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a# Periodic sinc function, also called the Dirichlet function.
The Dirichlet function is defined as::
diric(x, n) = sin(x * n/2) / (n * sin(x / 2)),
where `n` is a positive integer.
Parameters
----------
x : array_like
Input data
n : int
Integer defining the periodicity.
Returns
-------
diric : ndarray
Examples
--------
>>> import numpy as np
>>> from scipy import special
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-8*np.pi, 8*np.pi, num=201)
>>> plt.figure(figsize=(8, 8));
>>> for idx, n in enumerate([2, 3, 4, 9]):
... plt.subplot(2, 2, idx+1)
... plt.plot(x, special.diric(x, n))
... plt.title('diric, n={}'.format(n))
>>> plt.show()
The following example demonstrates that `diric` gives the magnitudes
(modulo the sign and scaling) of the Fourier coefficients of a
rectangular pulse.
Suppress output of values that are effectively 0:
>>> np.set_printoptions(suppress=True)
Create a signal `x` of length `m` with `k` ones:
>>> m = 8
>>> k = 3
>>> x = np.zeros(m)
>>> x[:k] = 1
Use the FFT to compute the Fourier transform of `x`, and
inspect the magnitudes of the coefficients:
>>> np.abs(np.fft.fft(x))
array([ 3. , 2.41421356, 1. , 0.41421356, 1. ,
0.41421356, 1. , 2.41421356])
Now find the same values (up to sign) using `diric`. We multiply
by `k` to account for the different scaling conventions of
`numpy.fft.fft` and `diric`:
>>> theta = np.linspace(0, 2*np.pi, m, endpoint=False)
>>> k * special.diric(theta, k)
array([ 3. , 2.41421356, 1. , -0.41421356, -1. ,
-0.41421356, 1. , 2.41421356])
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�u�W��U���f�,� -�E� �5�!� � �D� �5�!� � �D� �"�9�D� �!�U�C��B�H�T�N�N�D��F� 3�4�4�5�5�5�
�e�G��%�� �D� �4�� � �D� �4�� � �D� �4�� � �D� �!�T�3�t�D�y�>�>�4��9�-�.�.�.� �H� c � � t | � � rt | � � | k s| dk rt d� � �t | � � } t j | � � \ }}}}|d| dz � |d| � |d| � |d| � fS )a Compute zeros of integer-order Bessel functions Jn and Jn'.
Results are arranged in order of the magnitudes of the zeros.
Parameters
----------
nt : int
Number (<=1200) of zeros to compute
Returns
-------
zo[l-1] : ndarray
Value of the lth zero of Jn(x) and Jn'(x). Of length `nt`.
n[l-1] : ndarray
Order of the Jn(x) or Jn'(x) associated with lth zero. Of length `nt`.
m[l-1] : ndarray
Serial number of the zeros of Jn(x) or Jn'(x) associated
with lth zero. Of length `nt`.
t[l-1] : ndarray
0 if lth zero in zo is zero of Jn(x), 1 if it is a zero of Jn'(x). Of
length `nt`.
See Also
--------
jn_zeros, jnp_zeros : to get separated arrays of zeros.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
i� z Number must be integer <= 1200.r N)r r rb ra r# �jdzo)�ntrf �m�t�zos rj r= r= � s� � �D �B�<�<� <�E�"�I�I��O�O��d����:�;�;�;� �R���B��-��#�#�K�A�q�!�R�
�a��1��f�:�q��"��v�q��"��v�q��"��v� -�-r� c �4 � t |� � rt | � � st d� � �t | � � | k st |� � |k rt d� � �|dk rt d� � �t j t | � � |� � S )a� Compute nt zeros of Bessel functions Jn(x), Jn'(x), Yn(x), and Yn'(x).
Returns 4 arrays of length `nt`, corresponding to the first `nt`
zeros of Jn(x), Jn'(x), Yn(x), and Yn'(x), respectively. The zeros
are returned in ascending order.
Parameters
----------
n : int
Order of the Bessel functions
nt : int
Number (<=1200) of zeros to compute
Returns
-------
Jn : ndarray
First `nt` zeros of Jn
Jnp : ndarray
First `nt` zeros of Jn'
Yn : ndarray
First `nt` zeros of Yn
Ynp : ndarray
First `nt` zeros of Yn'
See Also
--------
jn_zeros, jnp_zeros, yn_zeros, ynp_zeros
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
Compute the first three roots of :math:`J_1`, :math:`J_1'`,
:math:`Y_1` and :math:`Y_1'`.
>>> from scipy.special import jnyn_zeros
>>> jn_roots, jnp_roots, yn_roots, ynp_roots = jnyn_zeros(1, 3)
>>> jn_roots, yn_roots
(array([ 3.83170597, 7.01558667, 10.17346814]),
array([2.19714133, 5.42968104, 8.59600587]))
Plot :math:`J_1`, :math:`J_1'`, :math:`Y_1`, :math:`Y_1'` and their roots.
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import jnyn_zeros, jvp, jn, yvp, yn
>>> jn_roots, jnp_roots, yn_roots, ynp_roots = jnyn_zeros(1, 3)
>>> fig, ax = plt.subplots()
>>> xmax= 11
>>> x = np.linspace(0, xmax)
>>> ax.plot(x, jn(1, x), label=r"$J_1$", c='r')
>>> ax.plot(x, jvp(1, x, 1), label=r"$J_1'$", c='b')
>>> ax.plot(x, yn(1, x), label=r"$Y_1$", c='y')
>>> ax.plot(x, yvp(1, x, 1), label=r"$Y_1'$", c='c')
>>> zeros = np.zeros((3, ))
>>> ax.scatter(jn_roots, zeros, s=30, c='r', zorder=5,
... label=r"$J_1$ roots")
>>> ax.scatter(jnp_roots, zeros, s=30, c='b', zorder=5,
... label=r"$J_1'$ roots")
>>> ax.scatter(yn_roots, zeros, s=30, c='y', zorder=5,
... label=r"$Y_1$ roots")
>>> ax.scatter(ynp_roots, zeros, s=30, c='c', zorder=5,
... label=r"$Y_1'$ roots")
>>> ax.hlines(0, 0, xmax, color='k')
>>> ax.set_ylim(-0.6, 0.6)
>>> ax.set_xlim(0, xmax)
>>> ax.legend(ncol=2, bbox_to_anchor=(1., 0.75))
>>> plt.tight_layout()
>>> plt.show()
�Arguments must be scalars.zArguments must be integers.r znt > 0)r rb r r# �jyzoru �rf r� s rj r? r? � s� � �V
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�a�����"�"�"� �=��Q���� $� $�$r� c �. � t | |� � d S )a/ Compute zeros of integer-order Bessel functions Jn.
Compute `nt` zeros of the Bessel functions :math:`J_n(x)` on the
interval :math:`(0, \infty)`. The zeros are returned in ascending
order. Note that this interval excludes the zero at :math:`x = 0`
that exists for :math:`n > 0`.
Parameters
----------
n : int
Order of Bessel function
nt : int
Number of zeros to return
Returns
-------
ndarray
First `nt` zeros of the Bessel function.
See Also
--------
jv: Real-order Bessel functions of the first kind
jnp_zeros: Zeros of :math:`Jn'`
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
Compute the first four positive roots of :math:`J_3`.
>>> from scipy.special import jn_zeros
>>> jn_zeros(3, 4)
array([ 6.3801619 , 9.76102313, 13.01520072, 16.22346616])
Plot :math:`J_3` and its first four positive roots. Note
that the root located at 0 is not returned by `jn_zeros`.
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import jn, jn_zeros
>>> j3_roots = jn_zeros(3, 4)
>>> xmax = 18
>>> xmin = -1
>>> x = np.linspace(xmin, xmax, 500)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, jn(3, x), label=r'$J_3$')
>>> ax.scatter(j3_roots, np.zeros((4, )), s=30, c='r',
... label=r"$J_3$_Zeros", zorder=5)
>>> ax.scatter(0, 0, s=30, c='k',
... label=r"Root at 0", zorder=5)
>>> ax.hlines(0, 0, xmax, color='k')
>>> ax.set_xlim(xmin, xmax)
>>> plt.legend()
>>> plt.show()
r �r? r� s rj r< r< D � � �x �a�� � �Q� � r� c �. � t | |� � d S )a� Compute zeros of integer-order Bessel function derivatives Jn'.
Compute `nt` zeros of the functions :math:`J_n'(x)` on the
interval :math:`(0, \infty)`. The zeros are returned in ascending
order. Note that this interval excludes the zero at :math:`x = 0`
that exists for :math:`n > 1`.
Parameters
----------
n : int
Order of Bessel function
nt : int
Number of zeros to return
Returns
-------
ndarray
First `nt` zeros of the Bessel function.
See Also
--------
jvp: Derivatives of integer-order Bessel functions of the first kind
jv: Float-order Bessel functions of the first kind
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
Compute the first four roots of :math:`J_2'`.
>>> from scipy.special import jnp_zeros
>>> jnp_zeros(2, 4)
array([ 3.05423693, 6.70613319, 9.96946782, 13.17037086])
As `jnp_zeros` yields the roots of :math:`J_n'`, it can be used to
compute the locations of the peaks of :math:`J_n`. Plot
:math:`J_2`, :math:`J_2'` and the locations of the roots of :math:`J_2'`.
