### 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 the zeros of Airy Functions Ai(x) and Ai’(x), a and a’ respectively, and the associated values of Ai(a’) and Ai’(a).
`bi_zeros`(nt) | Compute the zeros of Airy Functions Bi(x) and Bi’(x), b and b’ respectively, and the associated values of Ai(b’) and Ai’(b).
`itairy`(x) | Integrals of Airy functios

### Elliptic Functions and Integrals

`ellipj`(u, m) | Jacobian elliptic functions Calculates the Jacobian elliptic functions of parameter m between 0 and 1, and real u.
`ellipk`(m) | Complete elliptic integral of the first kind This function is defined as ..
`ellipkm1`(p) | Complete elliptic integral of the first kind around m = 1 This function is defined as ..
`ellipkinc`(phi, m) | Incomplete elliptic integral of the first kind This function is defined as ..
`ellipe`(m) | Complete elliptic integral of the second kind This function is defined as ..
`ellipeinc`(phi, m) | Incomplete elliptic integral of the second kind This function is defined as ..

### Bessel Functions

`jv`(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 Returns the Bessel function of the second kind of integer order n at x.
`yv`(v,z) | Bessel function of the second kind of real order Returns the Bessel function of the second kind of real order v at complex z.
`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 These are also sometimes called functions of the third kind.
`kv`(v,z) | Modified Bessel function of the second kind of real order v Returns the modified Bessel function of the second kind (sometimes called the third kind) for real order v at complex z.
`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 :Parameters: v 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) | Compute sequence of lambda functions with arbitrary order v and their derivatives.

#### Zeros of Bessel Functions

These are not universal functions:

`jnjnp_zeros`(nt) | Compute nt (<=1200) zeros of the Bessel functions Jn and Jn’ and arange them in order of their magnitudes.
`jnyn_zeros`(n, nt) | Compute nt zeros of the Bessel functions Jn(x), Jn’(x), Yn(x), and Yn’(x), respectively.
`jn_zeros`(n, nt) | Compute nt zeros of the Bessel function Jn(x).
`jnp_zeros`(n, nt) | Compute nt zeros of the Bessel function Jn’(x).
`yn_zeros`(n, nt) | Compute nt zeros of the Bessel function Yn(x).
`ynp_zeros`(n, nt) | Compute nt zeros of the Bessel function Yn’(x).
`y0_zeros`(nt[, complex]) | Returns nt (complex or real) zeros of Y0(z), z0, and the value of Y0’(z0) = -Y1(z0) at each zero.
`y1_zeros`(nt[, complex]) | Returns nt (complex or real) zeros of Y1(z), z1, and the value of Y1’(z1) = Y0(z1) at each zero.
`y1p_zeros`(nt[, complex]) | Returns nt (complex or real) zeros of Y1’(z), z1’, and the value of Y1(z1’) 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 Returns the Bessel function of the second kind of order 0 at x.
`y1`(x) | Bessel function of the second kind of order 1 Returns the Bessel function of the second kind of order 1 at x.
`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 0.
`k0`(x) | Modified Bessel function K of order 0 Modified Bessel function of the second kind (sometimes called the third kind) of order 0.
`k0e`(x) | Exponentially scaled modified Bessel function K of order 0 Defined as:: k0e(x) = exp(x) * k0(x).
`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 Returns simple integrals from 0 to x of the zeroth order Bessel functions j0 and y0.
`it2j0y0`(x) | Integrals related to Bessel functions of order 0
`iti0k0`(x) | Integrals of modified Bessel functions of order 0 Returns simple integrals from 0 to x of the zeroth order modified Bessel functions i0 and k0.
`it2i0k0`(x) | Integrals related to modified Bessel functions of order 0
`besselpoly`(a, lmb, nu) | Weighed integral of a Bessel function.

#### Derivatives of Bessel Functions

`jvp`(v, z[, n]) | Return the nth derivative of Jv(z) with respect to z.
`yvp`(v, z[, n]) | Return the nth derivative of Yv(z) with respect to z.
`kvp`(v, z[, n]) | Return the nth derivative of Kv(z) with respect to z.
`ivp`(v, z[, n]) | Return the nth derivative of Iv(z) with respect to z.
`h1vp`(v, z[, n]) | Return the nth derivative of H1v(z) with respect to z.
`h2vp`(v, z[, n]) | Return the nth derivative of H2v(z) with respect to z.

