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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`([flag]) | Sets or returns the error printing flag for special functions.

## Available functions

### Airy functions

`airy`(x[, out1, out2, out3, out4]) | (Ai,Aip,Bi,Bip)=airy(z) calculates the Airy functions and their derivatives
`airye`(x[, out1, out2, out3, out4]) | (Aie,Aipe,Bie,Bipe)=airye(z) calculates the exponentially scaled Airy functions and
`ai_zeros`(nt) | Compute the zeros of Airy Functions Ai(x) and Ai’(x), a and a’
`bi_zeros`(nt) | Compute the zeros of Airy Functions Bi(x) and Bi’(x), b and b’

### Elliptic Functions and Integrals

`ellipj`(x1, x2[, out1, out2, out3, out4]) | (sn,cn,dn,ph)=ellipj(u,m) calculates the Jacobian elliptic functions of
`ellipk`(...) | This function is rather imprecise around m==1.
`ellipkm1`(...) |
`ellipkinc`(...) |
`ellipe`(...) |
`ellipeinc`(...) |

### Bessel Functions

`jn`(x1, x2[, out]) | y=jv(v,z) returns the Bessel function of real order v at complex z.
`jv`(x1, x2[, out]) | y=jv(v,z) returns the Bessel function of real order v at complex z.
`jve`(...) |
`yn`(x1, x2[, out]) | y=yn(n,x) returns the Bessel function of the second kind of integer
`yv`(x1, x2[, out]) | y=yv(v,z) returns the Bessel function of the second kind of real
`yve`(...) |
`kn`(x1, x2[, out]) | y=kn(n,x) returns the modified Bessel function of the second kind (sometimes called the third kind) for
`kv`(x1, x2[, out]) | y=kv(v,z) returns the modified Bessel function of the second kind (sometimes called the third kind) for
`kve`(v,z) returns the exponentially scaled, ...) |
`iv`(x1, x2[, out]) | y=iv(v,z) returns the modified Bessel function of real order v of
`ive`(...) |
`hankel1`(x1, x2[, out]) | y=hankel1(v,z) returns the Hankel function of the first kind for real order v and complex argument z.
`hankel1e`(...) |
`hankel2`(x1, x2[, out]) | y=hankel2(v,z) returns the Hankel function of the second kind for real order v and complex argument z.
`hankel2e`(...) |

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’
`jnyn_zeros`(n, nt) | Compute nt zeros of the Bessel functions Jn(x), Jn’(x), Yn(x), and
`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
`y1_zeros`(nt[, complex]) | Returns nt (complex or real) zeros of Y1(z), z1, and the value
`y1p_zeros`(nt[, complex]) | Returns nt (complex or real) zeros of Y1’(z), z1’, and the value

#### Faster versions of common Bessel Functions

`j0`(x[, out]) | y=j0(x) returns the Bessel function of order 0 at x.
`j1`(x[, out]) | y=j1(x) returns the Bessel function of order 1 at x.
`y0`(x[, out]) | y=y0(x) returns the Bessel function of the second kind of order 0 at x.
`y1`(x[, out]) | y=y1(x) returns the Bessel function of the second kind of order 1 at x.
`i0`(x[, out]) | y=i0(x) returns the modified Bessel function of order 0 at x.
`i0e`(x[, out]) | y=i0e(x) returns the exponentially scaled modified Bessel function
`i1`(x[, out]) | y=i1(x) returns the modified Bessel function of order 1 at x.
`i1e`(x[, out]) | y=i1e(x) returns the exponentially scaled modified Bessel function
`k0`(x[, out]) | y=k0(x) returns the modified Bessel function of the second kind (sometimes called the third kind) of
`k0e`(x[, out]) | y=k0e(x) returns the exponentially scaled modified Bessel function
`k1`(x[, out]) | y=i1(x) returns the modified Bessel function of the second kind (sometimes called the third kind) of
`k1e`(...) |

#### Integrals of Bessel Functions

`itj0y0`(x[, out1, out2]) | (ij0,iy0)=itj0y0(x) returns simple integrals from 0 to x of the zeroth order
`it2j0y0`(x[, out1, out2]) | (ij0,iy0)=it2j0y0(x) returns the integrals int((1-j0(t))/t,t=0..x) and
`iti0k0`(x[, out1, out2]) | (ii0,ik0)=iti0k0(x) returns simple integrals from 0 to x of the zeroth order
`it2i0k0`(x[, out1, out2]) | (ii0,ik0)=it2i0k0(x) returns the integrals int((i0(t)-1)/t,t=0..x) and
`besselpoly`(x1, x2, x3[, out]) | y=besselpoly(a,lam,nu) returns the value of the integral:

#### 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
`sph_yn`(n, z) | Compute the spherical Bessel function yn(z) and its derivative for
`sph_jnyn`(n, z) | Compute the spherical Bessel functions, jn(z) and yn(z) and their
`sph_in`(n, z) | Compute the spherical Bessel function in(z) and its derivative for
`sph_kn`(n, z) | Compute the spherical Bessel function kn(z) and its derivative for
`sph_inkn`(n, z) | Compute the spherical Bessel functions, in(z) and kn(z) and their

#### Riccati-Bessel Functions

These are not universal functions:

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

### Struve Functions

`struve`(x1, x2[, out]) | y=struve(v,x) returns the Struve function Hv(x) of order v at x, x
`modstruve`(x1, x2[, out]) | y=modstruve(v,x) returns the modified Struve function Lv(x) of order
`itstruve0`(x[, out]) | y=itstruve0(x) returns the integral of the Struve function of order 0
`it2struve0`(x[, out]) | y=it2struve0(x) returns the integral of the Struve function of order 0
`itmodstruve0`(x[, out]) | y=itmodstruve0(x) returns the integral of the modified Struve function

### Raw Statistical Functions

See also

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

`bdtr`(...) |
`bdtrc`(...[, j]) |
`bdtri`(x1, x2, x3[, out]) | p=bdtri(k,n,y) finds the probability p such that the sum of the
`btdtr`(x1, x2, x3[, out]) | y=btdtr(a,b,x) returns the area from zero to x under the beta
`btdtri`(x1, x2, x3[, out]) | x=btdtri(a,b,p) returns the pth quantile of the beta distribution. It is
`fdtr`(x1, x2, x3[, out]) | y=fdtr(dfn,dfd,x) returns the area from zero to x under the F density
`fdtrc`(x1, x2, x3[, out]) | y=fdtrc(dfn,dfd,x) returns the complemented F distribution function.
`fdtri`(x1, x2, x3[, out]) | x=fdtri(dfn,dfd,p) finds the F density argument x such that
`gdtr`(x1, x2, x3[, out]) | y=gdtr(a,b,x) returns the integral from zero to x of the gamma
`gdtrc`(x1, x2, x3[, out]) | y=gdtrc(a,b,x) returns the integral from x to infinity of the gamma
`gdtria`(x1, x2, x3[, out]) |
`gdtrib`(x1, x2, x3[, out]) |
`gdtrix`(x1, x2, x3[, out]) |
`nbdtr`(x1, x2, x3[, out]) | y=nbdtr(k,n,p) returns the sum of the terms 0 through k of the
`nbdtrc`(x1, x2, x3[, out]) | y=nbdtrc(k,n,p) returns the sum of the terms k+1 to infinity of the
`nbdtri`(x1, x2, x3[, out]) | p=nbdtri(k,n,y) finds the argument p such that nbdtr(k,n,p)=y.
`pdtr`(x1, x2[, out]) | y=pdtr(k,m) returns the sum of the first k terms of the Poisson
`pdtrc`(x1, x2[, out]) | y=pdtrc(k,m) returns the sum of the terms from k+1 to infinity of the
`pdtri`(x1, x2[, out]) | m=pdtri(k,y) returns the Poisson variable m such that the sum
`stdtr`(...[, x]) |
`stdtridf`(x1, x2[, out]) | t=stdtridf(p,t) returns the argument df such that stdtr(df,t) is equal to p.
`stdtrit`(x1, x2[, out]) | t=stdtrit(df,p) returns the argument t such that stdtr(df,t) is equal to p.
`chdtr`(...[, t]) |
`chdtrc`(...[, t]) |
`chdtri`(x1, x2[, out]) | x=chdtri(v,p) returns the argument x such that chdtrc(v,x) is equal
`ndtr`(...) |
`ndtri`(x[, out]) | x=ndtri(y) returns the argument x for which the area udnder the
`smirnov`(x1, x2[, out]) | y=smirnov(n,e) returns the exact Kolmogorov-Smirnov complementary
`smirnovi`(x1, x2[, out]) | e=smirnovi(n,y) returns e such that smirnov(n,e) = y.
`kolmogorov`(x[, out]) | p=kolmogorov(y) returns the complementary cumulative distribution
`kolmogi`(x[, out]) | y=kolmogi(p) returns y such that kolmogorov(y) = p
`tklmbda`(x1, x2[, out]) |
`logit`(x[, out]) | Logit ufunc for ndarrays.
`expit`(x[, out]) | Expit ufunc for ndarrays.

