This is an archival dump of old wiki content --- see scipy.org for current material

This subpackage contains all the special functions.

### Airy Functions

• airy -- Airy functions and their derivatives.
• airye -- Exponentially scaled Airy functions
• ai_zeros -- [+]Zeros of Airy functions Ai(x) and Ai'(x)
• bi_zeros -- [+]Zeros of Airy functions Bi(x) and Bi'(x)

### Elliptic Functions and Integrals

• ellipj -- Jacobian elliptic functions
• ellipk -- Complete elliptic integral of the first kind.
• ellipkinc -- Incomplete elliptic integral of the first kind.
• ellipe -- Complete elliptic integral of the second kind.
• ellipeinc -- Incomplete elliptic integral of the second kind.

### Bessel Functions

• jn -- Bessel function of integer order and real argument.
• jv -- Bessel function of real-valued order and complex argument.
• jve -- Exponentially scaled Bessel function.
• yn -- Bessel function of second kind (integer order).
• yv -- Bessel function of the second kind (real-valued order).
• yve -- Exponentially scaled Bessel function of the second kind.
• kn -- Modified Bessel function of the third kind (integer order).
• kv -- Modified Bessel function of the third kind (real order).
• kve -- Exponentially scaled modified Bessel function of the third kind.
• iv -- Modified Bessel function.
• ive -- Exponentially scaled modified Bessel function.
• hankel1 -- Hankel function of the first kind.
• hankel1e -- Exponentially scaled Hankel function of the first kind.
• hankel2 -- Hankel function of the second kind.
• hankel2e -- Exponentially scaled Hankel function of the second kind.
• lmbda -- [+]Sequence of lambda functions with arbitrary order v.

#### Zeros of Bessel Functions

• jnjnp_zeros -- [+]Zeros of integer-order Bessel functions and derivatives sorted in order.
• jnyn_zeros -- [+]Zeros of integer-order Bessel functions and derivatives as separate arrays.
• jn_zeros -- [+]Zeros of Jn(x)
• jnp_zeros -- [+]Zeros of Jn'(x)
• yn_zeros -- [+]Zeros of Yn(x)
• ynp_zeros -- [+]Zeros of Yn'(x)
• y0_zeros -- [+]Complex zeros: Y0(z0)=0 and values of Y0'(z0)
• y1_zeros -- [+]Complex zeros: Y1(z1)=0 and values of Y1'(z1)
• y1p_zeros -- [+]Complex zeros of Y1'(z1')=0 and values of Y1(z1')

#### Faster versions of common Bessel Functions

• j0 -- Bessel function of order 0.
• j1 -- Bessel function of order 1.
• y0 -- Bessel function of second kind of order 0.
• y1 -- Bessel function of second kind of order 1.
• i0 -- Modified Bessel function of order 0.
• i0e -- Exponentially scaled modified Bessel function of order 0.
• i1 -- Modified Bessel function of order 1.
• i1e -- Exponentially scaled modified Bessel function of order 1.
• k0 -- Modified Bessel function of the third kind of order 0.
• k0e -- Exponentially scaled modified Bessel function of the third kind of order 0.
• k1 -- Modified Bessel function of the third kind of order 1.
• k1e -- Exponentially scaled modified Bessel function of the third kind of order 1.

#### Integrals of Bessel Functions

• itj0y0 -- Basic integrals of j0 and y0 from 0 to x.
• it2j0y0 -- Integrals of (1-j0(t))/t from 0 to x and y0(t)/t from x to inf.
• iti0k0 -- Basic integrals of i0 and k0 from 0 to x.
• it2i0k0 -- Integrals of (i0(t)-1)/t from 0 to x and k0(t)/t from x to inf.
• besselpoly -- Integral of a bessel function: Jv(2* a* x) * x[+]lambda from x=0 to 1.

#### Derivatives of Bessel Functions

• jvp -- Nth derivative of Jv(v,z)
• yvp -- Nth derivative of Yv(v,z)
• kvp -- Nth derivative of Kv(v,z)
• ivp -- Nth derivative of Iv(v,z)
• h1vp -- Nth derivative of H1v(v,z)
• h2vp -- Nth derivative of H2v(v,z)

#### Spherical Bessel Functions

• sph_jn -- [+]Sequence of spherical Bessel functions, jn(z)
• sph_yn -- [+]Sequence of spherical Bessel functions, yn(z)
• sph_jnyn -- [+]Sequence of spherical Bessel functions, jn(z) and yn(z)
• sph_in -- [+]Sequence of spherical Bessel functions, in(z)
• sph_kn -- [+]Sequence of spherical Bessel functions, kn(z)
• sph_inkn -- [+]Sequence of spherical Bessel functions, in(z) and kn(z)

#### Ricatti-Bessel Functions

• riccati_jn -- [+]Sequence of Ricatti-Bessel functions of first kind.
• riccati_yn -- [+]Sequence of Ricatti-Bessel functions of second kind.