#### Spherical Bessel Functions

These are not universal functions:

`sph_jn`(n, z) | Compute the spherical Bessel function jn(z) and its derivative for all orders up to and including n.
`sph_yn`(n, z) | Compute the spherical Bessel function yn(z) and its derivative for all orders up to and including n.
`sph_jnyn`(n, z) | Compute the spherical Bessel functions, jn(z) and yn(z) and their derivatives for all orders up to and including n.
`sph_in`(n, z) | Compute the spherical Bessel function in(z) and its derivative for all orders up to and including n.
`sph_kn`(n, z) | Compute the spherical Bessel function kn(z) and its derivative for all orders up to and including n.
`sph_inkn`(n, z) | Compute the spherical Bessel functions, in(z) and kn(z) and their derivatives for all orders up to and including n.

#### Riccati-Bessel Functions

These are not universal functions:

`riccati_jn`(n, x) | Compute the Ricatti-Bessel function of the first kind and its derivative for all orders up to and including n.
`riccati_yn`(n, x) | Compute the Ricatti-Bessel function of the second kind and its derivative for all orders up to and including n.

### Struve Functions

`struve`(v,x) | Struve function Computes the struve function Hv(x) of order v at x, x must be positive unless v is an integer.
`modstruve`(v, x) | Modified Struve function Returns the modified Struve function Lv(x) of order v at x, x must be positive unless v is an integer.
`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 Returns the area from zero to x under the F density function (also known as Snedcor’s density or the variance ratio density).
`fdtrc`(dfn, dfd, x) | F survival function Returns the complemented F distribution function.
`fdtri`(dfn, dfd, p) | Inverse to fdtr vs x Finds the F density argument x such that `fdtr(dfn, dfd, x) == p`.
`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 Returns the sum of the terms 0 through k of the negative binomial distribution:: sum((n+j-1)Cj p**n (1-p)**j,j=0..k).
`nbdtrc`(k,n,p) | Negative binomial survival function Returns the sum of the terms k+1 to infinity of the negative binomial distribution.
`nbdtri`(k, n, y) | Inverse of nbdtr vs p Finds the argument p such that `nbdtr(k,n,p) = y`.
`nbdtrik`(y,n,p) | Inverse of nbdtr vs k Finds the argument k such that `nbdtr(k,n,p) = y`.
`nbdtrin`(k,y,p) | Inverse of nbdtr vs n Finds the argument n such that `nbdtr(k,n,p) = y`.
`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 Returns the sum of the first k terms of the Poisson distribution: sum(exp(-m) * m**j / j!, j=0..k) = gammaincc( k+1, m).
`pdtrc`(k, m) | Poisson survival function Returns the sum of the terms from k+1 to infinity of the Poisson distribution: sum(exp(-m) * m**j / j!, j=k+1..inf) = gammainc( k+1, m).
`pdtri`(k,y) | Inverse to pdtr vs m Returns the Poisson variable m such that the sum from 0 to k of the Poisson density is equal to the given probability y: calculated by gammaincinv(k+1, y).
`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 Returns the argument df such that stdtr(df,t) is equal to p.
`stdtrit`(df,p) | Inverse of stdtr vs t Returns the argument t such that stdtr(df,t) is equal to p.
`chdtr`(v, x) | Chi square cumulative distribution function
`chdtrc`(v,x) | Chi square survival function
`chdtri`(v,p) | Inverse to chdtrc Returns the argument x such that `chdtrc(v,x) == p`.
`chdtriv`(p, x) | Inverse to chdtr vs v Returns the argument v such that `chdtr(v, x) == p`.
`ndtr`(x) | Gaussian cumulative distribution function
`log_ndtr`(x) | Logarithm of Gaussian cumulative distribution function
`ndtri`(y) | Inverse of ndtr vs x Returns the argument x for which the area under the Gaussian probability density function (integrated from minus infinity to x) is equal to y.
`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 Returns the exact Kolmogorov-Smirnov complementary cumulative distribution function (Dn+ or Dn-) for a one-sided test of equality between an empirical and a theoretical distribution.
`smirnovi`(n, y) | Inverse to smirnov Returns `e` such that `smirnov(n, e) = y`.
`kolmogorov`(y) | Complementary cumulative distribution function of Kolmogorov distribution Returns the complementary cumulative distribution function of Kolmogorov’s limiting distribution (Kn* for large n) of a two-sided test for equality between an empirical and a theoretical distribution.
`kolmogi`(p) | Inverse function to kolmogorov Returns y such that `kolmogorov(y) == p`.
`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.

