ó
    ˆ*£h�H  ã                   óè  • S r SSKrSSKrSSKrSrSrSrSrSrSr	S	r
S
rSrSrSrSrS rS rS r \R&                  " S5        S rS r\" \R2                  S 5      r\" \\R&                  5      r\" \\R0                  5      r\" \R4                  \R4                  5      r\" \R6                  \R6                  5      r\" \R8                  \R8                  5      r\" \R:                  \R:                  5      r\" \R<                  \R<                  5      r\" \R>                  \R>                  5      r\" \R@                  \R@                  5      r \" \RB                  \RB                  5      r!\" \RD                  \RD                  5      r"\" \RF                  \RF                  5      r#\" \RH                  S 5      r$\" \RJ                  S 5      r%\" S S 5      r&\" S S 5      r'S r(S r)S r*S r+S  r,\" \*\,5      r-\" \)\+5      r.S! r/S" r0S#r1\1* r2\1\1-
  r3S$r4\1S%S%S&S'S(S)S*S+S,S-S.S/S0S1S2S3S4S5S6S7S84r5\6" \55      S9-
  r7S:r8S;r9S< r:S= r;\" \:\;5      r<S> r=S? r>S@ r?SA r@/ SBQrASC rBSD rC\" \B\C5      rD/ SEQSSSF2   rE/ SGQSSSF2   rFSH rGSI rHSJ rISK rJSL rKSM rL/ SNQrMSOrNSZSP jrOSZSQ jrPSZSR jrQSS rR/ STQrS/ SUQSSSF2   rT/ SVQSSSF2   rU/ SWQrV/ SXQrWSY rXg! \\4 a    \R&                  r\R0                  r GN§f = f)[z÷
This module complements the math and cmath builtin modules by providing
fast machine precision versions of some additional functions (gamma, ...)
and wrapping math/cmath functions so that they can be called with either
real or complex arguments.
é    Ng-DTû!	@giW‹
¿@gÍ;fž ö?g¨ô—›wã@g¨ô—›wãù?gï9úþB.æ?gUµ»±k@g¶oüŒxâ?g]è—Oí?gm<BÎ{@ç!ð ;ó?g½¡çHÐPò?c                 ó:   ^ ^• UU 4S jnT R                   Ul         U$ )Nc                 óæ   >• [        U 5      [        L a  T" U 5      $ [        U 5      [        L a  T" U 5      $  [        U 5      n T" U 5      $ ! [        [        4 a    [        U 5      n T" U 5      s $ f = f©N)ÚtypeÚfloatÚcomplexÚ	TypeErrorÚ
ValueError©ÚxÚkwargsÚ	f_complexÚf_reals     €€ÚI/home/mande/repo/quber/.venv/lib/python3.13/site-packages/mpmath/math2.pyÚfÚ_mathfun_real.<locals>.f   sn   ø€ Ü�‹7”eÒÙ˜!“9ÐÜ�‹7”gÒÙ˜Q“<Ðð	 Ü�a“ˆAÙ˜!“9ÐøÜœ:Ð&ó 	 Ü˜“
ˆAÙ˜Q“<Òð	 ús   ·A
 Á
#A0Á/A0©Ú__name__©r   r   r   s   `` r   Ú_mathfun_realr      s   ù€ ö
 ð —‘€A„JØ€Hó    c                 ó:   ^ ^• UU 4S jnT R                   Ul         U$ )Nc                 óª   >• [        U 5      [        L a  T" U 5      $  T" [        U 5      5      $ ! [        [        4 a    T" [        U 5      5      s $ f = fr   )r   r	   r   r
