o
    "`F                     @   sf  d dl mZ G dd deZdd Zdd Zdd	 Zed
d Zedd Zedd Z	edd Z
edd Zedd Zedd Zedd Zedd Zedd Zedd Zed d! Zed"d# Zedhd%d&Zed'd( Zed)d* Zed+d, Zed-d. Zed/d0 Zed1d2 Zedid4d5Zedjd7d8Zed9d: Zed;d< Zed=d> Zed?d@ Z edAdB Z!edCdD Z"edEdF Z#edkdHdIZ$edJdK Z%edLdM Z&edNdO Z'edPdQ Z(dRdS Z)d3dGl*Z*d3dGl+Z+dTdU Z,dVdW Z-edidXdYZ.edhdZd[Z/djd\d]Z0ed^d_ Z1ed`da Z2edbdc Z3edjdddeZ4edjdfdgZ5dGS )l   )xrangec                   @   s   e Zd ZdZi ZdZdd Zedd Zdd Z	d	d
 Z
dd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd ZdS ) SpecialFunctionsa  
    This class implements special functions using high-level code.

    Elementary and some other functions (e.g. gamma function, basecase
    hypergeometric series) are assumed to be predefined by the context as
    "builtins" or "low-level" functions.
    gP?c              
   C   s.  | j }|jD ]}|j| \}}|||| q| d| _| d| _| d| _| d| _| d| _| d| _	| d| _
| d| _| d	| _| d
| _| d| _| d| _| d| _| d| _| d| _| d| _| d| _i | _| jdddddddd | | j| _d S )N)   r       r   )r   r   )   r   )r      )r      )r   r	   )   r   )r   r   )   r   )r
   r   )r   r   )r   r   )r   r   )r      )r
   r   )r
   r   argconjrootZpsiZzetaZfibZfac)Zphase	conjugateZnthrootZ	polygammaZhurwitzZ	fibonacciZ	factorial)	__class__defined_functions_wrap_specfunZ_mpqZmpq_1Zmpq_0Zmpq_1_2Zmpq_3_2Zmpq_1_4Zmpq_1_16Zmpq_3_16Zmpq_5_2Zmpq_3_4Zmpq_7_4Zmpq_5_4Zmpq_1_3Zmpq_2_3Zmpq_4_3Zmpq_1_6Zmpq_5_6Zmpq_5_3Z_misc_const_cache_aliasesupdateZmemoizeZzetazeroZzetazero_memoized)selfclsnamefwrap r   </usr/lib/python3/dist-packages/mpmath/functions/functions.py__init__   s@   
zSpecialFunctions.__init__c                 C   s   t | || d S N)setattr)r   r   r   r   r   r   r   r   =   s   zSpecialFunctions._wrap_specfunc                 C      t r   NotImplementedErrorctxnzr   r   r   _besseljD       zSpecialFunctions._besseljc                 C   r    r   r!   r$   r&   r   r   r   _erfE   r(   zSpecialFunctions._erfc                 C   r    r   r!   r)   r   r   r   _erfcF   r(   zSpecialFunctions._erfcc                 C   r    r   r!   )r$   r&   ar   r   r   _gamma_upper_intG   r(   z!SpecialFunctions._gamma_upper_intc                 C   r    r   r!   r#   r   r   r   _expint_intH   r(   zSpecialFunctions._expint_intc                 C   r    r   r!   r$   sr   r   r   _zetaI   r(   zSpecialFunctions._zetac                 C   r    r   r!   )r$   r0   r,   r%   ZderivativesZreflectr   r   r   _zetasum_fastJ   r(   zSpecialFunctions._zetasum_fastc                 C   r    r   r!   r)   r   r   r   _eiK   r(   zSpecialFunctions._eic                 C   r    r   r!   r)   r   r   r   _e1L   r(   zSpecialFunctions._e1c                 C   r    r   r!   r)   r   r   r   _ciM   r(   zSpecialFunctions._cic                 C   r    r   r!   r)   r   r   r   _siN   r(   zSpecialFunctions._sic                 C   r    r   r!   r/   r   r   r   _altzetaO   r(   zSpecialFunctions._altzetaN)__name__
__module____qualname____doc__r   ZTHETA_Q_LIMr   classmethodr   r'   r*   r+   r-   r.   r1   r2   r3   r4   r5   r6   r7   r   r   r   r   r      s&    +
r   c                 C      | dft j| j< | S )NTr   r   r8   r   r   r   r   defun_wrappedQ      r@   c                 C   r=   )NFr>   r?   r   r   r   defunU   rA   rB   c                 C   s   t t| j|  | S r   )r   r   r8   r?   r   r   r   defun_staticY   s   rC   c                 C      | j | | S r   )oneZtanr)   r   r   r   cot]      rF   c                 C   rD   r   )rE   Zcosr)   r   r   r   sec`   rG   rH   c                 C   rD   r   )rE   sinr)   r   r   r   cscc   rG   rJ   c                 C   rD   r   )rE   Ztanhr)   r   r   r   cothf   rG   rK   c                 C   rD   r   )rE   Zcoshr)   r   r   r   sechi   rG   rL   c                 C   rD   r   )rE   Zsinhr)   r   r   r   cschl   rG   rM   c                 C      |s| j d S | | j| S )N      ?)piZatanrE   r)   r   r   r   acoto      
rQ   c                 C      |  | j| S r   )ZacosrE   r)   r   r   r   asecv   rG   rT   c                 C   rS   r   )ZasinrE   r)   r   r   r   acscy   rG   rU   c                 C   rN   )Ny              ?)rP   ZatanhrE   r)   r   r   r   acoth|   rR   rV   c                 C   rS   r   )ZacoshrE   r)   r   r   r   asech   rG   rW   c                 C   rS   r   )ZasinhrE   r)   r   r   r   acsch   rG   rX   c                 C   sH   |  |}|r| |r|S | |r|dkr| jS | j S |t| S )Nr   )convertisnanZ_is_real_typerE   absr$   xr   r   r   sign   s   

