7
airy -- Airy functions and their derivatives.
8
airye -- Exponentially scaled Airy functions
9
ai_zeros -- **Zeros of Airy functions Ai(x) and Ai'(x)
10
bi_zeros -- **Zeros of Airy functions Bi(x) and Bi'(x)
12
Elliptic Functions and Integrals
14
ellipj -- Jacobian elliptic functions
15
ellipk -- Complete elliptic integral of the first kind.
16
ellipkinc -- Incomplete elliptic integral of the first kind.
17
ellipe -- Complete elliptic integral of the second kind.
18
ellipeinc -- Incomplete elliptic integral of the second kind.
22
jn -- Bessel function of integer order and real argument.
23
jv -- Bessel function of real-valued order and complex argument.
24
jve -- Exponentially scaled Bessel function.
25
yn -- Bessel function of second kind (integer order).
26
yv -- Bessel function of the second kind (real-valued order).
27
yve -- Exponentially scaled Bessel function of the second kind.
28
kn -- Modified Bessel function of the third kind (integer order).
29
kv -- Modified Bessel function of the third kind (real order).
30
kve -- Exponentially scaled modified Bessel function of the
32
iv -- Modified Bessel function.
33
ive -- Exponentially scaled modified Bessel function.
34
hankel1 -- Hankel function of the first kind.
35
hankel1e -- Exponentially scaled Hankel function of the first kind.
36
hankel2 -- Hankel function of the second kind.
37
hankel2e -- Exponentially scaled Hankel function of the second kind.
39
lmbda -- **Sequence of lambda functions with arbitrary order v.
41
Zeros of Bessel Functions
43
jnjnp_zeros -- **Zeros of integer-order Bessel functions and derivatives
45
jnyn_zeros -- **Zeros of integer-order Bessel functions and derivatives
47
jn_zeros -- **Zeros of Jn(x)
48
jnp_zeros -- **Zeros of Jn'(x)
49
yn_zeros -- **Zeros of Yn(x)
50
ynp_zeros -- **Zeros of Yn'(x)
51
y0_zeros -- **Complex zeros: Y0(z0)=0 and values of Y0'(z0)
52
y1_zeros -- **Complex zeros: Y1(z1)=0 and values of Y1'(z1)
53
y1p_zeros -- **Complex zeros of Y1'(z1')=0 and values of Y1(z1')
55
Faster versions of common Bessel Functions.
57
j0 -- Bessel function of order 0.
58
j1 -- Bessel function of order 1.
59
y0 -- Bessel function of second kind of order 0.
60
y1 -- Bessel function of second kind of order 1.
61
i0 -- Modified Bessel function of order 0.
62
i0e -- Exponentially scaled modified Bessel function of order 0.
63
i1 -- Modified Bessel function of order 1.
64
i1e -- Exponentially scaled modified Bessel function of order 1.
65
k0 -- Modified Bessel function of the third kind of order 0.
66
k0e -- Exponentially scaled modified Bessel function of the
67
third kind of order 0.
68
k1 -- Modified Bessel function of the third kind of order 1.
69
k1e -- Exponentially scaled modified Bessel function of the
70
third kind of order 1.
72
Integrals of Bessel Functions.
74
itj0y0 -- Basic integrals of j0 and y0 from 0 to x.
75
it2j0y0 -- Integrals of (1-j0(t))/t from 0 to x and
76
y0(t)/t from x to inf.
77
iti0k0 -- Basic integrals of i0 and k0 from 0 to x.
78
it2i0k0 -- Integrals of (i0(t)-1)/t from 0 to x and
79
k0(t)/t from x to inf.
80
besselpoly -- Integral of a bessel function: Jv(2*a*x) * x**lambda
83
Derivatives of Bessel Functions.
