def hyp0f1(v, z):
r"""Confluent hypergeometric limit function 0F1.
Parameters
----------
v, z : array_like
Input values.
Returns
-------
hyp0f1 : ndarray
The confluent hypergeometric limit function.
Notes
-----
This function is defined as:
.. math:: _0F_1(v,z) = \sum_{k=0}^{\inf}\frac{z^k}{(v)_k k!}.
It's also the limit as q -> infinity of ``1F1(q;v;z/q)``, and satisfies
the differential equation :math:``f''(z) + vf'(z) = f(z)`.
"""
v = atleast_1d(v)
z = atleast_1d(z)
v, z = np.broadcast_arrays(v, z)
arg = 2 * sqrt(abs(z))
old_err = np.seterr(all='ignore') # for z=0, a<1 and num=inf, next lines
num = where(z.real >= 0, iv(v - 1, arg), jv(v - 1, arg))
den = abs(z)**((v - 1.0) / 2)
num *= gamma(v)
np.seterr(**old_err)
num[z == 0] = 1
den[z == 0] = 1
return num / den
def polygamma(n, x):
"""Polygamma function which is the nth derivative of the digamma (psi)
function.
Parameters
----------
n : array_like of int
The order of the derivative of `psi`.
x : array_like
Where to evaluate the polygamma function.
Returns
-------
polygamma : ndarray
The result.
Examples
--------
>>> from scipy import special
>>> x = [2, 3, 25.5]
>>> special.polygamma(1, x)
array([ 0.64493407, 0.39493407, 0.03999467])
>>> special.polygamma(0, x) == special.psi(x)
array([ True, True, True], dtype=bool)
"""
n, x = asarray(n), asarray(x)
fac2 = (-1.0)**(n+1) * gamma(n+1.0) * zeta(n+1,x)
return where(n == 0, psi(x), fac2)
def polygamma(n, x):
"""Polygamma function which is the nth derivative of the digamma (psi)
function.
Parameters
----------
n : array_like of int
The order of the derivative of `psi`.
x : array_like
Where to evaluate the polygamma function.
Returns
-------
polygamma : ndarray
The result.
Examples
--------
>>> from scipy import special
>>> x = [2, 3, 25.5]
>>> special.polygamma(1, x)
array([ 0.64493407, 0.39493407, 0.03999467])
>>> special.polygamma(0, x) == special.psi(x)
array([ True, True, True], dtype=bool)
"""
n, x = asarray(n), asarray(x)
fac2 = (-1.0)**(n + 1) * gamma(n + 1.0) * zeta(n + 1, x)
return where(n == 0, psi(x), fac2)
def hyp0f1(v, z):
r"""Confluent hypergeometric limit function 0F1.
Parameters
----------
v, z : array_like
Input values.
Returns
-------
hyp0f1 : ndarray
The confluent hypergeometric limit function.
Notes
-----
This function is defined as:
.. math:: _0F_1(v,z) = \sum_{k=0}^{\inf}\frac{z^k}{(v)_k k!}.
It's also the limit as q -> infinity of ``1F1(q;v;z/q)``, and satisfies
the differential equation :math:``f''(z) + vf'(z) = f(z)`.
"""
v = atleast_1d(v)
z = atleast_1d(z)
v, z = np.broadcast_arrays(v, z)
arg = 2 * sqrt(abs(z))
old_err = np.seterr(all='ignore') # for z=0, a<1 and num=inf, next lines
num = where(z.real >= 0, iv(v - 1, arg), jv(v - 1, arg))
den = abs(z)**((v - 1.0) / 2)
num *= gamma(v)
np.seterr(**old_err)
num[z == 0] = 1
den[z == 0] = 1
return num / den
def lpmn(m, n, z):
"""Associated Legendre functions of the first kind, Pmn(z) and its
derivative, ``Pmn'(z)`` of order m and degree n. Returns two
arrays of size ``(m+1, n+1)`` containing ``Pmn(z)`` and ``Pmn'(z)`` for
all orders from ``0..m`` and degrees from ``0..n``.
Parameters
----------
m : int
``|m| <= n``; the order of the Legendre function.
n : int
where ``n >= 0``; the degree of the Legendre function. Often
called ``l`` (lower case L) in descriptions of the associated
Legendre function
z : float or complex
Input value.