### Struve Functions

• struve -- Struve function --- Hv(x)
• modstruve -- Modified struve function --- Lv(x)
• itstruve0 -- Integral of H0(t) from 0 to x
• it2struve0 -- Integral of H0(t)/t from x to Inf.
• itmodstruve0 -- Integral of L0(t) from 0 to x.

### Raw Statistical Functions (Friendly versions in scipy.stats)

• bdtr -- Sum of terms 0 through k of of the binomial pdf.
• bdtrc -- Sum of terms k+1 through n of the binomial pdf.
• bdtri -- Inverse of bdtr
• btdtr -- Integral from 0 to x of beta pdf.
• btdtri -- Quantiles of beta distribution
• fdtr -- Integral from 0 to x of F pdf.
• fdtrc -- Integral from x to infinity under F pdf.
• fdtri -- Inverse of fdtrc
• gdtr -- Integral from 0 to x of gamma pdf.
• gdtrc -- Integral from x to infinity under gamma pdf.
• gdtri -- Quantiles of gamma distribution
• nbdtr -- Sum of terms 0 through k of the negative binomial pdf.
• nbdtrc -- Sum of terms k+1 to infinity under negative binomial pdf.
• nbdtri -- Inverse of nbdtr
• pdtr -- Sum of terms 0 through k of the Poisson pdf.
• pdtrc -- Sum of terms k+1 to infinity of the Poisson pdf.
• pdtri -- Inverse of pdtr
• stdtr -- Integral from -infinity to t of the Student-t pdf.
• stdtri -- Inverse of stdtr (quantiles)
• chdtr -- Integral from 0 to x of the Chi-square pdf.
• chdtrc -- Integral from x to infnity of Chi-square pdf.
• chdtri -- Inverse of chdtrc.
• ndtr -- Integral from -infinity to x of standard normal pdf
• ndtri -- Inverse of ndtr (quantiles)
• smirnov -- Kolmogorov-Smirnov complementary CDF for one-sided test statistic (Dn+ or Dn-)
• smirnovi -- Inverse of smirnov.
• kolmogorov -- The complementary CDF of the (scaled) two-sided test statistic (Kn*) valid for large n.
• kolmogi -- Inverse of kolmogorov
• tklmbda -- Tukey-Lambda CDF

### Error Function and Fresnel Integrals

• erf -- Error function.
• erfc -- Complemented error function (1- erf(x))
• erfinv -- Inverse of error function
• erfcinv -- Inverse of erfc
• erf_zeros -- [+]Complex zeros of erf(z)
• fresnel -- Fresnel sine and cosine integrals.
• fresnel_zeros -- Complex zeros of both Fresnel integrals
• fresnelc_zeros -- [+]Complex zeros of fresnel cosine integrals
• fresnels_zeros -- [+]Complex zeros of fresnel sine integrals
• modfresnelp -- Modified Fresnel integrals F_+(x) and K_+(x)
• modfresnelm -- Modified Fresnel integrals F-(x) and K-(x)

### Legendre Functions

• lpn -- [+]Legendre Functions (polynomials) of the first kind
• lqn -- [+]Legendre Functions of the second kind.
• lpmn -- [+]Associated Legendre Function of the first kind.
• lqmn -- [+]Associated Legendre Function of the second kind.
• lpmv -- Associated Legendre Function of arbitrary non-negative degree v.
• sph_harm -- Spherical Harmonics (complex-valued) Y^m_n(theta,phi)
Orthogonal polynomials --- 15 types
These functions all return a polynomial class which can then be evaluated: vals = chebyt(n)(x) This class also has an attribute 'weights' which return 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 -- [+]Legendre polynomial P_n(x) (lpn -- for function).
• chebyt -- [+]Chebyshev polynomial T_n(x)
• chebyu -- [+]Chebyshev polynomial U_n(x)
• chebyc -- [+]Chebyshev polynomial C_n(x)
• chebys -- [+]Chebyshev polynomial S_n(x)
• jacobi -- [+]Jacobi polynomial P^(alpha,beta)_n(x)
• laguerre -- [+]Laguerre polynomial, L_n(x)
• genlaguerre -- [+]Generalized (Associated) Laguerre polynomial, L^alpha_n(x)
• hermite -- [+]Hermite polynomial H_n(x)
• hermitenorm -- [+]Normalized Hermite polynomial, He_n(x)
• gegenbauer -- [+]Gegenbauer (Ultraspherical) polynomials, C^(alpha)_n(x)
• sh_legendre -- [+]shifted Legendre polynomial, P*_n(x)
• sh_chebyt -- [+]shifted Chebyshev polynomial, T*_n(x)
• sh_chebyu -- [+]shifted Chebyshev polynomial, U*_n(x)
• sh_jacobi -- [+]shifted Jacobi polynomial, J*_n(x) = G^(p,q)_n(x)