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import jn, jnp_zeros, jvp
>>> j2_roots = jnp_zeros(2, 4)
>>> xmax = 15
>>> x = np.linspace(0, xmax, 500)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, jn(2, x), label=r'$J_2$')
>>> ax.plot(x, jvp(2, x, 1), label=r"$J_2'$")
>>> ax.hlines(0, 0, xmax, color='k')
>>> ax.scatter(j2_roots, np.zeros((4, )), s=30, c='r',
... label=r"Roots of $J_2'$", zorder=5)
>>> ax.set_ylim(-0.4, 0.8)
>>> ax.set_xlim(0, xmax)
>>> plt.legend()
>>> plt.show()
r r� r� s rj r> r> � r� r� c �. � t | |� � d S )a7 Compute zeros of integer-order Bessel function Yn(x).
Compute `nt` zeros of the functions :math:`Y_n(x)` on the interval
:math:`(0, \infty)`. The zeros are returned in ascending order.
Parameters
----------
n : int
Order of Bessel function
nt : int
Number of zeros to return
Returns
-------
ndarray
First `nt` zeros of the Bessel function.
See Also
--------
yn: Bessel function of the second kind for integer order
yv: Bessel function of the second kind for real order
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
Compute the first four roots of :math:`Y_2`.
>>> from scipy.special import yn_zeros
>>> yn_zeros(2, 4)
array([ 3.38424177, 6.79380751, 10.02347798, 13.20998671])
Plot :math:`Y_2` and its first four roots.
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import yn, yn_zeros
>>> xmin = 2
>>> xmax = 15
>>> x = np.linspace(xmin, xmax, 500)
>>> fig, ax = plt.subplots()
>>> ax.hlines(0, xmin, xmax, color='k')
>>> ax.plot(x, yn(2, x), label=r'$Y_2$')
>>> ax.scatter(yn_zeros(2, 4), np.zeros((4, )), s=30, c='r',
... label='Roots', zorder=5)
>>> ax.set_ylim(-0.4, 0.4)
>>> ax.set_xlim(xmin, xmax)
>>> plt.legend()
>>> plt.show()
rm r� r� s rj rZ rZ � s � �n �a�� � �Q� � r� c �. � t | |� � d S )a� Compute zeros of integer-order Bessel function derivatives Yn'(x).
Compute `nt` zeros of the functions :math:`Y_n'(x)` on the
interval :math:`(0, \infty)`. The zeros are returned in ascending
order.
Parameters
----------
n : int
Order of Bessel function
nt : int
Number of zeros to return
Returns
-------
ndarray
First `nt` zeros of the Bessel derivative function.
See Also
--------
yvp
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
Compute the first four roots of the first derivative of the
Bessel function of second kind for order 0 :math:`Y_0'`.
>>> from scipy.special import ynp_zeros
>>> ynp_zeros(0, 4)
array([ 2.19714133, 5.42968104, 8.59600587, 11.74915483])
Plot :math:`Y_0`, :math:`Y_0'` and confirm visually that the roots of
:math:`Y_0'` are located at local extrema of :math:`Y_0`.
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import yn, ynp_zeros, yvp
>>> zeros = ynp_zeros(0, 4)
>>> xmax = 13
>>> x = np.linspace(0, xmax, 500)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, yn(0, x), label=r'$Y_0$')
>>> ax.plot(x, yvp(0, x, 1), label=r"$Y_0'$")
>>> ax.scatter(zeros, np.zeros((4, )), s=30, c='r',
... label=r"Roots of $Y_0'$", zorder=5)
>>> for root in zeros:
... y0_extremum = yn(0, root)
... lower = min(0, y0_extremum)
... upper = max(0, y0_extremum)
... ax.vlines(root, lower, upper, color='r')
>>> ax.hlines(0, 0, xmax, color='k')
>>> ax.set_ylim(-0.6, 0.6)
>>> ax.set_xlim(0, xmax)
>>> plt.legend()
>>> plt.show()
� r� r� s rj r[ r[ � s � �@ �a�� � �Q� � r� Fc � � t | � � rt | � � | k s| dk rt d� � �d}| }t j | ||� � S )a� Compute nt zeros of Bessel function Y0(z), and derivative at each zero.
The derivatives are given by Y0'(z0) = -Y1(z0) at each zero z0.
Parameters
----------
nt : int
Number of zeros to return
complex : bool, default False
Set to False to return only the real zeros; set to True to return only
the complex zeros with negative real part and positive imaginary part.
Note that the complex conjugates of the latter are also zeros of the
function, but are not returned by this routine.
Returns
-------
z0n : ndarray
Location of nth zero of Y0(z)
y0pz0n : ndarray
Value of derivative Y0'(z0) for nth zero
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
Compute the first 4 real roots and the derivatives at the roots of
:math:`Y_0`:
>>> import numpy as np
>>> from scipy.special import y0_zeros
>>> zeros, grads = y0_zeros(4)
>>> with np.printoptions(precision=5):
... print(f"Roots: {zeros}")
... print(f"Gradients: {grads}")
Roots: [ 0.89358+0.j 3.95768+0.j 7.08605+0.j 10.22235+0.j]
Gradients: [-0.87942+0.j 0.40254+0.j -0.3001 +0.j 0.2497 +0.j]
Plot the real part of :math:`Y_0` and the first four computed roots.
>>> import matplotlib.pyplot as plt
>>> from scipy.special import y0
>>> xmin = 0
>>> xmax = 11
>>> x = np.linspace(xmin, xmax, 500)
>>> fig, ax = plt.subplots()
>>> ax.hlines(0, xmin, xmax, color='k')
>>> ax.plot(x, y0(x), label=r'$Y_0$')
>>> zeros, grads = y0_zeros(4)
>>> ax.scatter(zeros.real, np.zeros((4, )), s=30, c='r',
... label=r'$Y_0$_zeros', zorder=5)
>>> ax.set_ylim(-0.5, 0.6)
>>> ax.set_xlim(xmin, xmax)
>>> plt.legend(ncol=2)
>>> plt.show()
Compute the first 4 complex roots and the derivatives at the roots of
:math:`Y_0` by setting ``complex=True``:
>>> y0_zeros(4, True)
(array([ -2.40301663+0.53988231j, -5.5198767 +0.54718001j,
-8.6536724 +0.54841207j, -11.79151203+0.54881912j]),
array([ 0.10074769-0.88196771j, -0.02924642+0.5871695j ,
0.01490806-0.46945875j, -0.00937368+0.40230454j]))
r �*Arguments must be scalar positive integer.�r r rb r# �cyzo�r� �complex�kf�kcs rj rW rW ? sY � �J �B�<�<� G�E�"�I�I��O�O��q����E�F�F�F�
�B�
��B� �=��R�� $� $�$r� c � � t | � � rt | � � | k s| dk rt d� � �d}| }t j | ||� � S )a� Compute nt zeros of Bessel function Y1(z), and derivative at each zero.
The derivatives are given by Y1'(z1) = Y0(z1) at each zero z1.
Parameters
----------
nt : int
Number of zeros to return
complex : bool, default False
Set to False to return only the real zeros; set to True to return only
the complex zeros with negative real part and positive imaginary part.
Note that the complex conjugates of the latter are also zeros of the
function, but are not returned by this routine.
Returns
-------
z1n : ndarray
Location of nth zero of Y1(z)
y1pz1n : ndarray
Value of derivative Y1'(z1) for nth zero
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
Compute the first 4 real roots and the derivatives at the roots of
:math:`Y_1`:
>>> import numpy as np
>>> from scipy.special import y1_zeros
>>> zeros, grads = y1_zeros(4)
>>> with np.printoptions(precision=5):
... print(f"Roots: {zeros}")
... print(f"Gradients: {grads}")
Roots: [ 2.19714+0.j 5.42968+0.j 8.59601+0.j 11.74915+0.j]
Gradients: [ 0.52079+0.j -0.34032+0.j 0.27146+0.j -0.23246+0.j]
Extract the real parts:
>>> realzeros = zeros.real
>>> realzeros
array([ 2.19714133, 5.42968104, 8.59600587, 11.74915483])
Plot :math:`Y_1` and the first four computed roots.
>>> import matplotlib.pyplot as plt
>>> from scipy.special import y1
>>> xmin = 0
>>> xmax = 13
>>> x = np.linspace(xmin, xmax, 500)
>>> zeros, grads = y1_zeros(4)
>>> fig, ax = plt.subplots()
>>> ax.hlines(0, xmin, xmax, color='k')
>>> ax.plot(x, y1(x), label=r'$Y_1$')
>>> ax.scatter(zeros.real, np.zeros((4, )), s=30, c='r',
... label=r'$Y_1$_zeros', zorder=5)
>>> ax.set_ylim(-0.5, 0.5)
>>> ax.set_xlim(xmin, xmax)
>>> plt.legend()
>>> plt.show()
Compute the first 4 complex roots and the derivatives at the roots of
:math:`Y_1` by setting ``complex=True``:
>>> y1_zeros(4, True)
(array([ -0.50274327+0.78624371j, -3.83353519+0.56235654j,
-7.01590368+0.55339305j, -10.17357383+0.55127339j]),
array([-0.45952768+1.31710194j, 0.04830191-0.69251288j,
-0.02012695+0.51864253j, 0.011614 -0.43203296j]))
r r� r r� r� s rj rX rX � sY � �V �B�<�<� G�E�"�I�I��O�O��q����E�F�F�F�
�B�
��B� �=��R�� $� $�$r� c � � t | � � rt | � � | k s| dk rt d� � �d}| }t j | ||� � S )af Compute nt zeros of Bessel derivative Y1'(z), and value at each zero.
The values are given by Y1(z1) at each z1 where Y1'(z1)=0.