85
jvp -- Nth derivative of Jv(v,z)
86
yvp -- Nth derivative of Yv(v,z)
87
kvp -- Nth derivative of Kv(v,z)
88
ivp -- Nth derivative of Iv(v,z)
89
h1vp -- Nth derivative of H1v(v,z)
90
h2vp -- Nth derivative of H2v(v,z)
92
Spherical Bessel Functions
94
sph_jn -- **Sequence of spherical Bessel functions, jn(z)
95
sph_yn -- **Sequence of spherical Bessel functions, yn(z)
96
sph_jnyn -- **Sequence of spherical Bessel functions, jn(z) and yn(z)
97
sph_in -- **Sequence of spherical Bessel functions, in(z)
98
sph_kn -- **Sequence of spherical Bessel functions, kn(z)
99
sph_inkn -- **Sequence of spherical Bessel functions, in(z) and kn(z)
101
Ricatti-Bessel Functions
103
ricatti_jn -- **Sequence of Ricatti-Bessel functions of first kind.
104
ricatti_yn -- **Sequence of Ricatti-Bessel functions of second kind.
108
struve -- Struve function --- Hv(x)
109
modstruve -- Modified struve function --- Lv(x)
110
itstruve0 -- Integral of H0(t) from 0 to x
111
it2struve0 -- Integral of H0(t)/t from x to Inf.
112
itmodstruve0 -- Integral of L0(t) from 0 to x.
115
Raw Statistical Functions (Friendly versions in scipy.stats)
117
bdtr -- Sum of terms 0 through k of of the binomial pdf.
118
bdtrc -- Sum of terms k+1 through n of the binomial pdf.
119
bdtri -- Inverse of bdtr
120
btdtr -- Integral from 0 to x of beta pdf.
121
btdtri -- Quantiles of beta distribution
122
fdtr -- Integral from 0 to x of F pdf.
123
fdtrc -- Integral from x to infinity under F pdf.
124
fdtri -- Inverse of fdtrc
125
gdtr -- Integral from 0 to x of gamma pdf.
126
gdtrc -- Integral from x to infinity under gamma pdf.
127
gdtri -- Quantiles of gamma distribution
128
nbdtr -- Sum of terms 0 through k of the negative binomial pdf.
129
nbdtrc -- Sum of terms k+1 to infinity under negative binomial pdf.
130
nbdtri -- Inverse of nbdtr
131
pdtr -- Sum of terms 0 through k of the Poisson pdf.
132
pdtrc -- Sum of terms k+1 to infinity of the Poisson pdf.
133
pdtri -- Inverse of pdtr
134
stdtr -- Integral from -infinity to t of the Student-t pdf.
135
stdtri -- Inverse of stdtr (quantiles)
136
chdtr -- Integral from 0 to x of the Chi-square pdf.
137
chdtrc -- Integral from x to infnity of Chi-square pdf.
138
chdtri -- Inverse of chdtrc.
139
ndtr -- Integral from -infinity to x of standard normal pdf
140
ndtri -- Inverse of ndtr (quantiles)
141
smirnov -- Kolmogorov-Smirnov complementary CDF for one-sided
142
test statistic (Dn+ or Dn-)
143
smirnovi -- Inverse of smirnov.
144
kolmogorov -- The complementary CDF of the (scaled) two-sided test
145
statistic (Kn*) valid for large n.
146
kolmogi -- Inverse of kolmogorov
147
tklmbda -- Tukey-Lambda CDF
149
Gamma and Related Functions
151
gamma -- Gamma function.
152
gammaln -- Log of the absolute value of the gamma function.
153
gammainc -- Incomplete gamma integral.
154
gammaincinv -- Inverse of gammainc.
155
gammaincc -- Complemented incomplete gamma integral.
156
gammainccinv -- Inverse of gammaincc.
157
beta -- Beta function.
158
betaln -- Log of the absolute value of the beta function.
159
betainc -- Incomplete beta integral.
160
betaincinv -- Inverse of betainc.
161
betaincinva -- Inverse (in first argument, a) of betainc
162
betaincinvb -- Inverse (in first argument, b) of betainc
163
psi(digamma) -- Logarithmic derivative of the gamma function.
164
rgamma -- One divided by the gamma function.
165
polygamma -- Nth derivative of psi function.
167
Error Function and Fresnel Integrals
169
erf -- Error function.
170
erfc -- Complemented error function (1- erf(x))
171
erfinv -- Inverse of error function
172
erfcinv -- Inverse of erfc
173
erf_zeros -- **Complex zeros of erf(z)
174
fresnel -- Fresnel sine and cosine integrals.