### HyperGeometric Functions

• hyp2f1 -- Gauss hypergeometric function (2F1)
• hyp1f1 -- Confluent hypergeometric function (1F1)
• hyperu -- Confluent hypergeometric function (U)
• hyp0f1 -- Confluent hypergeometric limit function (0F1)
• hyp2f0 -- Hypergeometric function (2F0)
• hyp1f2 -- Hypergeometric function (1F2)
• hyp3f0 -- Hypergeometric function (3F0)

### Parabolic Cylinder Functions

• pbdv -- Parabolic cylinder function Dv(x) and derivative.
• pbvv -- Parabolic cylinder function Vv(x) and derivative.
• pbwa -- Parabolic cylinder function W(a,x) and derivative.
• pbdv_seq -- [+]Sequence of parabolic cylinder functions Dv(x)
• pbvv_seq -- [+]Sequence of parabolic cylinder functions Vv(x)
• pbdn_seq -- [+]Sequence of parabolic cylinder functions Dn(z), complex z

### Spheroidal Wave Functions

• pro_ang1 -- Prolate spheroidal angular function of the first kind
• obl_ang1 -- Oblate spheroidal angluar function of the first kind
• pro_cv -- Compute characteristic value for prolate functions
• obl_cv -- Compute characteristic value for oblate functions
• pro_cv_seq -- Compute sequence of prolate characteristic values
• obl_cv_seq -- Compute sequence of oblate characteristic values

The following functions require pre-computed characteristic values

• pro_ang1_cv -- Prolate spheroidal angular function of the first kind
• obl_ang1_cv -- Oblate spheroidal angluar function of the first kind

### Kelvin Functions

• kelvin -- All Kelvin functions (order 0) and derivatives.
• kelvin_zeros -- [+]Zeros of All Kelvin functions (order 0) and derivatives
• ber -- Kelvin function ber x
• bei -- Kelvin function bei x
• berp -- Derivative of Kelvin function ber x
• beip -- Derivative of Kelvin function bei x
• ker -- Kelvin function ker x
• kei -- Kelvin function kei x
• kerp -- Derivative of Kelvin function ker x
• keip -- Derivative of Kelvin function kei x
• ber_zeros -- [+]Zeros of Kelvin function bei x
• bei_zeros -- [+]Zeros of Kelvin function ber x
• berp_zeros -- [+]Zeros of derivative of Kelvin function ber x
• beip_zeros -- [+]Zeros of derivative of Kelvin function bei x
• ker_zeros -- [+]Zeros of Kelvin function kei x
• kei_zeros -- [+]Zeros of Kelvin function ker x
• kerp_zeros -- [+]Zeros of derivative of Kelvin function ker x
• keip_zeros -- [+]Zeros of derivative of Kelvin function kei x

### Other Special Functions

• expn -- Exponential integral.
• exp1 -- Exponential integral of order 1 (for complex argument)
• expi -- Another exponential integral -- Ei(x)
• dawsn -- Dawson's integral.
• shichi -- Hyperbolic sine and cosine integrals.
• sici -- Integral of the sinc and "cosinc" functions.
• spence -- Dilogarithm integral.
• zeta -- Riemann zeta function of two arguments.
• zetac -- 1.0 - standard Riemann zeta function.

### Convenience Functions

• cbrt -- Cube root.
• exp10 -- 10 raised to the x power.
• exp2 -- 2 raised to the x power.
• cosdg -- cosine of the angle given in degrees.
• sindg -- sine of the angle given in degrees.
• tandg -- tangent of the angle given in degrees.
• cotdg -- cotangent of the angle given in degrees.
• log1p -- log(1+x)
• expm1 -- exp(x)-1
• cosm1 -- cos(x)-1
• round -- round the argument to the nearest integer. If argument ends in 0.5 exactly, pick the nearest even integer.

[+] in the description indicates a function which is not a universal function and does not follow broadcasting and automatic array-looping rules.

### Error handling

Errors are handled by returning nans, or other appropriate values. Some of the special function routines will print an error message when an error occurs. By default this printing is disabled. To enable such messages use errprint(1) 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)
```

SciPy: SciPyPackages/Special (last edited 2015-10-24 17:48:23 by anonymous)