r^   r   c                 C   s2   |dkr	|  |S | |}| |}| ||S Nr   )Zagm1rY   Z_agm)r$   r,   br   r   r   agm   s
   


ra   c                 C   s,   |  |r	d| S |s|d S | || S r_   )isinfrI   r\   r   r   r   sinc   s
   
rc   c                 C   s2   |  |r	d| S |s|d S | || j|  S r_   )rb   ZsinpirP   r\   r   r   r   sincpi   s
   
rd   c                    sB   s j S   j k rdd   S   fdddS )NrO   r   c                      s   t  dgS N)iterexpr   r\   r   r   <lambda>       zexpm1.<locals>.<lambda>r   )zeromagprecsum_accuratelyr\   r   r\   r   expm1   s
   ro   c                 C   sH   |s| j S | || j k r|d|d   S | | jd|d| j dS )NrO   r   r   rm   )rk   rl   rm   logZfaddr\   r   r   r   log1p   s
   rr   c           
         s   | j }| j}  | }||}|dkr|S |s%r# dv r%| r%|S  | }|}|  }	|||	 | j k rH|	 |	 d d  S |  fdddS )Ni)r   rf   y              ?y             r   c                      s   t   dgS re   )rg   r   r]   yr   r   ri      s    zpowm1.<locals>.<lambda>r   )rl   rE   Zisintlnrm   rn   )
r$   r]   rt   rl   rE   wMZx1ZmagyZlnxr   rs   r   powm1   s   
rx   c                 C   sx   t |}t |}||; }|s| jS d| |kr| j S d| |kr$| jS d| d| kr0| j S | d| | | S )Nr   r   r   )intrE   jZexpjpimpf)r$   kr%   r   r   r   _rootof1   s   r}   r   c                 C   s   t |}| |}|rN|d@ r+d| |d kr+| |s+| |dk r+| | | S | j}z|  jd7  _| ||d| || }W || _|
 S || _w | ||S )Nr   r   r   
   )ry   rY   imrer   rm   r}   Z_nthroot)r$   r]   r%   r|   rm   vr   r   r   r      s   
0r   Fc                    st    j  j}z(  jd7  _|r fddtD }n fddtD }W | _n| _w dd |D S )Nr~   c                    s&   g | ]}|d kr  |qS r   r}   .0r|   r$   Zgcdr%   r   r   
<listcomp>   s   & zunitroots.<locals>.<listcomp>c                    s   g | ]}  |qS r   r   r   )r$   r%   r   r   r     s    c                 S   s   g | ]}|
 qS r   r   )r   r]   r   r   r   r     rj   )Z_gcdrm   range)r$   r%   Z	primitiverm   r   r   r   r   	unitroots   s   r   c                 C   s*   |  |}| |}| |}| ||S r   )rY   _re_imZatan2)r$   r]   r   r   r   r   r   r     s   