175
fresnel_zeros -- Complex zeros of both Fresnel integrals
176
fresnelc_zeros -- **Complex zeros of fresnel cosine integrals
177
fresnels_zeros -- **Complex zeros of fresnel sine integrals
178
modfresnelp -- Modified Fresnel integrals F_+(x) and K_+(x)
179
modfresnelm -- Modified Fresnel integrals F_-(x) and K_-(x)
183
lpn -- **Legendre Functions (polynomials) of the first kind
184
lqn -- **Legendre Functions of the second kind.
185
lpmn -- **Associated Legendre Function of the first kind.
186
lqmn -- **Associated Legendre Function of the second kind.
187
lpmv -- Associated Legendre Function of arbitrary non-negative
189
sph_harm -- Spherical Harmonics (complex-valued) Y^m_n(theta,phi)
191
Orthogonal polynomials --- 15 types
192
** These functions all return a polynomial class which can then be
193
evaluated: vals = chebyt(n)(x)
194
This class also has an attribute 'weights' which
195
return the roots, weights, and total weights for the appropriate
196
form of Gaussian quadrature. These are returned in an n x 3 array with roots
197
in the first column, weights in the second column, and total weights in the final
200
legendre -- **Legendre polynomial P_n(x) (lpn -- for function).
201
chebyt -- **Chebyshev polynomial T_n(x)
202
chebyu -- **Chebyshev polynomial U_n(x)
203
chebyc -- **Chebyshev polynomial C_n(x)
204
chebys -- **Chebyshev polynomial S_n(x)
205
jacobi -- **Jacobi polynomial P^(alpha,beta)_n(x)
206
laguerre -- **Laguerre polynomial, L_n(x)
207
genlaguerre -- **Generalized (Associated) Laguerre polynomial, L^alpha_n(x)
208
hermite -- **Hermite polynomial H_n(x)
209
hermitenorm -- **Normalized Hermite polynomial, He_n(x)
210
gegenbauer -- **Gegenbauer (Ultraspherical) polynomials, C^(alpha)_n(x)
211
sh_legendre -- **shifted Legendre polynomial, P*_n(x)
212
sh_chebyt -- **shifted Chebyshev polynomial, T*_n(x)
213
sh_chebyu -- **shifted Chebyshev polynomial, U*_n(x)
214
sh_jacobi -- **shifted Jacobi polynomial, J*_n(x) = G^(p,q)_n(x)
216
HyperGeometric Functions
218
hyp2f1 -- Gauss hypergeometric function (2F1)
219
hyp1f1 -- Confluent hypergeometric function (1F1)
220
hyperu -- Confluent hypergeometric function (U)
221
hyp0f1 -- Confluent hypergeometric limit function (0F1)
222
hyp2f0 -- Hypergeometric function (2F0)
223
hyp1f2 -- Hypergeometric function (1F2)
224
hyp3f0 -- Hypergeometric function (3F0)
226
Parabolic Cylinder Functions
228
pbdv -- Parabolic cylinder function Dv(x) and derivative.
229
pbvv -- Parabolic cylinder function Vv(x) and derivative.
230
pbwa -- Parabolic cylinder function W(a,x) and derivative.