Parameters
----------
nt : int
Number of zeros to return
complex : bool, default False
Set to False to return only the real zeros; set to True to return only
the complex zeros with negative real part and positive imaginary part.
Note that the complex conjugates of the latter are also zeros of the
function, but are not returned by this routine.
Returns
-------
z1pn : ndarray
Location of nth zero of Y1'(z)
y1z1pn : ndarray
Value of derivative Y1(z1) for nth zero
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
Compute the first four roots of :math:`Y_1'` and the values of
:math:`Y_1` at these roots.
>>> import numpy as np
>>> from scipy.special import y1p_zeros
>>> y1grad_roots, y1_values = y1p_zeros(4)
>>> with np.printoptions(precision=5):
... print(f"Y1' Roots: {y1grad_roots}")
... print(f"Y1 values: {y1_values}")
Y1' Roots: [ 3.68302+0.j 6.9415 +0.j 10.1234 +0.j 13.28576+0.j]
Y1 values: [ 0.41673+0.j -0.30317+0.j 0.25091+0.j -0.21897+0.j]
`y1p_zeros` can be used to calculate the extremal points of :math:`Y_1`
directly. Here we plot :math:`Y_1` and the first four extrema.
>>> import matplotlib.pyplot as plt
>>> from scipy.special import y1, yvp
>>> y1_roots, y1_values_at_roots = y1p_zeros(4)
>>> real_roots = y1_roots.real
>>> xmax = 15
>>> x = np.linspace(0, xmax, 500)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, y1(x), label=r'$Y_1$')
>>> ax.plot(x, yvp(1, x, 1), label=r"$Y_1'$")
>>> ax.scatter(real_roots, np.zeros((4, )), s=30, c='r',
... label=r"Roots of $Y_1'$", zorder=5)
>>> ax.scatter(real_roots, y1_values_at_roots.real, s=30, c='k',
... label=r"Extrema of $Y_1$", zorder=5)
>>> ax.hlines(0, 0, xmax, color='k')
>>> ax.set_ylim(-0.5, 0.5)
>>> ax.set_xlim(0, xmax)
>>> ax.legend(ncol=2, bbox_to_anchor=(1., 0.75))
>>> plt.tight_layout()
>>> plt.show()
r r� rm r� r� s rj rY rY � sY � �B �B�<�<� G�E�"�I�I��O�O��q����E�F�F�F�
�B�
��B� �=��R�� $� $�$r� c �� � t | � � } d} || |z
|� � }t d|dz � � D ].}||||z
dz z z |z }|| || |z
|dz z |� � z z
}�/|d|z z S )N� �?r rm � @)r �range)�v�zrf �L�phase�p�s�is rj �_bessel_diff_formular� % s� � �
��
�
�A� �A� ��!�A�#�q� � �A�
�1�a��c�]�]� � �� �Q�!�A�#�a�%�[� !�A� %�� �Q�q�q��1��q��s��A���
� ��� ��A��;�r� c �~ � t |d� � }|dk rt | |� � S t | ||t d� � S )a� Compute derivatives of Bessel functions of the first kind.
Compute the nth derivative of the Bessel function `Jv` with
respect to `z`.
Parameters
----------
v : array_like or float
Order of Bessel function
z : complex
Argument at which to evaluate the derivative; can be real or
complex.
n : int, default 1
Order of derivative. For 0 returns the Bessel function `jv` itself.
Returns
-------
scalar or ndarray
Values of the derivative of the Bessel function.
Notes
-----
The derivative is computed using the relation DLFM 10.6.7 [2]_.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions.
https://dlmf.nist.gov/10.6.E7
Examples
--------
Compute the Bessel function of the first kind of order 0 and
its first two derivatives at 1.
>>> from scipy.special import jvp
>>> jvp(0, 1, 0), jvp(0, 1, 1), jvp(0, 1, 2)
(0.7651976865579666, -0.44005058574493355, -0.3251471008130331)
Compute the first derivative of the Bessel function of the first
kind for several orders at 1 by providing an array for `v`.
>>> jvp([0, 1, 2], 1, 1)
array([-0.44005059, 0.3251471 , 0.21024362])
Compute the first derivative of the Bessel function of the first
kind of order 0 at several points by providing an array for `z`.
>>> import numpy as np
>>> points = np.array([0., 1.5, 3.])
>>> jvp(0, points, 1)
array([-0. , -0.55793651, -0.33905896])
Plot the Bessel function of the first kind of order 1 and its
first three derivatives.
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-10, 10, 1000)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, jvp(1, x, 0), label=r"$J_1$")
>>> ax.plot(x, jvp(1, x, 1), label=r"$J_1'$")
>>> ax.plot(x, jvp(1, x, 2), label=r"$J_1''$")
>>> ax.plot(x, jvp(1, x, 3), label=r"$J_1'''$")
>>> plt.legend()
>>> plt.show()
rf r rn )rk r r� �r� r� rf s rj r@ r@ 3 s@ � �N �A�s�#�#�A��A�v�v��!�Q�x�x��#�A�q�!�R��4�4�4r� c �~ � t |d� � }|dk rt | |� � S t | ||t d� � S )a� Compute derivatives of Bessel functions of the second kind.
Compute the nth derivative of the Bessel function `Yv` with
respect to `z`.
Parameters
----------
v : array_like of float
Order of Bessel function
z : complex
Argument at which to evaluate the derivative
n : int, default 1
Order of derivative. For 0 returns the BEssel function `yv`
See Also
--------
yv
Returns
-------
scalar or ndarray
nth derivative of the Bessel function.
Notes
-----
The derivative is computed using the relation DLFM 10.6.7 [2]_.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions.
https://dlmf.nist.gov/10.6.E7
Examples
--------
Compute the Bessel function of the second kind of order 0 and
its first two derivatives at 1.
>>> from scipy.special import yvp
>>> yvp(0, 1, 0), yvp(0, 1, 1), yvp(0, 1, 2)
(0.088256964215677, 0.7812128213002889, -0.8694697855159659)
Compute the first derivative of the Bessel function of the second
kind for several orders at 1 by providing an array for `v`.
>>> yvp([0, 1, 2], 1, 1)
array([0.78121282, 0.86946979, 2.52015239])
Compute the first derivative of the Bessel function of the
second kind of order 0 at several points by providing an array for `z`.
>>> import numpy as np
>>> points = np.array([0.5, 1.5, 3.])
>>> yvp(0, points, 1)
array([ 1.47147239, 0.41230863, -0.32467442])
Plot the Bessel function of the second kind of order 1 and its
first three derivatives.
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0, 5, 1000)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, yvp(1, x, 0), label=r"$Y_1$")
>>> ax.plot(x, yvp(1, x, 1), label=r"$Y_1'$")
>>> ax.plot(x, yvp(1, x, 2), label=r"$Y_1''$")
>>> ax.plot(x, yvp(1, x, 3), label=r"$Y_1'''$")
>>> ax.set_ylim(-10, 10)
>>> plt.legend()
>>> plt.show()
rf r rn )rk r r� r� s rj r\ r\ � s@ � �T �A�s�#�#�A��A�v�v��!�Q�x�x��#�A�q�!�R��4�4�4r� c � � t |d� � }|dk rt | |� � S d|z t | ||t d� � z S )a� Compute derivatives of real-order modified Bessel function Kv(z)
Kv(z) is the modified Bessel function of the second kind.
Derivative is calculated with respect to `z`.
Parameters
----------
v : array_like of float
Order of Bessel function
z : array_like of complex
Argument at which to evaluate the derivative
n : int, default 1
Order of derivative. For 0 returns the Bessel function `kv` itself.
Returns
-------
out : ndarray
The results
See Also
--------
kv
Notes
-----
The derivative is computed using the relation DLFM 10.29.5 [2]_.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 6.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions.
https://dlmf.nist.gov/10.29.E5
Examples
--------
Compute the modified bessel function of the second kind of order 0 and
its first two derivatives at 1.
>>> from scipy.special import kvp
>>> kvp(0, 1, 0), kvp(0, 1, 1), kvp(0, 1, 2)
(0.42102443824070834, -0.6019072301972346, 1.0229316684379428)
Compute the first derivative of the modified Bessel function of the second
kind for several orders at 1 by providing an array for `v`.
>>> kvp([0, 1, 2], 1, 1)
array([-0.60190723, -1.02293167, -3.85158503])
Compute the first derivative of the modified Bessel function of the
second kind of order 0 at several points by providing an array for `z`.
>>> import numpy as np
>>> points = np.array([0.5, 1.5, 3.])
>>> kvp(0, points, 1)
array([-1.65644112, -0.2773878 , -0.04015643])
Plot the modified bessel function of the second kind and its
first three derivatives.
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0, 5, 1000)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, kvp(1, x, 0), label=r"$K_1$")
>>> ax.plot(x, kvp(1, x, 1), label=r"$K_1'$")
>>> ax.plot(x, kvp(1, x, 2), label=r"$K_1''$")
>>> ax.plot(x, kvp(1, x, 3), label=r"$K_1'''$")
>>> ax.set_ylim(-2.5, 2.5)
>>> plt.legend()
>>> plt.show()
rf r rn r )rk r r� r� s rj rF rF � sJ � �T �A�s�#�#�A��A�v�v��!�Q�x�x���Q�w�-�a��A�r�1�=�=�=�=r� c �~ � t |d� � }|dk rt | |� � S t | ||t d� � S )a Compute derivatives of modified Bessel functions of the first kind.
Compute the nth derivative of the modified Bessel function `Iv`
with respect to `z`.