### 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 The gamma function is often referred to as the generalized factorial since `z*gamma(z) = gamma(z+1)` and `gamma(n+1) = n!` for natural number n.
`gammaln`(z) | Logarithm of absolute value of gamma function Defined as:: ln(abs(gamma(z))) ..
`gammasgn`(x) | Sign of the gamma function.
`gammainc`(a, x) | Incomplete gamma function Defined as:: 1 / gamma(a) * integral(exp(-t) * t**(a-1), t=0..x) a must be positive and x must be >= 0.
`gammaincinv`(a, y) | Inverse to gammainc Returns x such that `gammainc(a, x) = y`.
`gammaincc`(a,x) | Complemented incomplete gamma integral Defined as:: 1 / gamma(a) * integral(exp(-t) * t**(a-1), t=x..inf) = 1 - gammainc(a,x) a must be positive and x must be >= 0.
`gammainccinv`(a,y) | Inverse to gammaincc Returns x such that `gammaincc(a,x) == y`.
`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 The derivative of the logarithm of the gamma function evaluated at z (also called the digamma function).
`rgamma`(z) | Gamma function inverted
`polygamma`(n, x) | Polygamma function which is the nth derivative of the digamma (psi) function.
`multigammaln`(a, d) | Returns the log of multivariate gamma, also sometimes called the generalized gamma.
`digamma`(z) | Digamma function The derivative of the logarithm of the gamma function evaluated at z (also called the digamma function).
`poch`(z, m) | Rising factorial (z)_m The Pochhammer symbol (rising factorial), is defined as:: (z)_m = gamma(z + m) / gamma(z) For positive integer m it reads:: (z)_m = z * (z + 1) * ...

### 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 Returns the value of the Faddeeva function for complex argument:: exp(-z**2)*erfc(-i*z) ..
`dawsn`(x) | Dawson’s integral.
`fresnel`(z) | Fresnel sin and cos integrals
`fresnel_zeros`(nt) | Compute nt complex zeros of the 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 the error function erf(z).
`fresnelc_zeros`(nt) | Compute nt complex zeros of the cosine Fresnel integral C(z).
`fresnels_zeros`(nt) | Compute nt complex zeros of the 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) Computes the (associated) Legendre function of the first kind of order m and degree n,:: Pmn(z) = P_n^m(z) and its derivative, `Pmn'(z)`.
`lpn`(n, z) | Compute sequence of Legendre functions of the first kind (polynomials), Pn(z) and derivatives for all degrees from 0 to n (inclusive).
`lqn`(n, z) | Compute sequence of Legendre functions of the second kind, Qn(z) and derivatives for all degrees from 0 to n (inclusive).
`lpmn`(m, n, z) | Associated Legendre function of the first kind, Pmn(z) Computes the associated Legendre function of the first kind of order m and degree n,:: Pmn(z) = P_n^m(z) and its derivative, `Pmn'(z)`.
`lqmn`(m, n, z) | Associated Legendre functions of the second kind, Qmn(z) and its derivative, `Qmn'(z)` of order m and degree n.

### Ellipsoidal Harmonics

`ellip_harm`(h2, k2, n, p, s[, signm, signn]) | Ellipsoidal harmonic functions E^p_n(l) These are also known as Lame functions of the first kind, and are solutions to the Lame equation: ..
`ellip_harm_2`(h2, k2, n, p, s) | Ellipsoidal harmonic functions F^p_n(l) These are also known as Lame functions of the second kind, and are solutions to the Lame equation: ..
`ellip_normal`(h2, k2, n, p) | Ellipsoidal harmonic normalization constants gamma^p_n The normalization constant is defined as ..