   r   r   s     €€r   r   Ú_mathfun.<locals>.f+   sP   ø€ Ü�‹7”gÒÙ˜Q“<Ðð	)Ùœ% ›(Ó#Ð#øÜœ:Ð&ó 	)ÙœW Q›ZÓ(Ò(ð	)ús   �. ®!AÁAr   r   s   `` r   Ú_mathfunr   *   s   ù€ ö)ð —‘€A„JØ€Hr   c                 ó:   ^ ^• UU 4S jnT R                   Ul         U$ )Nc                  ób   >•  T" S U  5       6 $ ! [         [        4 a    T" S U  5       6 s $ f = f)Nc              3   ó8   #   • U  H  n[        U5      v •  M     g 7fr   )r   ©Ú.0r   s     r   Ú	<genexpr>Ú(_mathfun_n.<locals>.f.<locals>.<genexpr>8   s   é € Ð3ªd¨œE !ŸH˜Hªdùó   ‚c              3   ó8   #   • U  H  n[        U5      v •  M     g 7fr   ©r	   r    s     r   r"   r#   :   s   é € Ð8²4¨aœw qŸz˜z²4ùr$   )r
   r   )Úargsr   r   r   s     €€r   r   Ú_mathfun_n.<locals>.f6   s>   ø€ ð	:ÙÑ3©dÓ3Ð4Ð4øÜœ:Ð&ó 	:ÙÑ8±4Ó8Ð9Ò9ð	:ús   ƒ �.­.r   r   s   `` r   Ú
_mathfun_nr)   5   s   ù€ ö:ð
 —‘€A„JØ€Hr   g       Àc                 óP   • U S::  a  [        S5      e[        R                  " U 5      $ ©Nç        zmath domain error)r   ÚmathÚlog©r   s    r   Úmath_logr0   B   s#   € Ø�‹8ÜÐ0Ó1Ð1Ü�xŠx˜‹{Ðr   c                 óP   • U S:  a  [        S5      e[        R                  " U 5      $ r+   )r   r-   Úsqrtr/   s    r   Ú	math_sqrtr3   F   s#   € Øˆs‹7ÜÐ0Ó1Ð1Ü�yŠy˜‹|Ðr   c                 ó   • [        U 5      U-  $ r   r&   )r   Úys     r   Ú<lambda>r6   N   s   € ¬G°A«J¸ªMr   c                 ó’   • [        [        R                  " U R                  5      [        R                  " U R                  5      5      $ r   )r	   r-   ÚfloorÚrealÚimag©Úzs    r   r6   r6   `   s%   € Œg”d—j’j §¡Ó(¬$¯*ª*°Q·V±VÓ*<Ô=r   c                 ó’   • [        [        R                  " U R                  5      [        R                  " U R                  5      5      $ r   )r	   r-   Úceilr9   r:   r;   s    r   r6   r6   b   s%   € Œg”d—i’i §¡Ó'¬¯ª°1·6±6Ó):Ô;r   c                 óZ   • [         R                  " U 5      [         R                  " U 5      4$ r   )r-   ÚcosÚsinr/   s    r   r6   r6   e   s   € ¤4§8¢8¨A£;´·²¸³Ñ"<r   c                 óZ   • [         R                  " U 5      [         R                  " U 5      4$ r   )Úcmathr@   rA   r;   s    r   r6   r6   f   s   € ¤5§9¢9¨Q£<´·²¸1³Ñ">r   c                 ó   • U S-  $ ©NgUUUUUUÕ?© r/   s    r   r6   r6   h   s   € ˜!˜dš)r   c                 ó   • U S-  $ rE   rF   r;   s    r   r6   r6   h   s   € ¨q°4ªyr   c                 ór   • SU-  n [        U 5      U-  $ ! [        [        4 a    [        U 5      U-  s $ f = f©Nç      ð?)r   r   r