### Gamma and Related Functions

`gamma`(x[, out]) | y=gamma(z) returns the gamma function of the argument. The gamma
`gammaln`(...) | y=gammaln(z) returns the base e logarithm of the absolute value of the
`gammasgn`(x[, out]) | y=gammasgn(x) returns the sign of the gamma function.
`gammainc`(x1, x2[, out]) | y=gammainc(a,x) returns the incomplete gamma integral defined as
`gammaincinv`(x1, x2[, out]) | gammaincinv(a, y) returns x such that gammainc(a, x) = y.
`gammaincc`(x1, x2[, out]) | y=gammaincc(a,x) returns the complemented incomplete gamma integral
`gammainccinv`(x1, x2[, out]) | x=gammainccinv(a,y) returns x such that gammaincc(a,x) = y.
`beta`(...) |
`betaln`(x1, x2[, out]) | y=betaln(a,b) returns the natural logarithm of the absolute value of
`betainc`(a, b, x) | Compute the incomplete beta integral of the arguments, evaluated from zero to x:: gamma(a+b) / (gamma(a)*gamma(b)) * integral(t**(a-1) (1-t)**(b-1), t=0..x).
`betaincinv`(a,b,y) | Compute x such that betainc(a,b,x) = y.
`psi`(x[, out]) | y=psi(z) is the derivative of the logarithm of the gamma function
`rgamma`(x[, out]) | y=rgamma(z) returns one divided by the gamma function of x.
`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.

### Error Function and Fresnel Integrals

`erf`(z) | Returns the error function of complex argument.
`erfc`(x[, out]) | y=erfc(x) returns 1 - erf(x).
`erfcx`(x[, out]) | Scaled complementary error function, exp(x^2) erfc(x) ..
`erfi`(x[, out]) | Imaginary error function, -i erf(i z) ..
`erfinv`(y) |
`erfcinv`(y) |
`wofz`(...) | y=wofz(z) returns the value of the fadeeva function for complex argument
`dawsn`(x[, out]) | y=dawsn(x) returns dawson’s integral: exp(-x**2) *
`fresnel`(x[, out1, out2]) | (ssa,cca)=fresnel(z) returns the fresnel sin and cos integrals: integral(sin(pi/2
`fresnel_zeros`(nt) | Compute nt complex zeros of the sine and cosine Fresnel integrals
`modfresnelp`(x[, out1, out2]) | (fp,kp)=modfresnelp(x) returns the modified fresnel integrals F_+(x) and K_+(x)
`modfresnelm`(x[, out1, out2]) | (fm,km)=modfresnelp(x) returns the modified fresnel integrals `F_-(x)` and `K_-(x)`

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`(x1, x2, x3[, out]) | y=lpmv(m,v,x) returns the associated legendre function of integer order
`sph_harm` | Compute spherical harmonics.

These are not universal functions:

`lpn`(n, z) | Compute sequence of Legendre functions of the first kind (polynomials),
`lqn`(n, z) | Compute sequence of Legendre functions of the second kind,
`lpmn`(m, n, z) | Associated Legendre functions of the first kind, Pmn(z) and its
`lqmn`(m, n, z) | Associated Legendre functions of the second kind, Qmn(z) and its

### Orthogonal polynomials

The following functions evaluate values of orthogonal polynomials:

`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 orthopoly1d objects, which functions 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.

`legendre`(n[, monic]) | Returns the nth order Legendre polynomial, P_n(x), orthogonal over
`chebyt`(n[, monic]) | Return nth order Chebyshev polynomial of first kind, Tn(x). Orthogonal
`chebyu`(n[, monic]) | Return nth order Chebyshev polynomial of second kind, Un(x). Orthogonal
`chebyc`(n[, monic]) | Return nth order Chebyshev polynomial of first kind, Cn(x). Orthogonal
`chebys`(n[, monic]) | Return nth order Chebyshev polynomial of second kind, Sn(x). Orthogonal
`jacobi`(n, alpha, beta[, monic]) | Returns the nth order Jacobi polynomial, P^(alpha,beta)_n(x)
`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
`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)