Parameters
----------
v : array_like or float
Order of Bessel function
z : array_like
Argument at which to evaluate the derivative; can be real or
complex.
n : int, default 1
Order of derivative. For 0, returns the Bessel function `iv` itself.
Returns
-------
scalar or ndarray
nth derivative of the modified Bessel function.
See Also
--------
iv
Notes
-----
The derivative is computed using the relation DLFM 10.29.5 [2]_.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 6.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions.
https://dlmf.nist.gov/10.29.E5
Examples
--------
Compute the modified Bessel function of the first kind of order 0 and
its first two derivatives at 1.
>>> from scipy.special import ivp
>>> ivp(0, 1, 0), ivp(0, 1, 1), ivp(0, 1, 2)
(1.2660658777520084, 0.565159103992485, 0.7009067737595233)
Compute the first derivative of the modified Bessel function of the first
kind for several orders at 1 by providing an array for `v`.
>>> ivp([0, 1, 2], 1, 1)
array([0.5651591 , 0.70090677, 0.29366376])
Compute the first derivative of the modified Bessel function of the
first kind of order 0 at several points by providing an array for `z`.
>>> import numpy as np
>>> points = np.array([0., 1.5, 3.])
>>> ivp(0, points, 1)
array([0. , 0.98166643, 3.95337022])
Plot the modified Bessel function of the first kind of order 1 and its
first three derivatives.
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-5, 5, 1000)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, ivp(1, x, 0), label=r"$I_1$")
>>> ax.plot(x, ivp(1, x, 1), label=r"$I_1'$")
>>> ax.plot(x, ivp(1, x, 2), label=r"$I_1''$")
>>> ax.plot(x, ivp(1, x, 3), label=r"$I_1'''$")
>>> plt.legend()
>>> plt.show()
rf r r )rk r r� r� s rj r; r; # s@ � �T �A�s�#�#�A��A�v�v��!�Q�x�x��#�A�q�!�R��3�3�3r� c �~ � t |d� � }|dk rt | |� � S t | ||t d� � S )au Compute derivatives of Hankel function H1v(z) with respect to `z`.
Parameters
----------
v : array_like
Order of Hankel function
z : array_like
Argument at which to evaluate the derivative. Can be real or
complex.
n : int, default 1
Order of derivative. For 0 returns the Hankel function `h1v` itself.
Returns
-------
scalar or ndarray
Values of the derivative of the Hankel function.
See Also
--------
hankel1
Notes
-----
The derivative is computed using the relation DLFM 10.6.7 [2]_.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions.
https://dlmf.nist.gov/10.6.E7
Examples
--------
Compute the Hankel function of the first kind of order 0 and
its first two derivatives at 1.
>>> from scipy.special import h1vp
>>> h1vp(0, 1, 0), h1vp(0, 1, 1), h1vp(0, 1, 2)
((0.7651976865579664+0.088256964215677j),
(-0.44005058574493355+0.7812128213002889j),
(-0.3251471008130329-0.8694697855159659j))
Compute the first derivative of the Hankel function of the first kind
for several orders at 1 by providing an array for `v`.
>>> h1vp([0, 1, 2], 1, 1)
array([-0.44005059+0.78121282j, 0.3251471 +0.86946979j,
0.21024362+2.52015239j])
Compute the first derivative of the Hankel function of the first kind
of order 0 at several points by providing an array for `z`.
>>> import numpy as np
>>> points = np.array([0.5, 1.5, 3.])
>>> h1vp(0, points, 1)
array([-0.24226846+1.47147239j, -0.55793651+0.41230863j,
-0.33905896-0.32467442j])
rf r rn )rk r r� r� s rj r9 r9 t �A � �| �A�s�#�#�A��A�v�v��q�!�}�}� �#�A�q�!�W�b�9�9�9r� c �~ � t |d� � }|dk rt | |� � S t | ||t d� � S )ax Compute derivatives of Hankel function H2v(z) with respect to `z`.
Parameters
----------
v : array_like
Order of Hankel function
z : array_like
Argument at which to evaluate the derivative. Can be real or
complex.
n : int, default 1
Order of derivative. For 0 returns the Hankel function `h2v` itself.
Returns
-------
scalar or ndarray
Values of the derivative of the Hankel function.
See Also
--------
hankel2
Notes
-----
The derivative is computed using the relation DLFM 10.6.7 [2]_.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 5.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions.
https://dlmf.nist.gov/10.6.E7
Examples
--------
Compute the Hankel function of the second kind of order 0 and
its first two derivatives at 1.
>>> from scipy.special import h2vp
>>> h2vp(0, 1, 0), h2vp(0, 1, 1), h2vp(0, 1, 2)
((0.7651976865579664-0.088256964215677j),
(-0.44005058574493355-0.7812128213002889j),
(-0.3251471008130329+0.8694697855159659j))
Compute the first derivative of the Hankel function of the second kind
for several orders at 1 by providing an array for `v`.
>>> h2vp([0, 1, 2], 1, 1)
array([-0.44005059-0.78121282j, 0.3251471 -0.86946979j,
0.21024362-2.52015239j])
Compute the first derivative of the Hankel function of the second kind
of order 0 at several points by providing an array for `z`.
>>> import numpy as np
>>> points = np.array([0.5, 1.5, 3.])
>>> h2vp(0, points, 1)
array([-0.24226846-1.47147239j, -0.55793651-0.41230863j,
-0.33905896+0.32467442j])
rf r rn )rk r r� r� s rj r: r: � r� r� c � � t | � � rt |� � st d� � �t | dd�� � } | dk rd}n| }t j ||� � \ }}}|d| dz � |d| dz � fS )a� Compute Ricatti-Bessel function of the first kind and its derivative.
The Ricatti-Bessel function of the first kind is defined as :math:`x
j_n(x)`, where :math:`j_n` is the spherical Bessel function of the first
kind of order :math:`n`.
This function computes the value and first derivative of the
Ricatti-Bessel function for all orders up to and including `n`.
Parameters
----------
n : int
Maximum order of function to compute
x : float
Argument at which to evaluate
Returns
-------
jn : ndarray
Value of j0(x), ..., jn(x)
jnp : ndarray
First derivative j0'(x), ..., jn'(x)
Notes
-----
The computation is carried out via backward recurrence, using the
relation DLMF 10.51.1 [2]_.
Wrapper for a Fortran routine created by Shanjie Zhang and Jianming
Jin [1]_.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions.
https://dlmf.nist.gov/10.51.E1
�arguments must be scalars.rf F�rh r r N)r rb rk r# �rctj�rf rx �n1�nm�jn�jnps rj rU rU � � � �R
�Q�K�K� 7�H�Q�K�K� 7��5�6�6�6��A�s�5�1�1�1�A� �Q���
���
���-��A�&�&�K�B��C�
�f��!��f�:�s�6�Q�q�S�6�{� "�"r� c � � t | � � rt |� � st d� � �t | dd�� � } | dk rd}n| }t j ||� � \ }}}|d| dz � |d| dz � fS )a� Compute Ricatti-Bessel function of the second kind and its derivative.
The Ricatti-Bessel function of the second kind is defined as :math:`x
y_n(x)`, where :math:`y_n` is the spherical Bessel function of the second
kind of order :math:`n`.
This function computes the value and first derivative of the function for
all orders up to and including `n`.
Parameters
----------
n : int
Maximum order of function to compute
x : float
Argument at which to evaluate
Returns
-------
yn : ndarray
Value of y0(x), ..., yn(x)
ynp : ndarray
First derivative y0'(x), ..., yn'(x)
Notes
-----
The computation is carried out via ascending recurrence, using the
relation DLMF 10.51.1 [2]_.
Wrapper for a Fortran routine created by Shanjie Zhang and Jianming
Jin [1]_.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions.
https://dlmf.nist.gov/10.51.E1
r� rf Fr� r r N)r rb rk r# �rctyr� s rj rV rV 2 r� r� c � � t | � � | k s| dk st | � � st d� � �t j | � � S )a� Compute the first nt zero in the first quadrant, ordered by absolute value.
Zeros in the other quadrants can be obtained by using the symmetries erf(-z) = erf(z) and
erf(conj(z)) = conj(erf(z)).
Parameters
----------
nt : int
The number of zeros to compute
Returns
-------
The locations of the zeros of erf : ndarray (complex)
Complex values at which zeros of erf(z)
Examples
--------
>>> from scipy import special
>>> special.erf_zeros(1)
array([1.45061616+1.880943j])
Check that erf is (close to) zero for the value returned by erf_zeros
>>> special.erf(special.erf_zeros(1))
array([4.95159469e-14-1.16407394e-16j])
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r �)Argument must be positive scalar integer.)r r rb r# �cerzo�r� s rj r1 r1 f sE � �F �b� � �R���R�1�W�W�X�b�\�\�W��D�E�E�E� �>�"� � � r� c � � t | � � | k s| dk st | � � st d� � �t j d| � � S )a6 Compute nt complex zeros of cosine Fresnel integral C(z).
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r r� r �r r rb r# �fcszor� s rj r7 r7 � �F � � �b� � �R���R�1�W�W�X�b�\�\�W��D�E�E�E� �>�!�R� � � r� c � � t | � � | k s| dk st | � � st d� � �t j d| � � S )a4 Compute nt complex zeros of sine Fresnel integral S(z).
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r r� rm r� r� s rj r8 r8 � r� r� c �� � t | � � | k s| dk st | � � st d� � �t j d| � � t j d| � � fS )aI Compute nt complex zeros of sine and cosine Fresnel integrals S(z) and C(z).
