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 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 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 ivp(v,z,n=1):
"""Return the nth derivative of Iv(z) with respect to z.
"""
if not isinstance(n,types.IntType) or (n<0):
raise ValueError("n must be a non-negative integer.")
if n == 0:
return iv(v,z)
else:
return bessel_diff_formula(v, z, n, iv, 1)
def ivp(v, z, n=1):
"""Return the nth derivative of Iv(z) with respect to z.
"""
if not isinstance(n, types.IntType) or (n < 0):
raise ValueError("n must be a non-negative integer.")
if n == 0:
return iv(v, z)
else:
return bessel_diff_formula(v, z, n, iv, 1)