   r	   ©r   ÚnÚrs      r   ÚnthrootrN   j   s@   € Ø
ˆ1‰€AðÜ�Q‹x˜1‰}ÐøÜœ	Ð"ó Ü�q‹z˜Q‰Òðús   ‡ •6µ6c                 óF  • U S:  a  [        U * 5      * $ [        U S5      u  pU[        -  nUS-  nUS:X  a  [        R                  " U5      $ US:X  a  [        R
                  " U5      $ US:X  a  [        R                  " U5      * $ US:X  a  [        R
                  " U5      * $ g ©Nr   ç      à?é   é   é   é   )Ú_sinpi_realÚdivmodÚpir-   rA   r@   rK   s      r   rV   rV   q   s�   € Øˆ1ƒuÜ˜Q˜B“ÐÐÜ�!�S‹>�D€AØŒ�G€AØˆ�F€AØˆAƒv”d—h’h˜q“kÐ!ØˆAƒv”d—h’h˜q“kÐ!ØˆAƒv”t—x’x “{�lÐ"ØˆAƒv”t—x’x “{�lÐ"€vr   c                 ó2  • U S:  a  U * n [        U S5      u  pU[        -  nUS-  nUS:X  a  [        R                  " U5      $ US:X  a  [        R                  " U5      * $ US:X  a  [        R                  " U5      * $ US:X  a  [        R                  " U5      $ g rP   )rW   rX   r-   r@   rA   rK   s      r   Ú_cospi_realrZ   |   s…   € Øˆ1ƒuØˆBˆÜ�!�S‹>�D€AØŒ�G€AØˆ�F€AØˆAƒv”d—h’h˜q“kÐ!ØˆAƒv”t—x’x “{�lÐ"ØˆAƒv”t—x’x “{�lÐ"ØˆAƒv”d—h’h˜q“kÐ!€vr   c                 ó”  • U R                   S:  a  [        U * 5      * $ [        U R                   S5      u  p[        [	        X R
                  5      -  n US-  nUS:X  a  [        R                  " U 5      $ US:X  a  [        R                  " U 5      $ US:X  a  [        R                  " U 5      * $ US:X  a  [        R                  " U 5      * $ g rP   )	r9   Ú_sinpi_complexrW   rX   r	   r:   rC   rA   r@   ©r<   rL   rM   s      r   r\   r\   ‡   s¡   € Ø‡v�v�ƒzÜ ˜rÓ"Ð"Ð"Ü�!—&‘&˜#Ó�D€AÜ
Œ7�1—f‘fÓÑ€AØˆ�F€AØˆAƒv”e—i’i “lÐ"ØˆAƒv”e—i’i “lÐ"ØˆAƒv”u—y’y “|�mÐ#ØˆAƒv”u—y’y “|�mÐ#€vr   c                 ó€  • U R                   S:  a  U * n [        U R                   S5      u  p[        [        X R                  5      -  n US-  nUS:X  a  [
        R                  " U 5      $ US:X  a  [
        R                  " U 5      * $ US:X  a  [
        R                  " U 5      * $ US:X  a  [
        R                  " U 5      $ g rP   )r9   rW   rX   r	   r:   rC   r@   rA   r]   s      r   Ú_cospi_complexr_   ’   s˜   € Ø‡v�v�ƒzØˆBˆÜ�!—&‘&˜#Ó�D€AÜ
Œ7�1—f‘fÓÑ€AØˆ�F€AØˆAƒv”e—i’i “lÐ"ØˆAƒv”u—y’y “|�mÐ#ØˆAƒv”u—y’y “|�mÐ#ØˆAƒv”e—i’i “lÐ"€vr   c                 ó¼   •  [        U 5      [        U 5      -  $ ! [         a8    [        U 5      R                  S:”  a   g[        U 5      R                  S:  a   ge f = f)Né