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

`hyp2f1`(x1, x2, x3, x4[, out]) | y=hyp2f1(a,b,c,z) returns the gauss hypergeometric function
`hyp1f1`(x1, x2, x3[, out]) | y=hyp1f1(a,b,x) returns the confluent hypergeometeric function
`hyperu`(x1, x2, x3[, out]) | y=hyperu(a,b,x) returns the confluent hypergeometric function of the
`hyp0f1`(v, z) | Confluent hypergeometric limit function 0F1.
`hyp2f0`(x1, x2, x3, x4[, out1, out2]) | (y,err)=hyp2f0(a,b,x,type) returns (y,err) with the hypergeometric function 2F0 in y and an error estimate in err. The input type determines a convergence factor and
`hyp1f2`(x1, x2, x3, x4[, out1, out2]) | (y,err)=hyp1f2(a,b,c,x) returns (y,err) with the hypergeometric function 1F2 in y and an error estimate in err.
`hyp3f0`(x1, x2, x3, x4[, out1, out2]) | (y,err)=hyp3f0(a,b,c,x) returns (y,err) with the hypergeometric function 3F0 in y and an error estimate in err.

### Parabolic Cylinder Functions

`pbdv`(x1, x2[, out1, out2]) | (d,dp)=pbdv(v,x) returns (d,dp) with the parabolic cylinder function Dv(x) in
`pbvv`(x1, x2[, out1, out2]) | (v,vp)=pbvv(v,x) returns (v,vp) with the parabolic cylinder function Vv(x) in
`pbwa`(x1, x2[, out1, out2]) | (w,wp)=pbwa(a,x) returns (w,wp) with the parabolic cylinder function W(a,x) in

These are not universal functions:

`pbdv_seq`(v, x) | Compute sequence of parabolic cylinder functions Dv(x) and
`pbvv_seq`(v, x) | Compute sequence of parabolic cylinder functions Dv(x) and
`pbdn_seq`(n, z) | Compute sequence of parabolic cylinder functions Dn(z) and

### Mathieu and Related Functions

`mathieu_a`(x1, x2[, out]) | lmbda=mathieu_a(m,q) returns the characteristic value for the even solution,
`mathieu_b`(x1, x2[, out]) | lmbda=mathieu_b(m,q) returns the characteristic value for the odd solution,

These are not universal functions:

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

The following return both function and first derivative:

`mathieu_cem`(x1, x2, x3[, out1, out2]) | (y,yp)=mathieu_cem(m,q,x) returns the even Mathieu function, ce_m(x,q),
`mathieu_sem`(x1, x2, x3[, out1, out2]) | (y,yp)=mathieu_sem(m,q,x) returns the odd Mathieu function, se_m(x,q),
`mathieu_modcem1`(x1, x2, x3[, out1, out2]) | (y,yp)=mathieu_modcem1(m,q,x) evaluates the even modified Mathieu function
`mathieu_modcem2`(x1, x2, x3[, out1, out2]) | (y,yp)=mathieu_modcem2(m,q,x) evaluates the even modified Mathieu function
`mathieu_modsem1`(x1, x2, x3[, out1, out2]) | (y,yp)=mathieu_modsem1(m,q,x) evaluates the odd modified Mathieu function
`mathieu_modsem2`(x1, x2, x3[, out1, out2]) | (y,yp)=mathieu_modsem2(m,q,x) evaluates the odd modified Mathieu function

### Spheroidal Wave Functions

`pro_ang1`(x1, x2, x3, x4[, out1, out2]) | (s,sp)=pro_ang1(m,n,c,x) computes the prolate sheroidal angular function
`pro_rad1`(x1, x2, x3, x4[, out1, out2]) | (s,sp)=pro_rad1(m,n,c,x) computes the prolate sheroidal radial function
`pro_rad2`(x1, x2, x3, x4[, out1, out2]) | (s,sp)=pro_rad2(m,n,c,x) computes the prolate sheroidal radial function
`obl_ang1`(x1, x2, x3, x4[, out1, out2]) | (s,sp)=obl_ang1(m,n,c,x) computes the oblate sheroidal angular function
`obl_rad1`(x1, x2, x3, x4[, out1, out2]) | (s,sp)=obl_rad1(m,n,c,x) computes the oblate sheroidal radial function
`obl_rad2`(x1, x2, x3, x4[, out1, out2]) | (s,sp)=obl_rad2(m,n,c,x) computes the oblate sheroidal radial function
`pro_cv`(x1, x2, x3[, out]) | cv=pro_cv(m,n,c) computes the characteristic value of prolate spheroidal
`obl_cv`(x1, x2, x3[, out]) | cv=obl_cv(m,n,c) computes the characteristic value of oblate spheroidal
`pro_cv_seq`(m, n, c) | Compute a sequence of characteristic values for the prolate
`obl_cv_seq`(m, n, c) | Compute a sequence of characteristic values for the oblate