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r r� rm r r� r� s rj r6 r6 � sW � � �b� � �R���R�1�W�W�X�b�\�\�W��D�E�E�E� �>�!�R� � �(�.��B�"7�"7� 7�7r� � c �. � t j ||| � � S )a| Compute the generalized (associated) Laguerre polynomial of degree n and order k.
The polynomial :math:`L^{(k)}_n(x)` is orthogonal over ``[0, inf)``,
with weighting function ``exp(-x) * x**k`` with ``k > -1``.
Notes
-----
`assoc_laguerre` is a simple wrapper around `eval_genlaguerre`, with
reversed argument order ``(x, n, k=0.0) --> (n, k, x)``.
)r �eval_genlaguerre)rx rf �ks rj r& r& � s � � � #�A�q�!� ,� ,�,r� c �� � t | � � t |� � }} d| dz z t | dz � � z t | dz |� � z }t | dk t |� � |� � S )a. Polygamma functions.
Defined as :math:`\psi^{(n)}(x)` where :math:`\psi` is the
`digamma` function. See [dlmf]_ for details.
Parameters
----------
n : array_like
The order of the derivative of the digamma function; must be
integral
x : array_like
Real valued input
Returns
-------
ndarray
Function results
See Also
--------
digamma
References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
https://dlmf.nist.gov/5.15
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)
g �r r� r )r r r] r r )rf rx �fac2s rj rS rS � sc � �L �1�:�:�w�q�z�z�q�A� �A�a�C�=�5��3��<�<� '�$�q��s�A�,�,� 6�D� ��a���Q���� &� &�&r� c � � t | � � rt |� � st d� � �|dk rt d� � �| t | � � k s| dk rt d� � �|dk r2ddt |� � z z d|z z
d t |� � z |z z }n1d
d t |� � z z d |z z
d
t |� � z |z z }t |d| z z � � }|dk rt j dt d� � d}t t | � � � � } | dz rd}t | |� � }t j
|| ||� � }|d|� S )a� Fourier coefficients for even Mathieu and modified Mathieu functions.
The Fourier series of the even solutions of the Mathieu differential
equation are of the form
.. math:: \mathrm{ce}_{2n}(z, q) = \sum_{k=0}^{\infty} A_{(2n)}^{(2k)} \cos 2kz
.. math:: \mathrm{ce}_{2n+1}(z, q) = \sum_{k=0}^{\infty} A_{(2n+1)}^{(2k+1)} \cos (2k+1)z
This function returns the coefficients :math:`A_{(2n)}^{(2k)}` for even
input m=2n, and the coefficients :math:`A_{(2n+1)}^{(2k+1)}` for odd input
m=2n+1.
Parameters
----------
m : int
Order of Mathieu functions. Must be non-negative.
q : float (>=0)
Parameter of Mathieu functions. Must be non-negative.
Returns
-------
Ak : ndarray
Even or odd Fourier coefficients, corresponding to even or odd m.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/28.4#i
�m and q must be scalars.r �q >=0zm must be an integer >=0.r � @������ L@�fffff�`@�����̬V@� 1@�������@�T㥛� �?笭�_vOn?� �?� � Too many predicted coefficients.rm N) r rb r r
ra �warnings�warn�RuntimeWarningr r# �fcoef)r� �q�qm�km�kd�a�fcs rj rL rL � s^ � �F
�Q�K�K� 5�H�Q�K�K� 5��3�4�4�4� �A�����!�!�!� �U�1�X�X�
�
�1�q�5�5��4�5�5�5� �Q���
�4��Q���<�
�%��'�
)�D��a���L��N�
:���
�C��Q���K�
�$�q�&�
(�5��a���=��?�
:�� �R�#�a�%�Z� � �B� �C�x�x��
�8�.�!�L�L�L�
�B� �E�!�H�H�
�
�A��1�u� �
���!�Q���A� � ��A�q�!� $� $�B�
�c�r�c�7�Nr� c � � t | � � rt |� � st d� � �|dk rt d� � �| t | � � k s| dk rt d� � �|dk r2ddt |� � z z d|z z
d t |� � z |z z }n1d
d t |� � z z d |z z
d
t |� � z |z z }t |d| z z � � }|dk rt j dt d� � d}t t | � � � � } | dz rd}t | |� � }t j
|| ||� � }|d|� S )aA Fourier coefficients for even Mathieu and modified Mathieu functions.
The Fourier series of the odd solutions of the Mathieu differential
equation are of the form
.. math:: \mathrm{se}_{2n+1}(z, q) = \sum_{k=0}^{\infty} B_{(2n+1)}^{(2k+1)} \sin (2k+1)z
.. math:: \mathrm{se}_{2n+2}(z, q) = \sum_{k=0}^{\infty} B_{(2n+2)}^{(2k+2)} \sin (2k+2)z
This function returns the coefficients :math:`B_{(2n+2)}^{(2k+2)}` for even
input m=2n+2, and the coefficients :math:`B_{(2n+1)}^{(2k+1)}` for odd
input m=2n+1.
Parameters
----------
m : int
Order of Mathieu functions. Must be non-negative.
q : float (>=0)
Parameter of Mathieu functions. Must be non-negative.
Returns
-------
Bk : ndarray
Even or odd Fourier coefficients, corresponding to even or odd m.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r� r r� zm must be an integer > 0r r� r� r� r� r� r� r� r� r� r� r� rm � r� N) r rb r r
ra r� r� r� r r# r� )r� r� r� r� r� �br� s rj rM rM 3 s^ � �B
�Q�K�K� 5�H�Q�K�K� 5��3�4�4�4� �A�����!�!�!� �U�1�X�X�
�
�1��6�6��3�4�4�4� �Q���
�4��Q���<�
�%��'�
)�D��a���L��N�
:���
�C��Q���K�
�$�q�&�
(�5��a���=��?�
:�� �R�#�a�%�Z� � �B� �C�x�x��
�8�.�!�L�L�L�
�B� �E�!�H�H�
�
�A��1�u� �
���!�Q���A� � ��A�q�!� $� $�B�
�c�r�c�7�Nr� c
�T � t | � � rt | � � |k rt d� � �t |� � r|dk rt d� � �t |� � st d� � �t |� � rt d� � �| dk r�| }t d|dz �d|dz �f \ }}t j d�� � 5 t |� � dk rEt ||k d d
|z t ||z
dz � � z t ||z dz � � z � � }n>t ||k d t ||z
dz � � t ||z dz � � z � � }d d d � � n # 1 swxY w Y n| }t j
|||� � \ }}| dk r
||z }||z }||fS ) a[ Sequence of associated Legendre functions of the first kind.
Computes the associated Legendre function of the first kind of order m and
degree n, ``Pmn(z)`` = :math:`P_n^m(z)`, and its derivative, ``Pmn'(z)``.
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``.
This function takes a real argument ``z``. For complex arguments ``z``
use clpmn instead.
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
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
See Also
--------
clpmn: associated Legendre functions of the first kind for complex z
Notes
-----
In the interval (-1, 1), Ferrer's function of the first kind is
returned. The phase convention used for the intervals (1, inf)
and (-inf, -1) is such that the result is always real.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/14.3
�m must be <= n.r �!n must be a non-negative integer.�z must be scalar.z)Argument must be real. Use clpmn instead.r �ignore��allr� rn N) r ru rb r r r �errstater r r# rH ) r� rf r� �mp�mf�nf�fixarrr� �pds rj rH rH l s � �` �A�;�;� ,�3�q�6�6�A�:�:��*�+�+�+� �A�;�;� >�1�q�5�5��<�=�=�=� �A�;�;� .��,�-�-�-���|�|� F��D�E�E�E� �A����R���q��A��v�q��1��u�}�%���B�
�
�(�
+�
+�
+� N� N��1�v�v��z�z� �r�B�w�� "�R�x�%��2��a��.�.� 8�5��B��q��>�>� I�K� K��� �r�B�w��U�2�b�5��7�^�^�e�B�r�E�!�G�n�n�-L�M�M�� N� N� N� N� N� N� N� N� N� N� N���� N� N� N� N�� �� �M�"�a�� #� #�E�A�r� �A���
��J��
�&�[�� �b�5�Ls � BE/�/E3�6E3r� c
�l � t | � � rt | � � |k rt d� � �t |� � r|dk rt d� � �t |� � st d� � �|dk s|dk st d� � �| dk r�| }t d|dz �d|dz �f \ }}t j d �
� � 5 |dk rEt
||k d d |z t ||z
dz � � z t ||z dz � � z � � }n>t
||k d t ||z
dz � � t ||z dz � � z � � }d
d
d
� � n # 1 swxY w Y n| }t j ||t |� � t |� � |� � \ }} | dk r
||z }| |z } || fS )a� Associated Legendre function of the first kind for complex arguments.
Computes the associated Legendre function of the first kind of order m and
degree n, ``Pmn(z)`` = :math:`P_n^m(z)`, and its derivative, ``Pmn'(z)``.
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.
type : int, optional
takes values 2 or 3
2: cut on the real axis ``|x| > 1``
3: cut on the real axis ``-1 < x < 1`` (default)
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``
See Also
--------
lpmn: associated Legendre functions of the first kind for real z
Notes
-----
By default, i.e. for ``type=3``, phase conventions are chosen according
to [1]_ such that the function is analytic. The cut lies on the interval
(-1, 1). Approaching the cut from above or below in general yields a phase
factor with respect to Ferrer's function of the first kind
(cf. `lpmn`).