r   c                 C   s   t | |S r   )r[   rY   r\   r   r   r   fabs  s   r   c                 C   s   |  |}t|dr|jS |S )Nreal)rY   hasattrr   r\   r   r   r   r     s   

r   c                 C   s    |  |}t|dr|jS | jS )Nimag)rY   r   r   rk   r\   r   r   r   r     s   

r   c                 C   s,   |  |}z| W S  ty   | Y S w r   )rY   r   AttributeErrorr\   r   r   r   r      s   

r   c                 C   s   |  || |fS r   )r   r   r)   r   r   r   polar(     r   c                 C   s   || j | |  S r   )ZmpcZcos_sin)r$   rZphir   r   r   rect,  r   r   Nc                 C   s8   |d u r	|  |S | jd }| j ||d| j ||d S )N   rp   )ru   rm   )r$   r]   r`   wpr   r   r   rq   0  s   

rq   c                 C   s   |  |dS )Nr~   )rq   r\   r   r   r   log107  s   r   c                 C   s   |  ||  | S r   )rY   )r$   r]   rt   r   r   r   fmod;  r   r   c                 C   s
   || j  S r   Zdegreer\   r   r   r   degrees?     
r   c                 C   s
   || j  S r   r   r\   r   r   r   radiansC  r   r   c                 C   sv   |s|s|S | j | S || jkr"|dkr|S |d| | j | j  S || j kr6| d| d | j | j  S | |S )Nr   r   r   )ZninfinfrP   rz   ru   )r$   r&   r|   r   r   r   _lambertw_specialG  s   



r   c                 C   s  d}t | drt| j}| j}|rd|dk  }t|}nt| }d}d}|s(d}t||} |dkrd|  k r;dk rn nd|  k rGdk rn nt|r{|d	krWd
d| d   S |dkrcdd| d   S |dk rodd| d   S |dk r{dd| d   S |dk r|dkrdd| d   S dd| d   S d}|s||kr|} |dk rdd | | d!   d"| |   S |d!k r| S d#d$|   S |s|dkrt|}t|}nt| }t|}n|dkrXd}|s||  k rdk rn n|} |dkr|d%k rd&|  k rdk rn ndd | | d!   d"| |   S |s7d|  kr'dk r7n nt| }|t|  S |dkrL|sL|dk rLt| d' }nt| d( }t|}|| ||  ||d)  d)|d)    S )*Nr   r   rf   g        g      g      @g      g      @      ?yx&1?p=
ף?yh|?5?ʡEƿy      ?      @g      ?y)\(?&1?y      ?L7A`y      ?      ?yx&1?p=
ףyh|?5?ʡE?y      ?      g      пy)\(?&1ʿy      ?L7A`?y      ?      g      y'1ZԿq=
ףp?yM`"r   y'1ZԿq=
ףpyM`"?g2,6V׿gɿg4@rO   g}tp?g?g333333?g?g333333y        -DT!	@y        -DT!@r   )r   floatr   r   complexmathrq   cmath)r&   r|   Z	imag_signr]   rt   r   L1L2r   r   r   _lambertw_approx_hybridZ  s`   



0 
0 "
(r   c                    s    }d|  k rdk rn nd|  k rdk rn n|dk r	td dk r	|dksJ|d	kr= dksJ|dkr	 dk r	  fd
d}  | }  j|7  _ d j d  }  j|8  _ d	 dd d d	d}|dkr| } j}	t	t
d|D ]`vr݈ fddt	dD |< d d  d |d  d   d  | d  d  d   <  |  }
|	|
7 }	  |
| k r|	df  S d7 q  j|d 7  _|	dfS |dks|d	krt|dfS |dkr7|d	k r,d  dfS  } |}n=|d	krc scd   k rQdk rcn n  }| |  dfS  d j |  } |}|| ||  ||d  d|d    dfS )z
    Return rough approximation for W_k(z) from an asymptotic series,
    sufficiently accurate for the Halley iteration to converge to
    the correct value.
    ii  ii  r   g,6V?g?r   rf   c                      s     dgS re   )rh   r   r)   r   r   ri     s    z"_lambertw_series.<locals>.<lambda>r   r   c                 3   s(    | ]}|  d  |   V  qdS )r   Nr   )r   rz   )lur   r   	<genexpr>  s   & z#_lambertw_series.<locals>.<genexpr>r   TFg,6V׿y               @)rl   r[   r   rn   rm   Zsqrter{   rk   r   maxZfsumr   ru   r   rP   )r$   r&   r|   tolZmagzZdeltaZcancellationpr,   r0   Ztermr   r   r   )r$   r   r   r&   r   _lambertw_series  sT   
6$T