The following functions require pre-computed characteristic value:

`pro_ang1_cv`(x1, x2, x3, x4, x5[, out1, out2]) | (s,sp)=pro_ang1_cv(m,n,c,cv,x) computes the prolate sheroidal angular function
`pro_rad1_cv`(x1, x2, x3, x4, x5[, out1, out2]) | (s,sp)=pro_rad1_cv(m,n,c,cv,x) computes the prolate sheroidal radial function
`pro_rad2_cv`(x1, x2, x3, x4, x5[, out1, out2]) | (s,sp)=pro_rad2_cv(m,n,c,cv,x) computes the prolate sheroidal radial function
`obl_ang1_cv`(x1, x2, x3, x4, x5[, out1, out2]) | (s,sp)=obl_ang1_cv(m,n,c,cv,x) computes the oblate sheroidal angular function
`obl_rad1_cv`(x1, x2, x3, x4, x5[, out1, out2]) | (s,sp)=obl_rad1_cv(m,n,c,cv,x) computes the oblate sheroidal radial function
`obl_rad2_cv`(x1, x2, x3, x4, x5[, out1, out2]) | (s,sp)=obl_rad2_cv(m,n,c,cv,x) computes the oblate sheroidal radial function

### Kelvin Functions

`kelvin`(x[, out1, out2, out3, out4]) | (Be, Ke, Bep, Kep)=kelvin(x) returns the tuple (Be, Ke, Bep, Kep) which containes
`kelvin_zeros`(nt) | Compute nt zeros of all the Kelvin functions returned in a
`ber`(x[, out]) | y=ber(x) returns the Kelvin function ber x
`bei`(x[, out]) | y=bei(x) returns the Kelvin function bei x
`berp`(x[, out]) | y=berp(x) returns the derivative of the Kelvin function ber x
`beip`(x[, out]) | y=beip(x) returns the derivative of the Kelvin function bei x
`ker`(x[, out]) | y=ker(x) returns the Kelvin function ker x
`kei`(x[, out]) | y=kei(x) returns the Kelvin function ker x
`kerp`(x[, out]) | y=kerp(x) returns the derivative of the Kelvin function ker x
`keip`(x[, out]) | y=keip(x) returns the derivative of the Kelvin function kei x

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

### Other Special Functions

`binom`(n, k) | Binomial coefficient
`expn`(x1, x2[, out]) | y=expn(n,x) returns the exponential integral for integer n and
`exp1`(x[, out]) | y=exp1(z) returns the exponential integral (n=1) of complex argument
`expi`(x[, out]) | y=expi(x) returns an exponential integral of argument x defined as
`shichi`(x[, out1, out2]) | (shi,chi)=shichi(x) returns the hyperbolic sine and cosine integrals:
`sici`(x[, out1, out2]) | (si,ci)=sici(x) returns in si the integral of the sinc function from 0 to x:
`spence`(...) |
`lambertw`(z[, k, tol]) | Lambert W function [R332].
`zeta`(...) |
`zetac`(...[, k]) |

### Convenience Functions

`cbrt`(x[, out]) | y=cbrt(x) returns the real cube root of x.
`exp10`(x[, out]) | y=exp10(x) returns 10 raised to the x power.
`exp2`(x[, out]) | y=exp2(x) returns 2 raised to the x power.
`radian`(x1, x2, x3[, out]) | y=radian(d,m,s) returns the angle given in (d)egrees, (m)inutes, and
`cosdg`(x[, out]) | y=cosdg(x) calculates the cosine of the angle x given in degrees.
`sindg`(x[, out]) | y=sindg(x) calculates the sine of the angle x given in degrees.
`tandg`(x[, out]) | y=tandg(x) calculates the tangent of the angle x given in degrees.
`cotdg`(x[, out]) | y=cotdg(x) calculates the cotangent of the angle x given in degrees.
`log1p`(x[, out]) | y=log1p(x) calculates log(1+x) for use when x is near zero.
`expm1`(x[, out]) | y=expm1(x) calculates exp(x) - 1 for use when x is near zero.
`cosm1`(x[, out]) | y=calculates cos(x) - 1 for use when x is near zero.
`round`(x[, out]) | y=Returns the nearest integer to x as a double precision

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