231
pbdv_seq -- **Sequence of parabolic cylinder functions Dv(x)
232
pbvv_seq -- **Sequence of parabolic cylinder functions Vv(x)
233
pbdn_seq -- **Sequence of parabolic cylinder functions Dn(z), complex z
235
mathieu and Related Functions (and derivatives)
237
mathieu_a -- Characteristic values for even solution (ce_m)
238
mathieu_b -- Characteristic values for odd solution (se_m)
239
mathieu_even_coef -- **sequence of expansion coefficients for even solution
240
mathieu_odd_coef -- **sequence of expansion coefficients for odd solution
241
** All the following return both function and first derivative **
242
mathieu_cem -- Even mathieu function
243
mathieu_sem -- Odd mathieu function
244
mathieu_modcem1 -- Even modified mathieu function of the first kind
245
mathieu_modcem2 -- Even modified mathieu function of the second kind
246
mathieu_modsem1 -- Odd modified mathieu function of the first kind
247
mathieu_modsem2 -- Odd modified mathieu function of the second kind
249
Spheroidal Wave Functions
251
pro_ang1 -- Prolate spheroidal angular function of the first kind
252
pro_rad1 -- Prolate spheroidal radial function of the first kind
253
pro_rad2 -- Prolate spheroidal radial function of the second kind
254
obl_ang1 -- Oblate spheroidal angluar function of the first kind
255
obl_rad1 -- Oblate spheroidal radial function of the first kind
256
obl_rad2 -- Oblate spheroidal radial function of the second kind
257
pro_cv -- Compute characteristic value for prolate functions
258
obl_cv -- Compute characteristic value for oblate functions
259
pro_cv_seq -- Compute sequence of prolate characteristic values
260
obl_cv_seq -- Compute sequence of oblate characteristic values
261
** The following functions require pre-computed characteristic values **
262
pro_ang1_cv -- Prolate spheroidal angular function of the first kind
263
pro_rad1_cv -- Prolate spheroidal radial function of the first kind
264
pro_rad2_cv -- Prolate spheroidal radial function of the second kind
265
obl_ang1_cv -- Oblate spheroidal angluar function of the first kind
266
obl_rad1_cv -- Oblate spheroidal radial function of the first kind
267
obl_rad2_cv -- Oblate spheroidal radial function of the second kind
271
kelvin -- All Kelvin functions (order 0) and derivatives.
272
kelvin_zeros -- **Zeros of All Kelvin functions (order 0) and derivatives
273
ber -- Kelvin function ber x
274
bei -- Kelvin function bei x
275
berp -- Derivative of Kelvin function ber x
276
beip -- Derivative of Kelvin function bei x
277
ker -- Kelvin function ker x
278
kei -- Kelvin function kei x
279
kerp -- Derivative of Kelvin function ker x
280
keip -- Derivative of Kelvin function kei x
281
ber_zeros -- **Zeros of Kelvin function bei x
282
bei_zeros -- **Zeros of Kelvin function ber x
283
berp_zeros -- **Zeros of derivative of Kelvin function ber x
284
beip_zeros -- **Zeros of derivative of Kelvin function bei x
285
ker_zeros -- **Zeros of Kelvin function kei x
286
kei_zeros -- **Zeros of Kelvin function ker x
287
kerp_zeros -- **Zeros of derivative of Kelvin function ker x
288
keip_zeros -- **Zeros of derivative of Kelvin function kei x
290
Other Special Functions
292
expn -- Exponential integral.
293
exp1 -- Exponential integral of order 1 (for complex argument)
294
expi -- Another exponential integral -- Ei(x)
295
wofz -- Fadeeva function.
296
dawsn -- Dawson's integral.
297
shichi -- Hyperbolic sine and cosine integrals.
298
sici -- Integral of the sinc and "cosinc" functions.
299
spence -- Dilogarithm integral.
300
zeta -- Riemann zeta function of two arguments.
301
zetac -- 1.0 - standard Riemann zeta function.
303
Convenience Functions
306
exp10 -- 10 raised to the x power.
307
exp2 -- 2 raised to the x power.
308
radian -- radian angle given degrees, minutes, and seconds.
309
cosdg -- cosine of the angle given in degrees.
310
sindg -- sine of the angle given in degrees.
311
tandg -- tangent of the angle given in degrees.
312
cotdg -- cotangent of the angle given in degrees.
316
round -- round the argument to the nearest integer. If argument
317
ends in 0.5 exactly, pick the nearest even integer.
319
** in the description indicates a function which is not a universal
320
function and does not follow broadcasting and automatic
325
Errors are handled by returning nans, or other appropriate values.
326
Some of the special function routines will print an error message
327
when an error occurs. By default this printing
328
is disabled. To enable such messages use errprint(1)
329
To disable such messages use errprint(0).
332
>>> print scipy.special.bdtr(-1,10,0.3)
333
>>> scipy.special.errprint(1)
334
>>> print scipy.special.bdtr(-1,10,0.3)
339
global_symbols = ['isinf','isfinite','isnan']
added
removed
Lib/special/info_special.py