8
,r   c                 C   s  |  |}t|}| |st| ||S | j}|  jd| |p d 7  _| j}|d }t| |||\}}|s| d}tdD ]7}	| 	|}
||
 }|| }||||
 || | || |     }| || | || kru|} n|}q@|	dkr| 
d|  || _|
 S )Nr   r   r
   r   d   z1Lambert W iteration failed to converge for z = %s)rY   ry   Zisnormalr   rm   rl   r   r{   r   rh   warn)r$   r&   r|   rm   r   r   rv   ZdoneZtwoiZewZwewZwewzZwnr   r   r   lambertw  s0   



(r   c                 C   s   |  |}|s| |r|S t|dS | |s(| |s(| |s(| |r,|| S |dkr2|S |dkr<||d  S |dkrE| |S t| ||d| | S )Nr   r   r   T)rY   rZ   typerb   rd   _polyexprh   )r$   r%   r]   r   r   r   bell  s   

(r   c                    s     fdd} j |ddS )Nc                  3   s@    r	  V  } d}	 | |  V  |d7 }|  | } qr_   )rd   )tr|   r$   extrar%   r]   r   r   _terms  s   z_polyexp.<locals>._termsr   )Z
check_step)rn   )r$   r%   r]   r   r   r   r   r   r     s   	r   c                 C   s   |  |s|  |s| |s| |r|| S |dkr || S |dkr)| |S |dkr4| || S |dkrC| || |d  S t| ||S )Nr   r   r   )rb   rZ   ro   rh   r   )r$   r0   r&   r   r   r   polyexp  s   (r   c                 C   s  t |}|dk rtd| j}|dkr|S |dkr|| S |dkr%|| S d}d}d}d}td|d D ]6}|| sj| || }	| || }
|
rQ||
|	 9 }q4|	dkr^||9 }|d7 }q4|	dkrj||9 }|d7 }q4|r||krw|d9 }|S ||9 }|| }|S )Nr   zn cannot be negativer   r   rf   )ry   
ValueErrorrE   r   Zmoebiusrx   )r$   r%   r&   r   Za_prodZb_prodZ	num_zerosZ	num_polesdrv   r`   r   r   r   
cyclotomic  sD   
r   c                 C   s  t |}|dk r| jS |d dkr ||d @ dkr| j
 S | jS dD ]*}|| sL|| d}}|dkrEt||\}}|rA| j  S |dks3| |  S q"| |rW| |S |dkr]td}	 t |d|  d }|dk rq| jS || |kr| |r| |S |d7 }q`)a  
    Evaluates the von Mangoldt function `\Lambda(n) = \log p`
    if `n = p^k` a power of a prime, and `\Lambda(n) = 0` otherwise.

    **Examples**

        >>> from mpmath import *
        >>> mp.dps = 25; mp.pretty = True
        >>> [mangoldt(n) for n in range(-2,3)]
        [0.0, 0.0, 0.0, 0.0, 0.6931471805599453094172321]
        >>> mangoldt(6)
        0.0
        >>> mangoldt(7)
        1.945910149055313305105353
        >>> mangoldt(8)
        0.6931471805599453094172321
        >>> fsum(mangoldt(n) for n in range(101))
        94.04531122935739224600493
        >>> fsum(mangoldt(n) for n in range(10001))
        10013.39669326311478372032

    r   r   r   )
r   r
   r                        l       73Me'r   rO   )ry   rk   Zln2divmodru   Zisprimer"   )r$   r%   r   qr   r|   r   r   r   mangoldt;  s>   




r   c                 C   *   |  t|t|}|rt|S | |S r   )Z
_stirling1ry   r{   r$   r%   r|   exactr   r   r   r   	stirling1w     
r   c                 C   r   r   )Z
_stirling2ry   r{   r   r   r   r   	stirling2  r   r   r   )r   )Fr   )6Zlibmp.backendr   objectr   r@   rB   rC   rF   rH   rJ   rK   rL   rM   rQ   rT   rU   rV   rW   rX   r^   ra   rc   rd   ro   rr   rx   r}   r   r   r   r   r   r   r   r   r   rq   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   <module>   s    N















	













?6

	
*
;