   ù              ð?ù       €      ð¿)ÚsinpiÚcospiÚOverflowErrorr	   r:   r/   s    r   Útanpirg       sR   € ðÜ�Q‹xœ% ›(Ñ"Ð"øÜó Ü�1‹:�?‰?˜RÓÙÜ�1‹:�?‰?˜RÓÙØðúó   ‚ ™#A¾AÁAc                 ó¼   •  [        U 5      [        U 5      -  $ ! [         a8    [        U 5      R                  S:”  a   g[        U 5      R                  S:  a   ge f = f)Nra   rc   rb   )re   rd   rf   r	   r:   r/   s    r   Úcotpirj   ª   sR   € ðÜ�Q‹xœ% ›(Ñ"Ð"øÜó Ü�1‹:�?‰?˜RÓÙÜ�1‹:�?‰?˜RÓÙØðúrh   g      ðg      °<rJ   ç       @ç      @g      8@g      ^@g     €†@g     °³@g     °ã@g     &Ag    €¯KAg    ¨ƒAg    üŒ¼Ag   ÀŒ2÷Ag   (;L4Bg  €uwsBg  €uw³Bg  Øìî7ôBg  sÊì¾6Cg �h0¹{Cg ZA¾³áÀCrS   é   )	gPùÿÿÿÿï?gÃ¶)$…@gÔŽŽ¬“ÀgØ´Óa–ˆ@g]/£Q®fÀg�œ‹Â)@gU(+™²¼Á¿gO2NNQðä>gï®?�P5„>c                 óŒ  • [        U 5      nX:X  a$  US::  a  [        S5      eU[        ::  a	  [        U   $ U S:  a!  [        [        U 5      [        SU -
  5      -  -  $ U S-  n [        S   n[        S[        S-   5       H  nU[        U   X-   -  -  nM     U [        -   S-   nSX@S-   -  -  [        R                  " U* 5      -  U-  $ )Nr   zgamma function polerQ   rS   rJ   rT   ç'ö“@)ÚintÚZeroDivisionErrorÚ_max_exact_gammaÚ_exact_gammarX   rV   Ú_gamma_realÚ
_lanczos_pÚrangeÚ
_lanczos_gr-   Úexp)r   Ú_intxrM   ÚiÚts        r   rt   rt   Æ   sÊ   € Ü�‹F€EØƒzØ�A‹:ä#Ð$9Ó:Ð:ØÔ$Ó$Ü Ñ&Ð&Øˆ3ƒwä”[ “^¤K°°!±Ó$4Ñ4Ñ5Ð5à	ˆS‰ˆÜ�q‰MˆÜ�qœ* Q™,Ö'ˆAØ”˜A‘ ¡Ñ$Ñ$ŠAñ (à”
‰N˜SÑ ˆØ&¨¨s©U©Ñ3´d·h²hÀ¸r³lÑBÀQÑFÐFr   c                 ó–  • U R                   (       d  [        [        U R                  5      5      $ U R                  S:  a!  [        [        U 5      [        SU -
  5      -  -  $ U S-  n [        S   n[        S[        S-   5       H  nU[        U   X-   -  -  nM     U [        -   S-   nSX0S-   -  -  [        R                  " U* 5      -  U-  $ )NrQ   rS   rJ   r   rT   ro   )r:   r	   rt   r9   rX   r\   Ú_gamma_complexru   rv   rw   rC   rx   )r   rM   rz   r{   s       r   r}   r}   Ù   sµ   € Ø�6�6Ü”{ 1§6¡6Ó*Ó+Ð+Ø‡v�v�ƒ|ä”^ AÓ&¤~°a¸±cÓ':Ñ:Ñ;Ð;à	ˆS‰ˆÜ�q‰MˆÜ�qœ* Q™,Ö'ˆAØ”˜A‘ ¡Ñ$Ñ$ŠAñ (à”
‰N˜SÑ ˆØ&¨¨s©U©Ñ3´e·i²iÀÀ³mÑCÀaÑGÐGr   c                 óJ   •  S[        U 5      -  $ ! [         a    U S-  s $ f = f)NrJ   r,   )Úgammarq   r/   s    r   Úrgammar€   é   s-   € ðØ”%˜“(‰{ÐøÜó Ø�‰uŠðús   ‚ �"¡"c                 ó   • [        U S-   5      $ rI   )r   r/   s    r   Ú	factorialr‚   ï   s   € Ü��3‘‹<Ðr   c                 óª   • [        U 5      [        L a  [        R                  " SU 5      $ [        R                  " U R                  U R