For ``type=2`` a cut at ``|x| > 1`` is chosen. Approaching the real values
on the interval (-1, 1) in the complex plane yields Ferrer's function
of the first kind.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] NIST Digital Library of Mathematical Functions
https://dlmf.nist.gov/14.21
r� r r� r� rm r� ztype must be either 2 or 3.r r� r� r� rn N) r ru rb r r r� r r r# r- r r )
r� rf r� �typer� r� r� r� r� r� s
rj r- r- � s � �n �A�;�;� ,�3�q�6�6�A�:�:��*�+�+�+� �A�;�;� >�1�q�5�5��<�=�=�=� �A�;�;� .��,�-�-�-� �A�I�I������6�7�7�7� �A����R���q��A��v�q��1��u�}�%���B�
�
�(�
+�
+�
+� N� N��q�y�y� �r�B�w�� "�R�x�%��2��a��.�.� 8�5��B��q��>�>� I�K� K��� �r�B�w��U�2�b�5��7�^�^�e�B�r�E�!�G�n�n�-L�M�M�� N� N� N� N� N� N� N� N� N� N� N���� N� N� N� N�� �� �N�2�q�$�q�'�'�4��7�7�D� 9� 9�E�A�r� �A���
��J��
�&�[�� �b�5�Ls � B
E � E#�&E#c � � t | � � r| dk rt d� � �t |� � r|dk rt d� � �t |� � st d� � �t | � � } t |� � }t d| � � }t d|� � }t |� � rt j |||� � \ }}nt j |||� � \ }}|d| dz �d|dz �f |d| dz �d|dz �f fS )aa Sequence of associated Legendre functions of the second kind.
Computes the associated Legendre function of the second kind of order m and
degree n, ``Qmn(z)`` = :math:`Q_n^m(z)`, and its derivative, ``Qmn'(z)``.
Returns two arrays of size ``(m+1, n+1)`` containing ``Qmn(z)`` and
``Qmn'(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 : complex
Input value.
Returns
-------
Qmn_z : (m+1, n+1) array
Values for all orders 0..m and degrees 0..n
Qmn_d_z : (m+1, n+1) array
Derivatives for all orders 0..m and degrees 0..n
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r z!m must be a non-negative integer.r� r� r N)r rb ra �maxr r# �clqmnrJ )r� rf r� �mm�nnr� �qds rj rJ rJ s � �B �A�;�;� >�1�q�5�5��<�=�=�=� �A�;�;� >�1�q�5�5��<�=�=�=� �A�;�;� .��,�-�-�-� �A���A� �A���A�
�Q����B� �Q����B���|�|� )���r�2�q�)�)���2�2��
�b�"�a�(�(���2� �V�q��s�V�V�q��s�V�^� �b��1�Q�3���1�Q�3�� � 0� 0�0r� c �� � t | � � r| dk rt d� � �t | � � } | dk rd}n| }t j t |� � � � d| dz � S )aD Bernoulli numbers B0..Bn (inclusive).
Parameters
----------
n : int
Indicated the number of terms in the Bernoulli series to generate.
Returns
-------
ndarray
The Bernoulli numbers ``[B(0), B(1), ..., B(n)]``.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] "Bernoulli number", Wikipedia, https://en.wikipedia.org/wiki/Bernoulli_number
Examples
--------
>>> import numpy as np
>>> from scipy.special import bernoulli, zeta
>>> bernoulli(4)
array([ 1. , -0.5 , 0.16666667, 0. , -0.03333333])
The Wikipedia article ([2]_) points out the relationship between the
Bernoulli numbers and the zeta function, ``B_n^+ = -n * zeta(1 - n)``
for ``n > 0``:
>>> n = np.arange(1, 5)
>>> -n * zeta(1 - n)
array([ 0.5 , 0.16666667, -0. , -0.03333333])
Note that, in the notation used in the wikipedia article,
`bernoulli` computes ``B_n^-`` (i.e. it used the convention that
``B_1`` is -1/2). The relation given above is for ``B_n^+``, so the
sign of 0.5 does not match the output of ``bernoulli(4)``.
r r� rm Nr )r rb ra r# �bernob�rf r� s rj r* r* > sm � �R �A�;�;� >�1�q�5�5��<�=�=�=� �A���A� �A���
���
�� �?�3�r�7�7� #� #�F�a��c�F� +�+r� c � � t | � � r| dk rt d� � �t | � � } | dk rd}n| }t j |� � d| dz � S )a Euler numbers E(0), E(1), ..., E(n).
The Euler numbers [1]_ are also known as the secant numbers.
Because ``euler(n)`` returns floating point values, it does not give
exact values for large `n`. The first inexact value is E(22).
Parameters
----------
n : int
The highest index of the Euler number to be returned.
Returns
-------
ndarray
The Euler numbers [E(0), E(1), ..., E(n)].
The odd Euler numbers, which are all zero, are included.
References
----------
.. [1] Sequence A122045, The On-Line Encyclopedia of Integer Sequences,
https://oeis.org/A122045
.. [2] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
Examples
--------
>>> import numpy as np
>>> from scipy.special import euler
>>> euler(6)
array([ 1., 0., -1., 0., 5., 0., -61.])
>>> euler(13).astype(np.int64)
array([ 1, 0, -1, 0, 5, 0, -61,
0, 1385, 0, -50521, 0, 2702765, 0])
>>> euler(22)[-1] # Exact value of E(22) is -69348874393137901.
-69348874393137976.0
r r� rm Nr )r rb ra r# �eulerbr s rj r2 r2 q sg � �T �A�;�;� >�1�q�5�5��<�=�=�=� �A���A� �A���
���
�� �?�2� � �v��1��v� &�&r� c �J � t | � � rt |� � st d� � �t | dd�� � } | dk rd}n| }t |� � rt j ||� � \ }}nt j ||� � \ }}|d| dz � |d| dz � fS )a� Legendre function of the first kind.
Compute sequence of Legendre functions of the first kind (polynomials),
Pn(z) and derivatives for all degrees from 0 to n (inclusive).
See also special.legendre for polynomial class.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r� rf Fr� r N)r rb rk r r# �clpnrI )rf r� r� �pnr� s rj rI rI � s� � �
�Q�K�K� 7�H�Q�K�K� 7��5�6�6�6��A�s�5�1�1�1�A� �A���
���
����|�|� %�� �r�1�%�%���B�B�� �b�!�$�$���B�
�f��!��f�:�r�&�A�a�C�&�z� !�!r� c �J � t | � � rt |� � st d� � �t | dd�� � } | dk rd}n| }t |� � rt j ||� � \ }}nt j ||� � \ }}|d| dz � |d| dz � fS )a� Legendre function of the second kind.
Compute sequence of Legendre functions of the second kind, Qn(z) and
derivatives for all degrees from 0 to n (inclusive).
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r� rf Fr� r N)r rb rk r r# �clqn�lqnb)rf r� r� �qnr� s rj rK rK � s� � �
�Q�K�K� 7�H�Q�K�K� 7��5�6�6�6��A�s�5�1�1�1�A� �A���
���
����|�|� &�� �r�1�%�%���B�B�� �r�1�%�%���B�
�f��!��f�:�r�&�A�a�C�&�z� !�!r� c � � d}t | � � rt | � � | k s| dk rt d� � �t j | |� � S )a�
Compute `nt` zeros and values of the Airy function Ai and its derivative.
Computes the first `nt` zeros, `a`, of the Airy function Ai(x);
first `nt` zeros, `ap`, of the derivative of the Airy function Ai'(x);
the corresponding values Ai(a');
and the corresponding values Ai'(a).
Parameters
----------
nt : int
Number of zeros to compute
Returns
-------
a : ndarray
First `nt` zeros of Ai(x)
ap : ndarray
First `nt` zeros of Ai'(x)
ai : ndarray
Values of Ai(x) evaluated at first `nt` zeros of Ai'(x)
aip : ndarray
Values of Ai'(x) evaluated at first `nt` zeros of Ai(x)
Examples
--------
>>> from scipy import special
>>> a, ap, ai, aip = special.ai_zeros(3)
>>> a
array([-2.33810741, -4.08794944, -5.52055983])
>>> ap
array([-1.01879297, -3.24819758, -4.82009921])
>>> ai
array([ 0.53565666, -0.41901548, 0.38040647])
>>> aip
array([ 0.70121082, -0.80311137, 0.86520403])
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r r �%nt must be a positive integer scalar.�r r rb r# �airyzo�r� r� s rj r% r% � �P � �Z
�B� �B�<�<� B�E�"�I�I��O�O��q����@�A�A�A� �?�2�r� "� "�"r� c � � d}t | � � rt | � � | k s| dk rt d� � �t j | |� � S )a�
Compute `nt` zeros and values of the Airy function Bi and its derivative.
Computes the first `nt` zeros, b, of the Airy function Bi(x);
first `nt` zeros, b', of the derivative of the Airy function Bi'(x);
the corresponding values Bi(b');
and the corresponding values Bi'(b).