Returns
-------
Pmn_z : (m+1, n+1) array
Values for all orders 0..m and degrees 0..n
Pmn_d_z : (m+1, n+1) array
Derivatives for all orders 0..m and degrees 0..n
"""
if not isscalar(m) or (abs(m) > n):
raise ValueError("m must be <= n.")
if not isscalar(n) or (n < 0):
raise ValueError("n must be a non-negative integer.")
if not isscalar(z):
raise ValueError("z must be scalar.")
if (m < 0):
mp = -m
mf, nf = mgrid[0:mp + 1, 0:n + 1]
sv = errprint(0)
fixarr = where(mf > nf, 0.0,
(-1)**mf * gamma(nf - mf + 1) / gamma(nf + mf + 1))
sv = errprint(sv)
else:
mp = m
if iscomplex(z):
p, pd = specfun.clpmn(mp, n, real(z), imag(z))
else:
p, pd = specfun.lpmn(mp, n, z)
if (m < 0):
p = p * fixarr
pd = pd * fixarr
return p, pd
def lpmn(m,n,z):
"""Associated Legendre functions of the first kind, Pmn(z) and its
derivative, ``Pmn'(z)`` of order m and degree n. Returns two
arrays of size ``(m+1, n+1)`` containing ``Pmn(z)`` and ``Pmn'(z)`` for
all orders from ``0..m`` and degrees from ``0..n``.
Parameters
----------
m : int
``|m| <= n``; the order of the Legendre function.
n : int
where ``n >= 0``; the degree of the Legendre function. Often
called ``l`` (lower case L) in descriptions of the associated
Legendre function
z : float or complex
Input value.
Returns
-------
Pmn_z : (m+1, n+1) array
Values for all orders 0..m and degrees 0..n
Pmn_d_z : (m+1, n+1) array
Derivatives for all orders 0..m and degrees 0..n
"""
if not isscalar(m) or (abs(m)>n):
raise ValueError("m must be <= n.")
if not isscalar(n) or (n<0):
raise ValueError("n must be a non-negative integer.")
if not isscalar(z):
raise ValueError("z must be scalar.")
if (m < 0):
mp = -m
mf,nf = mgrid[0:mp+1,0:n+1]
sv = errprint(0)
fixarr = where(mf>nf,0.0,(-1)**mf * gamma(nf-mf+1) / gamma(nf+mf+1))
sv = errprint(sv)
else:
mp = m
if iscomplex(z):
p,pd = specfun.clpmn(mp,n,real(z),imag(z))
else:
p,pd = specfun.lpmn(mp,n,z)
if (m < 0):
p = p * fixarr
pd = pd * fixarr
return p,pd
def polygamma(n, x):
"""Polygamma function which is the nth derivative of the digamma (psi)
function."""
n, x = asarray(n), asarray(x)
cond = (n==0)
fac2 = (-1.0)**(n+1) * gamma(n+1.0) * zeta(n+1,x)
if sometrue(cond,axis=0):
return where(cond, psi(x), fac2)
return fac2
def hyp0f1(v,z):
"""Confluent hypergeometric limit function 0F1.
Limit as q->infinity of 1F1(q;a;z/q)
"""
z = asarray(z)
if issubdtype(z.dtype, complexfloating):
arg = 2*sqrt(abs(z))
num = where(z>=0, iv(v-1,arg), jv(v-1,arg))
den = abs(z)**((v-1.0)/2)
else:
num = iv(v-1,2*sqrt(z))
den = z**((v-1.0)/2.0)
num *= gamma(v)
return where(z==0,1.0,num/ asarray(den))
def la_roots(n, alpha, mu=0):
"""[x,w] = la_roots(n,alpha)
Returns the roots (x) of the nth order generalized (associated) Laguerre
polynomial, L^(alpha)_n(x), and weights (w) to use in Gaussian quadrature over
[0,inf] with weighting function exp(-x) x**alpha with alpha > -1.
"""
if not all(alpha > -1):
raise ValueError("alpha > -1")
if n < 1:
raise ValueError("n must be positive.")
(p, q) = (alpha, 0.0)
sbn_La = lambda k: -sqrt(k * (k + p)) # from recurrence relation
an_La = lambda k: 2 * k + p + 1
mu0 = cephes.gamma(alpha + 1) # integral of weight over interval
val = gen_roots_and_weights(n, an_La, sbn_La, mu0)
if mu:
return val + [mu0]
else:
return val
def la_roots(n, alpha, mu=0):
"""[x,w] = la_roots(n,alpha)
Returns the roots (x) of the nth order generalized (associated) Laguerre
polynomial, L^(alpha)_n(x), and weights (w) to use in Gaussian quadrature over
[0,inf] with weighting function exp(-x) x**alpha with alpha > -1.
"""
if not all(alpha > -1):
raise ValueError("alpha > -1")
if n < 1:
raise ValueError("n must be positive.")
(p,q) = (alpha,0.0)
sbn_La = lambda k: -sqrt(k*(k + p)) # from recurrence relation
an_La = lambda k: 2*k + p + 1
mu0 = cephes.gamma(alpha+1) # integral of weight over interval
val = gen_roots_and_weights(n,an_La,sbn_La,mu0)
if mu:
return val + [mu0]
else:
return val