### Orthogonal polynomials

The following functions evaluate values of orthogonal polynomials:

`assoc_laguerre`(x, n[, k]) | Returns the n-th 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 Returns the nth-order Legendre polynomial, P_n(x), orthogonal over [-1, 1] with weight function 1.
`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 nth order Chebyshev polynomial of first kind, Cn(x).
`chebys`(n[, monic]) | Return nth order Chebyshev polynomial of second kind, Sn(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.

### 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 The parameter type determines a convergence factor and can be either 1 or 2.
`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 Returns (d,dp) the parabolic cylinder function Dv(x) in d and the derivative, Dv’(x) in dp.
`pbvv`(v,x) | Parabolic cylinder function V Returns the parabolic cylinder function Vv(x) in v and the derivative, Vv’(x) in vp.
`pbwa`(a,x) | Parabolic cylinder function W Returns the parabolic cylinder function W(a,x) in w and the derivative, W’(a,x) in wp.

These are not universal functions:

`pbdv_seq`(v, x) | Compute sequence of parabolic cylinder functions Dv(x) and their derivatives for Dv0(x)..Dv(x) with v0=v-int(v).
`pbvv_seq`(v, x) | Compute sequence of parabolic cylinder functions Dv(x) and their derivatives for Dv0(x)..Dv(x) with v0=v-int(v).
`pbdn_seq`(n, z) | Compute sequence of parabolic cylinder functions Dn(z) and their derivatives for D0(z)..Dn(z).

### Mathieu and Related Functions

`mathieu_a`(m,q) | Characteristic value of even Mathieu functions Returns the characteristic value for the even solution, `ce_m(z,q)`, of Mathieu’s equation.
`mathieu_b`(m,q) | Characteristic value of odd Mathieu functions Returns the characteristic value for the odd solution, `se_m(z,q)`, of Mathieu’s equation.

These are not universal functions:

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

The following return both function and first derivative:

`mathieu_cem`(m,q,x) | Even Mathieu function and its derivative Returns the even Mathieu function, `ce_m(x,q)`, of order m and parameter q evaluated at x (given in degrees).
`mathieu_sem`(m, q, x) | Odd Mathieu function and its derivative Returns the odd Mathieu function, se_m(x,q), of order m and parameter q evaluated at x (given in degrees).
`mathieu_modcem1`(m, q, x) | Even modified Mathieu function of the first kind and its derivative Evaluates the even modified Mathieu function of the first kind, `Mc1m(x,q)`, and its derivative at x for order m and parameter q.
`mathieu_modcem2`(m, q, x) | Even modified Mathieu function of the second kind and its derivative Evaluates the even modified Mathieu function of the second kind, Mc2m(x,q), and its derivative at x (given in degrees) for order m and parameter q.
`mathieu_modsem1`(m,q,x) | Odd modified Mathieu function of the first kind and its derivative Evaluates the odd modified Mathieu function of the first kind, Ms1m(x,q), and its derivative at x (given in degrees) for order m and parameter q.
`mathieu_modsem2`(m, q, x) | Odd modified Mathieu function of the second kind and its derivative Evaluates the odd modified Mathieu function of the second kind, Ms2m(x,q), and its derivative at x (given in degrees) for order m and parameter q.

### Spheroidal Wave Functions

`pro_ang1`(m,n,c,x) | Prolate spheroidal angular function of the first kind and its derivative Computes the prolate spheroidal angular function of the first kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and `|x| < 1.0`.
`pro_rad1`(m,n,c,x) | Prolate spheroidal radial function of the first kind and its derivative Computes the prolate spheroidal radial function of the first kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and `|x| < 1.0`.
`pro_rad2`(m,n,c,x) | Prolate spheroidal radial function of the secon kind and its derivative Computes the prolate spheroidal radial function of the second kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and `|x| < 1.0`.
`obl_ang1`(m, n, c, x) | Oblate spheroidal angular function of the first kind and its derivative Computes the oblate spheroidal angular function of the first kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and `|x| < 1.0`.
`obl_rad1`(m,n,c,x) | Oblate spheroidal radial function of the first kind and its derivative Computes the oblate spheroidal radial function of the first kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and `|x| < 1.0`.
`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 Computes the characteristic value of prolate spheroidal wave functions of order m,n (n>=m) and spheroidal parameter c.
`obl_cv`(m, n, c) | Characteristic value of oblate spheroidal function Computes the characteristic value of oblate spheroidal wave functions of order m,n (n>=m) and spheroidal parameter c.
`pro_cv_seq`(m, n, c) | Compute a sequence of characteristic values for the prolate spheroidal wave functions for mode m and n’=m..n and spheroidal parameter c.
`obl_cv_seq`(m, n, c) | Compute a sequence of characteristic values for the oblate spheroidal wave functions for mode m and n’=m..n and spheroidal parameter c.