Parameters
----------
nt : int
Number of zeros to compute
Returns
-------
b : ndarray
First `nt` zeros of Bi(x)
bp : ndarray
First `nt` zeros of Bi'(x)
bi : ndarray
Values of Bi(x) evaluated at first `nt` zeros of Bi'(x)
bip : ndarray
Values of Bi'(x) evaluated at first `nt` zeros of Bi(x)
Examples
--------
>>> from scipy import special
>>> b, bp, bi, bip = special.bi_zeros(3)
>>> b
array([-1.17371322, -3.2710933 , -4.83073784])
>>> bp
array([-2.29443968, -4.07315509, -5.51239573])
>>> bi
array([-0.45494438, 0.39652284, -0.36796916])
>>> bip
array([ 0.60195789, -0.76031014, 0.83699101])
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
rm r r
r r s rj r, r, r r� c � � t | � � rt |� � st d� � �| dk rt d� � �t | � � }| |z
}|dk rd}n|}||z }| t | � � k rt j ||� � \ }}}nt j ||� � \ }}}|d|dz � |d|dz � fS )a� Jahnke-Emden Lambda function, Lambdav(x).
This function is defined as [2]_,
.. math:: \Lambda_v(x) = \Gamma(v+1) \frac{J_v(x)}{(x/2)^v},
where :math:`\Gamma` is the gamma function and :math:`J_v` is the
Bessel function of the first kind.
Parameters
----------
v : float
Order of the Lambda function
x : float
Value at which to evaluate the function and derivatives
Returns
-------
vl : ndarray
Values of Lambda_vi(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v.
dl : ndarray
Derivatives Lambda_vi'(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
.. [2] Jahnke, E. and Emde, F. "Tables of Functions with Formulae and
Curves" (4th ed.), Dover, 1945
r� r zargument must be > 0.r N)r rb ra r r# �lamv�lamn) r� rx rf �v0r� �v1�vm�vl�dls rj rG rG C s� � �@
�Q�K�K� 7�H�Q�K�K� 7��5�6�6�6� �A����0�1�1�1� �A���A�
�Q��B� �A���
���
�� �b��B� �U�1�X�X�
�
� �]�2�q�)�)�
��B��� �]�2�q�)�)�
��B��
�f��!��f�:�r�&�A�a�C�&�z� !�!r� c � � t | � � rt |� � st d� � �t | � � }| |z
}|dk rd}n|}||z }t j ||� � \ }}}} |d|dz � |d|dz � fS )a� Parabolic cylinder functions Dv(x) and derivatives.
Parameters
----------
v : float
Order of the parabolic cylinder function
x : float
Value at which to evaluate the function and derivatives
Returns
-------
dv : ndarray
Values of D_vi(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v.
dp : ndarray
Derivatives D_vi'(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 13.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r� r N)r rb ra r# �pbdv�
r� rx rf r r� r �dv�dp�pdf�pdds
rj rP rP u s� � �0
�Q�K�K� 7�H�Q�K�K� 7��5�6�6�6� �A���A�
�1��B� �A���
���
�� �b��B� �}�R��+�+��B��C��
�e�r�!�t�e�9�b��"�Q�$��i� � r� c � � t | � � rt |� � st d� � �t | � � }| |z
}|dk rd}n|}||z }t j ||� � \ }}}} |d|dz � |d|dz � fS )a� Parabolic cylinder functions Vv(x) and derivatives.
Parameters
----------
v : float
Order of the parabolic cylinder function
x : float
Value at which to evaluate the function and derivatives
Returns
-------
dv : ndarray
Values of V_vi(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v.
dp : ndarray
Derivatives V_vi'(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 13.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r� r N)r rb ra r# �pbvvr s
rj rQ rQ � s� � �0
�Q�K�K� 7�H�Q�K�K� 7��5�6�6�6� �A���A�
�1��B� �Q���
���
�� �b��B� �}�R��+�+��B��C��
�e�r�!�t�e�9�b��"�Q�$��i� � r� c �4 � t | � � rt |� � st d� � �t | � � | k rt d� � �t | � � dk rd}n| }t j ||� � \ }}|d|dz � |d|dz � fS )a� Parabolic cylinder functions Dn(z) and derivatives.
Parameters
----------
n : int
Order of the parabolic cylinder function
z : complex
Value at which to evaluate the function and derivatives
Returns
-------
dv : ndarray
Values of D_i(z), for i=0, ..., i=n.
dp : ndarray
Derivatives D_i'(z), for i=0, ..., i=n.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996, chapter 13.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r� zn must be an integer.r N)r rb r ru r# �cpbdn)rf r� r� �cpb�cpds rj rO rO � s� � �0
�Q�K�K� 7�H�Q�K�K� 7��5�6�6�6�
�a���A�
�
��0�1�1�1� �A���!� � �
���
���~�b�!�$�$�H�C�� �u��1��u�:�s�5�B�q�D�5�z� !�!r� c � � t | � � rt | � � | k s| dk rt d� � �t j | d� � S )a Compute nt zeros of the Kelvin function ber.
Parameters
----------
nt : int
Number of zeros to compute. Must be positive.
Returns
-------
ndarray
First `nt` zeros of the Kelvin function.
See Also
--------
ber
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r �#nt must be positive integer scalar.r �r r rb r# �klvnzor� s rj r) r) � �J � �0 �B�<�<� @�E�"�I�I��O�O��q����>�?�?�?� �?�2�q� !� !�!r� c � � t | � � rt | � � | k s| dk rt d� � �t j | d� � S )a Compute nt zeros of the Kelvin function bei.
Parameters
----------
nt : int
Number of zeros to compute. Must be positive.
Returns
-------
ndarray
First `nt` zeros of the Kelvin function.
See Also
--------
bei
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r r) rm r* r� s rj r' r' r, r� c � � t | � � rt | � � | k s| dk rt d� � �t j | d� � S )a Compute nt zeros of the Kelvin function ker.
Parameters
----------
nt : int
Number of zeros to compute. Must be positive.
Returns
-------
ndarray
First `nt` zeros of the Kelvin function.
See Also
--------
ker
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r r) r� r* r� s rj rD rD r, r� c � � t | � � rt | � � | k s| dk rt d� � �t j | d� � S )a Compute nt zeros of the Kelvin function kei.
Parameters
----------
nt : int
Number of zeros to compute. Must be positive.
Returns
-------
ndarray
First `nt` zeros of the Kelvin function.
See Also
--------
kei
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r r) r� r* r� s rj rA rA : r, r� c � � t | � � rt | � � | k s| dk rt d� � �t j | d� � S )a. Compute nt zeros of the derivative of the Kelvin function ber.
Parameters
----------
nt : int
Number of zeros to compute. Must be positive.
Returns
-------
ndarray
First `nt` zeros of the derivative of the Kelvin function.
See Also
--------
ber, berp
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r r) � r* r� s rj r+ r+ W r, r� c � � t | � � rt | � � | k s| dk rt d� � �t j | d� � S )a. Compute nt zeros of the derivative of the Kelvin function bei.
Parameters
----------
nt : int
Number of zeros to compute. Must be positive.
Returns
-------
ndarray
First `nt` zeros of the derivative of the Kelvin function.
See Also
--------
bei, beip
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r r) � r* r� s rj r( r( t r, r� c � � t | � � rt | � � | k s| dk rt d� � �t j | d� � S )a. Compute nt zeros of the derivative of the Kelvin function ker.
Parameters
----------
nt : int
Number of zeros to compute. Must be positive.
Returns
-------
ndarray
First `nt` zeros of the derivative of the Kelvin function.
See Also
--------
ker, kerp
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r r) � r* r� s rj rE rE � r, r� c � � t | � � rt | � � | k s| dk rt d� � �t j | d� � S )a. Compute nt zeros of the derivative of the Kelvin function kei.
Parameters
----------
nt : int
Number of zeros to compute. Must be positive.
Returns
-------
ndarray
First `nt` zeros of the derivative of the Kelvin function.
See Also
--------
kei, keip
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r r) � r* r� s rj rB rB � r, r� c � � t | � � rt | � � | k s| dk rt d� � �t j | d� � t j | d� � t j | d� � t j | d� � t j | d� � t j | d� � t j | d � � t j | d
� � fS ) a� Compute nt zeros of all Kelvin functions.
Returned in a length-8 tuple of arrays of length nt. The tuple contains
the arrays of zeros of (ber, bei, ker, kei, ber', bei', ker', kei').
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r r) r rm r� r� r1 r3 r5 r7 r* r� s rj rC rC � s� � � �B�<�<� @�E�"�I�I��O�O��q����>�?�?�?� �O�B�� "� "� �O�B�� "� "� �O�B�� "� "� �O�B�� "� "� �O�B�� "� "� �O�B�� "� "� �O�B�� "� "� �O�B�� "� "� $� $r� c �n � t | � � r t |� � rt |� � st d� � �|t |� � k s| t | � � k rt d� � �|| z
dk rt d� � �|| z
dz }t j | ||d� � d d|� S )a� Characteristic values for prolate spheroidal wave functions.
Compute a sequence of characteristic values for the prolate
spheroidal wave functions for mode m and n'=m..n and spheroidal
parameter c.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r� �Modes must be integers.�� �(Difference between n and m is too large.r N�r rb r r# �segv�r� rf �c�maxLs rj rT rT � s� � �
�Q�K�K� 7�H�Q�K�K� 7�H�Q�K�K� 7��5�6�6�6� �U�1�X�X�
�
�1��a���=�=��2�3�3�3� �!��c� � ��C�D�D�D� �Q�3�q�5�D� �=��A�q�!� $� $�Q� '���� .�.r� c �n � t | � � r t |� � rt |� � st d� � �|t |� � k s| t | � � k rt d� � �|| z
dk rt d� � �|| z
dz }t j | ||d� � d d|� S )a� Characteristic values for oblate spheroidal wave functions.
Compute a sequence of characteristic values for the oblate
spheroidal wave functions for mode m and n'=m..n and spheroidal
parameter c.
