, k[, exact]) | Multifactorial of n of order k, n(!!...!).
`shichi`(x) | Hyperbolic sine and cosine integrals
`sici`(x) | Sine and cosine integrals
`spence`(x) | Dilogarithm integral
`lambertw`(z[, k, tol]) | Lambert W function [R745].
`zeta`(x, q) | Hurwitz zeta function
`zetac`(x) | Riemann zeta function minus 1.

### Convenience Functions

`cbrt`(x) | Cube root of x
`exp10`(x) | 10**x
`exp2`(x) | 2**x
`radian`(d, m, s) | Convert from degrees to radians
`cosdg`(x) | Cosine of the angle x given in degrees.
`sindg`(x) | Sine of angle given in degrees
`tandg`(x) | Tangent of angle x given in degrees.
`cotdg`(x) | Cotangent of the angle x given in degrees.
`log1p`(x) | Calculates log(1+x) for use when x is near zero
`expm1`(x) | exp(x) - 1 for use when x is near zero.
`cosm1`(x) | cos(x) - 1 for use when x is near zero.
`round`(x) | Round to nearest integer
`xlogy`(x, y) | Compute `x*log(y)` so that the result is 0 if `x = 0`.
`xlog1py`(x, y) | Compute `x*log1p(y)` so that the result is 0 if `x = 0`.
`exprel`(x) | Relative error exponential, (exp(x)-1)/x, for use when x is near zero.
`sinc`(x) | Return the sinc function.