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 Computes the prolate spheroidal angular function of the first kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and `|x| < 1.0`.
`pro_rad1_cv`(m,n,c,cv,x) | Prolate spheroidal radial function pro_rad1 for precomputed characteristic value Computes the prolate spheroidal radial function of the first kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and `|x| < 1.0`.
`pro_rad2_cv`(m,n,c,cv,x) | Prolate spheroidal radial function pro_rad2 for precomputed characteristic value Computes the prolate spheroidal radial function of the second kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and `|x| < 1.0`.
`obl_ang1_cv`(m, n, c, cv, x) | Oblate spheroidal angular function obl_ang1 for precomputed characteristic value Computes the oblate spheroidal angular function of the first kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and `|x| < 1.0`.
`obl_rad1_cv`(m,n,c,cv,x) | Oblate spheroidal radial function obl_rad1 for precomputed characteristic value Computes the oblate spheroidal radial function of the first kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and `|x| < 1.0`.
`obl_rad2_cv`(m,n,c,cv,x) | Oblate spheroidal radial function obl_rad2 for precomputed characteristic value Computes the oblate spheroidal radial function of the second kind and its derivative (with respect to x) for mode parameters m>=0 and n>=m, spheroidal parameter c and `|x| < 1.0`.

### Kelvin Functions

`kelvin`(x) | Kelvin functions as complex numbers :Returns: Be, Ke, Bep, Kep The tuple (Be, Ke, Bep, Kep) contains complex numbers representing the real and imaginary Kelvin functions and their derivatives evaluated at x.
`kelvin_zeros`(nt) | Compute nt zeros of all the Kelvin functions returned in a length 8 tuple of arrays of length nt.
`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 Start with a_0=a and b_0=b and iteratively compute a_{n+1} = (a_n+b_n)/2 b_{n+1} = sqrt(a_n*b_n) until a_n=b_n.
`bernoulli`(n) | Return an array of the Bernoulli numbers B0..Bn
`binom`(n, k) | Binomial coefficient
`diric`(x, n) | Return the periodic sinc function, also called the Dirichlet function.
`euler`(n) | Return an array of the Euler numbers E0..En (inclusive)
`expn`(n, x) | Exponential integral E_n Returns the exponential integral for integer n and non-negative x and n:: integral(exp(-x*t) / t**n, t=1..inf).
`exp1`(z) | Exponential integral E_1 of complex argument z :: integral(exp(-z*t)/t,t=1..inf).
`expi`(x) | Exponential integral Ei Defined as:: integral(exp(t)/t,t=-inf..x) See `expn` for a different exponential integral.
`factorial`(n[, exact]) | The factorial function, n! = special.gamma(n+1).
`factorial2`(n[, exact]) | Double factorial.
`factorialk`(n, k[, exact]) | n(!!...!) = multifactorial of order k
`shichi`(x) | Hyperbolic sine and cosine integrals :Returns: shi `integral(sinh(t)/t, t=0..x)` chi `eul + ln x + integral((cosh(t)-1)/t, t=0..x)` where `eul` is Euler’s constant.
`sici`(x) | Sine and cosine integrals :Returns: si `integral(sin(t)/t, t=0..x)` ci `eul + ln x + integral((cos(t) - 1)/t, t=0..x)` where `eul` is Euler’s constant.
`spence`(x) | Dilogarithm integral
`lambertw`(z[, k, tol]) | Lambert W function [R596].
`zeta`(x, q) | Hurwitz zeta function The Riemann zeta function of two arguments (also known as the Hurwitz zeta funtion).
`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 Returns the angle given in (d)egrees, (m)inutes, and (s)econds in 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 Returns the nearest integer to x as a double precision floating point result.
`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.