References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
Functions", John Wiley and Sons, 1996.
https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
r� r: r; r< r rn Nr= r? s rj rN rN � s� � �
�Q�K�K� 7�H�Q�K�K� 7�H�Q�K�K� 7��5�6�6�6� �U�1�X�X�
�
�1��a���=�=��2�3�3�3� �!��c� � ��C�D�D�D� �Q�3�q�5�D� �=��A�q�"� %� %�a� (��$�� /�/r� c � � |rt | |z dz
|||�� � S |ret | � � | k st |� � |k r/|r t j dt d�� � nt | |� � S t | |� � S t
|� � t
| � � } }|| k | dk z |dk z }t | |� � }t |t j
� � rd|| < n|st j d� � }|S )a1 The number of combinations of N things taken k at a time.
This is often expressed as "N choose k".
Parameters
----------
N : int, ndarray
Number of things.
k : int, ndarray
Number of elements taken.
exact : bool, optional
For integers, if `exact` is False, then floating point precision is
used, otherwise the result is computed exactly. For non-integers, if
`exact` is True, the inputs are currently cast to integers, though
this behavior is deprecated (see below).
repetition : bool, optional
If `repetition` is True, then the number of combinations with
repetition is computed.
legacy : bool, optional
If `legacy` is True and `exact` is True, then non-integral arguments
are cast to ints; if `legacy` is False, the result for non-integral
arguments is unaffected by the value of `exact`.
.. deprecated:: 1.9.0
Non-integer arguments are currently being cast to integers when
`exact=True`. This behaviour is deprecated and the default will
change to avoid the cast in SciPy 1.11.0. To opt into the future
behavior set `legacy=False`. If you want to keep the
argument-casting but silence this warning, cast your inputs
directly, e.g. ``comb(int(your_N), int(your_k), exact=True)``.
Returns
-------
val : int, float, ndarray
The total number of combinations.
See Also
--------
binom : Binomial coefficient considered as a function of two real
variables.
Notes
-----
- Array arguments accepted only for exact=False case.
- If N < 0, or k < 0, then 0 is returned.
- If k > N and repetition=False, then 0 is returned.
Examples
--------
>>> import numpy as np
>>> from scipy.special import comb
>>> k = np.array([3, 4])
>>> n = np.array([10, 10])
>>> comb(n, k, exact=False)
array([ 120., 210.])
>>> comb(10, 3, exact=True)
120
>>> comb(10, 3, exact=True, repetition=True)
220
r )�legacyah Non-integer arguments are currently being cast to integers when exact=True. This behaviour is deprecated and the default will change to avoid the cast in SciPy 1.11.0. To opt into the future behavior set legacy=False. If you want to keep the argument-casting but silence this warning, cast your inputs directly, e.g. comb(int(your_N), int(your_k), exact=True).rm )�
stacklevelr ) r. ra r� r� �DeprecationWarningr$ r r" �
isinstancerr �ndarray�float64)�Nr� �exact�
repetitionrD �cond�valss rj r. r.
s � �| � 8��A��E�A�I�q�%��7�7�7�7� � � �q�6�6�Q�;�;�#�a�&�&�A�+�+��
"��
�G� '�1� � � � � � �A�q�z�z�!���A��� ��q�z�z�7�1�:�:�1���Q��1��6�"�a�1�f�-���Q��{�{�� �d�B�J� '� '� !��D�$��K�K�� !��:�a�=�=�D�� r� c �z � |r9|| k s | dk s|dk rdS d}t | |z
dz | dz � � D ]}||z }�|S t |� � t | � � } }|| k | dk z |dk z }t | |z
dz |� � }t |t j � � rd|| < n|st j d� � }|S )ac Permutations of N things taken k at a time, i.e., k-permutations of N.
It's also known as "partial permutations".
Parameters
----------
N : int, ndarray
Number of things.
k : int, ndarray
Number of elements taken.
exact : bool, optional
If `exact` is False, then floating point precision is used, otherwise
exact long integer is computed.
Returns
-------
val : int, ndarray
The number of k-permutations of N.
Notes
-----
- Array arguments accepted only for exact=False case.
- If k > N, N < 0, or k < 0, then a 0 is returned.
Examples
--------
>>> import numpy as np
>>> from scipy.special import perm
>>> k = np.array([3, 4])
>>> n = np.array([10, 10])
>>> perm(n, k)
array([ 720., 5040.])
>>> perm(10, 3, exact=True)
720
r r )r� r r! rG rr rH rI )rJ r� rK �valr� rM rN s rj rR rR p
s� � �J
� �
��E�E�q�1�u�u�!�a�%�%��1����q�1�u�q�y�!�a�%�(�(� � �A� �1�H�C�C��
��q�z�z�7�1�:�:�1���Q��1��6�"�a�1�f�-���A��E�A�I�q�!�!�� �d�B�J� '� '� !��D�$��K�K�� !��:�a�=�=�D�� r� c � � | dz |k r,|| z dz }t | |� � t |dz |� � z S | |k r| S | |z S )z�
Product of a range of numbers.
Returns the product of
lo * (lo+1) * (lo+2) * ... * (hi-2) * (hi-1) * hi
= hi! / (lo-1)!
Breaks into smaller products first for speed:
_range_prod(2, 9) = ((2*3)*(4*5))*((6*7)*(8*9))
r rm )� _range_prod)�lo�hi�mids rj rR rR �
sX � �
�A�v��{�{��B�w�1�n���2�s�#�#�k�#��'�2�&>�&>�>�>� �R�x�x�� �
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The factorial of a number or array of numbers.
The factorial of non-negative integer `n` is the product of all
positive integers less than or equal to `n`::
n! = n * (n - 1) * (n - 2) * ... * 1
Parameters
----------
n : int or array_like of ints
Input values. If ``n < 0``, the return value is 0.
exact : bool, optional
If True, calculate the answer exactly using long integer arithmetic.
If False, result is approximated in floating point rapidly using the
`gamma` function.
Default is False.
Returns
-------
nf : float or int or ndarray
Factorial of `n`, as integer or float depending on `exact`.
Notes
-----
For arrays with ``exact=True``, the factorial is computed only once, for
the largest input, with each other result computed in the process.
The output dtype is increased to ``int64`` or ``object`` if necessary.
With ``exact=False`` the factorial is approximated using the gamma
function:
.. math:: n! = \Gamma(n+1)
Examples
--------
>>> import numpy as np
>>> from scipy.special import factorial
>>> arr = np.array([3, 4, 5])
>>> factorial(arr, exact=False)
array([ 6., 24., 120.])
>>> factorial(arr, exact=True)
array([ 6, 24, 120])
>>> factorial(5, exact=True)
120
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� � z � � |S ) a Double factorial.
This is the factorial with every second value skipped. E.g., ``7!! = 7 * 5
* 3 * 1``. It can be approximated numerically as::
n!! = special.gamma(n/2+1)*2**((m+1)/2)/sqrt(pi) n odd
= 2**(n/2) * (n/2)! n even
Parameters
----------
n : int or array_like
Calculate ``n!!``. Arrays are only supported with `exact` set
to False. If ``n < -1``, the return value is 0.
Otherwise if ``n <= 0``, the return value is 1.
exact : bool, optional
The result can be approximated rapidly using the gamma-formula
above (default). If `exact` is set to True, calculate the
answer exactly using integer arithmetic.
Returns
-------
nff : float or int
Double factorial of `n`, as an int or a float depending on
`exact`.
Examples
--------
>>> from scipy.special import factorial2
>>> factorial2(7, exact=False)
array(105.00000000000001)
>>> factorial2(7, exact=True)
105
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This is the multifactorial of n skipping k values. For example,
factorialk(17, 4) = 17!!!! = 17 * 13 * 9 * 5 * 1
In particular, for any integer ``n``, we have
factorialk(n, 1) = factorial(n)
factorialk(n, 2) = factorial2(n)
Parameters
----------
n : int
Calculate multifactorial. If ``n < 1 - k``, the return value is 0.
Otherwise if ``n <= 0``, the return value is 1.
k : int
Order of multifactorial.
exact : bool, optional
If exact is set to True, calculate the answer exactly using
integer arithmetic.
Returns
-------
val : int
Multifactorial of `n`.
Raises
------
NotImplementedError
Raises when exact is False
Examples
--------
>>> from scipy.special import factorialk
>>> factorialk(5, 1, exact=True)
120
>>> factorialk(5, 3, exact=True)
10
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Riemann or Hurwitz zeta function.
Parameters
----------
x : array_like of float
Input data, must be real
q : array_like of float, optional
Input data, must be real. Defaults to Riemann zeta.
out : ndarray, optional
Output array for the computed values.
Returns
-------
out : array_like
Values of zeta(x).
Notes
-----
The two-argument version is the Hurwitz zeta function
.. math::
\zeta(x, q) = \sum_{k=0}^{\infty} \frac{1}{(k + q)^x};
see [dlmf]_ for details. The Riemann zeta function corresponds to
the case when ``q = 1``.
See Also
--------
zetac
References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
https://dlmf.nist.gov/25.11#i
Examples
--------
>>> import numpy as np
>>> from scipy.special import zeta, polygamma, factorial
Some specific values:
>>> zeta(2), np.pi**2/6
(1.6449340668482266, 1.6449340668482264)
>>> zeta(4), np.pi**4/90
(1.0823232337111381, 1.082323233711138)
Relation to the `polygamma` function:
>>> m = 3
>>> x = 1.25
>>> polygamma(m, x)
array(2.782144009188397)
>>> (-1)**(m+1) * factorial(m) * zeta(m+1, x)
2.7821440091883969
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