                  5      $ )Nr,   )r   r   r-   Úatan2r:   r9   r/   s    r   Úargr…   ò   s8   € ÜˆAƒw”%ÒÜ�zŠz˜#˜aÓ Ð Ü�:Š:�a—f‘f˜QŸV™VÓ$Ð$r   c                 ó\  • [        U 5      [        [        4;  a   [        U 5      n  U R
                  nU R                  nUS:  a×  [        U 5      S:  a,  [        [        U 5      5      nUS:X  a  UR                  5       nU$ SU -
  n UR
                  nUR                  n[        U5      nUS:X  a  SnOUS:  a  SnOSn[        * S-  [        U5      -  S[        U5      -
  -  [        -   [        [        XG-
  5      5      -
  [!        U5      -
  S[        -  U-  U-  -   $ U S:X  d  U S:X  a  U S-  $ Sn	[        U 5      S	:  a$  U	[        U 5      -  n	U S-  n [        U 5      S	:  a  M$  S
U S-
  [        U 5      -  -   U -
  n
SU -  nX»-  nU
SU-  -  o«U-  nU
SU-  -  o«U-  nU
SU-  -  o«U-  nU
SU-  -  o«U-  nU
SU-  -  o«U-  nU
SU-  -  o«U-  nU
SU-  -  o«U-  nU
SU-  -  n
X©-   $ ! [        [        4 a    [        U 5      n  GNÚf = f! [         a    U nSn GNÖf = f! [         a    UnSn GNŒf = f)Nr,   rQ   r   rS   éÿÿÿÿrb   rJ   rk   é   gµ¾dÈñgí?çUUUUUUµ?glÁlÁf¿g  J?g8�8�C¿g#ÿ+•K?g}<™Ù°j_¿g¤A¤Az?g˜SË†Bž¿)r   r   r	   r   r
   r9   r:   ÚAttributeErrorÚabsr.   r   Ú	conjugater8   rX   Úlogpird   Úloggamma)r   ÚxrealÚximagÚvr<   ÚreÚimÚrefloorÚimsignÚpÚsrM   Úr2s                r   rŽ   rŽ   ø   s“  € ÜˆAƒw”uœgÐ&Ó&ð	Ü�a“ˆAðØ—‘ˆØ—‘ˆð ˆsƒ{Üˆq‹6�C‹<Ü”E˜!“H“ˆAØ˜‹zØ—K‘K“M�ØˆHØˆa‰Cˆð	Ø—‘ˆBØ—‘ˆBô ˜“)ˆØ�‹9Ø‰FØ�#‹XØ‰FàˆFÜ��B‘œ˜G›Ñ$ a¬¨F«¡mÑ4´uÑ<Ü”�a‘iÓ Ó!ñ"Ü$,¨Q£Kñ0Ø24´R±%¸±-ÀÑ2FñGð 	GàˆCƒx�1˜“8Ø�‰sˆ
Ø
€AÜ
ˆa‹&�2‹+Ø	ŒS�‹V‰ˆØ	ˆS‰ˆô ˆa‹&�2�+ð 	  #¡¤s¨1£v™~Ñ-°Ñ1€AØ
ˆ1‰€AØ	
‰€BØÐ	  Ñ	"Ñ"€A¨¡G AØÐ	" 1Ñ	$Ñ$€A¨2¡g aØÐ	" 1Ñ	$Ñ$€A¨2¡g aØÐ	" 1Ñ	$Ñ$€A¨2¡g aØÐ	" 1Ñ	$Ñ$€A¨2¡g aØÐ	" 1Ñ	$Ñ$€A¨2¡g aØÐ	! !Ñ	#Ñ#€A¨"¡W QØÐ	  Ñ	"Ñ"€AØ‰5€Løôc œIÐ&ó 	Ü˜“
‹Að	ûô
 ó ØˆØ‹ðûô ó 	ØˆBØ‹Bð	ús4   ›G" §H ÂH Ç"HÈ HÈHÈHÈH+È*H+)
r‰   g�¿gAAp?gq¿g|ðÁ?g–Y™•Y™•¿r‰   g^^^^^^Ü¿gèæŸ›n@g¥LQÊt:Àc                 ó0  • [        U 5      nX:X  a  US::  a  [        S5      eU S:  a  SU -
  n [        [        U 5      -  nOSnU S:  a  USU -  -  nU S-  n U S:  a  M  U S-  nUn[         H  nX%U-  -  nUS:  a    OXC-  nM     U[        U 5      -   SU -  -
  $ )	Nr   zpolygamma polerQ   rJ   r,   ç      $@éþÿÿÿç#B’¡œÇ;)rp   rq   rX   rj   Ú
_psi_coeffr0   )r   ry   r—   Úx2r{   Úcs         r   Ú_digamma_realr    ;  sº   € Ü�‹F€EØƒzØ�A‹:Ü#Ð$4Ó5Ð5Øˆ3ƒwØ�‰EˆÜŒu�Q‹x‰K‰àˆØ
ˆd‹(Ø	ˆS�‰U‰
ˆØ	ˆS‰ˆð ˆd�(ð 
