# 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 errprint({flag}) sets 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(x[, out]) y=ellipkm1(1 - m) returns the complete integral of the first kind: ellipkinc(x1, x2[, out]) y=ellipkinc(phi,m) returns the incomplete elliptic integral of the first ellipe(x[, out]) y=ellipe(m) returns the complete integral of the second kind: ellipeinc(x1, x2[, out]) y=ellipeinc(phi,m) returns the incomplete elliptic integral of the

### 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(x1, x2[, out]) y=jve(v,z) returns the exponentially scaled Bessel function of real order 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(x1, x2[, out]) y=yve(v,z) returns the exponentially scaled Bessel function of the second 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(x1, x2[, out]) y=kve(v,z) returns the exponentially scaled, modified Bessel function iv(x1, x2[, out]) y=iv(v,z) returns the modified Bessel function of real order v of ive(x1, x2[, out]) y=ive(v,z) returns the exponentially scaled modified Bessel function of 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(x1, x2[, out]) y=hankel1e(v,z) returns the exponentially scaled Hankel function of the first 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(x1, x2[, out]) y=hankel2e(v,z) returns the exponentially scaled Hankel function of the second

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(x[, out]) y=k1e(x) returns the exponentially scaled modified Bessel function

#### 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¶

scipy.stats: Friendly versions of these functions.

 bdtr(x1, x2, x3[, out]) y=bdtr(k,n,p) returns the sum of the terms 0 through k of the bdtrc(x1, x2, x3[, out]) y=bdtrc(k,n,p) returns the sum of the terms k+1 through n of the 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(x1, x2[, out]) p=stdtr(df,t) returns the integral from minus infinity to t of the Student t 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(x1, x2[, out]) p=chdtr(v,x) Returns the area under the left hand tail (from 0 to x) of the Chi chdtrc(x1, x2[, out]) p=chdtrc(v,x) returns the area under the right hand tail (from x to chdtri(x1, x2[, out]) x=chdtri(v,p) returns the argument x such that chdtrc(v,x) is equal ndtr(x[, out]) y=ndtr(x) returns the area under the standard Gaussian probability 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]) NULL expit(x[, out]) NULL

### Error Function and Fresnel Integrals¶

 erf(x[, out]) y=erf(z) returns the error function of complex argument defined as erfc(x[, out]) y=erfc(x) returns 1 - erf(x). erfinv(y) erfcinv(y) 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) amd 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 Evaluate Legendre polynomial at a point. eval_chebyt Evaluate Chebyshev T polynomial at a point. eval_chebyu Evaluate Chebyshev U polynomial at a point. eval_chebyc Evaluate Chebyshev C polynomial at a point. eval_chebys Evaluate Chebyshev S polynomial at a point. eval_jacobi Evaluate Jacobi polynomial at a point. eval_laguerre Evaluate Laguerre polynomial at a point. eval_genlaguerre Evaluate generalized Laguerre polynomial at a point. eval_hermite Evaluate Hermite polynomial at a point. eval_hermitenorm Evaluate normalized Hermite polynomial at a point. eval_gegenbauer Evaluate Gegenbauer polynomial at a point. eval_sh_legendre Evaluate shifted Legendre polynomial at a point. eval_sh_chebyt Evaluate shifted Chebyshev T polynomial at a point. eval_sh_chebyu Evaluate shifted Chebyshev U polynomial at a point. eval_sh_jacobi 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

### Spheroidal Wave Functions¶

The following functions require pre-computed characteristic value:

### 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 length 8 tuple of arrays of length nt. 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¶

 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 wofz(x[, out]) 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) * 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(x[, out]) y=spence(x) returns the dilogarithm integral: -integral(log t / lambertw(z[, k, tol]) Lambert W function. zeta(x1, x2[, out]) y=zeta(x,q) returns the Riemann zeta function of two arguments: zetac(x[, out]) y=zetac(x) returns 1.0 - the Riemann zeta function: sum(k**(-x), k=2..inf)

### 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