ˆB‰€BØ
€AßˆØ	ˆq‰S‰ˆØˆu‹9ÙØ	‰Šñ	 ð
 Œx˜‹{‰?˜S ™UÑ"Ð"r   c                 ó¬  • U R                   (       d  [        [        U R                  5      5      $ U R                  S:  a  SU -
  n [        [        U 5      -  nOSn[        U 5      S:  a  USU -  -  nU S-  n [        U 5      S:  a  M  U S-  nUn[         H  nXU-  -  n[        U5      S:  a    OX2-  nM!     U[        R                  " U 5      -   SU -  -
  $ )NrQ   rJ   r,   rš   r›   rœ   )
r:   r	   r    r9   rX   rj   r‹   r�   rC   r.   )r   r—   rž   r{   rŸ   s        r   Ú_digamma_complexr¢   Q  sÉ   € Ø�6�6Ü”} Q§V¡VÓ,Ó-Ð-Ø‡v�v�ƒ|Ø�‰EˆÜŒu�Q‹x‰K‰àˆÜ
ˆa‹&�4‹-Ø	ˆS�‰U‰
ˆØ	ˆS‰ˆô ˆa‹&�4�-ð 
ˆB‰€BØ
€AßˆØ	ˆq‰S‰ˆÜˆq‹6�E‹>ÙØ	‰Šñ	 ð
 Œu�yŠy˜‹|Ñ˜c !™eÑ#Ð#r   )	gðÈS  ð?g}^Ð.@gCG­bÓ@gß¸ÊÐGƒ÷?g1”›R5å?gßµÎy¡ÈÊ?gLy×ß~F§?goO$ÀñÔy?gÅ•X”ø3=?r‡   )
rJ   ga	oŽ
@ghrÃ—›@g‚íÕ-ý@g¯¡…ÉF@gvª˜#oó?g›Œþ,Ø?gEã¦°uº´?góŸ¿‘ä†?gaôn$háI?c                 ó6   • U S   nU SS   H
  nX1U-  -   nM     U$ )Nr   rS   rF   )Úcoeffsr   r–   rŸ   s       r   Ú_polyvalr¥   ‚  s+   € Øˆq‰	€AØ�A�B‹ZˆØ�!‘‰GŠñ à€Hr   c                 ó¾   • X -  nU =p#Sn[        U5      S:”  aA  X1U-  -  nX#XD-   S-   -  -  nUS-  nX1U-  -  nX#XD-   S-   -  -  nUS-  n[        U5      S:”  a  MA  SU-  $ )NrS   ç—ÔFFõg<gm›BP×ò?)r‹   )r   rž   r—   r{   rL   s        r   Ú_erf_taylorr¨   ˆ  s„   € à	
‰€BØ€I€AØ	€AÜ
ˆa‹&�5‹.Ø	�‰T‰	ˆØ	�‘�A‘‰Y‰ˆØ	ˆQ‰ˆØ	�‰T‰	ˆØ	�‘�A‘‰Y‰ˆØ	ˆQ‰ˆô ˆa‹&�5�.ð ! Ñ"Ð"r   c                 ód   • [        U * U -  5      [        [        U 5      -  [        [        U 5      -  $ r   )rx   r¥   Ú_erfc_coeff_PÚ_erfc_coeff_Qr/   s    r   Ú	_erfc_midr¬   –  s*   € ä�ˆr�!‰t‹9”Xœm¨AÓ.Ñ.¬x¼ÀaÓ/HÑHÐHr   c                 óÎ   • X -  n[        U* 5      U -  S-  nSU-  =p4Sn[        SSS5       H3  nXT-  nXCUS-   -  -  nXT-  nXCUS-   -  -  n[        U5      S:  d  M0    XR-  $    XR-  $ )	Ngm›BP×â?rQ   rJ   rS   é   rR   rT   r§   )rx   rv   r‹   )r   rž   r‘   rM   r{   r—   rL   s          r   Ú_erfc_asympr¯   š  s‹   € à	
‰€BÜˆRˆC‹�‰
Ð)Ñ)€AØ�"‰HÐ€AØ€AÜ�1�R˜Ž]ˆØ	‰ˆØ	�!�A‘#‰Y‰ˆØ	‰ˆØ	�!�A‘#‰Y‰ˆÜˆq‹6�E�>ØØ‰5€Lñ ð ‰5€Lr   c                 ó˜   • [        U 5      n X :w  a  U $ U S:  a  [        U * 5      * $ U S:¼  a  U S:¼  a  gS[        U 5      -
  $ [        U 5      $ )z
erf of a real number.
r,   rJ   rl   )r   Úerfr¬   r¨   r/   s    r   r±   r±   ©  sU   € ô 	ˆa‹€AØƒvØˆØˆ3ƒwÜ�Q�B“ˆxˆØˆCƒxØ�‹8ØØ”Y˜q“\Ñ!Ð!Ü�q‹>Ðr   c                 ó¾   • [        U 5      n X :w  a  U $ U S:  a  U S:  a  gS[        U * 5      -
  $ U S:”  a  [        U 5      $ U S:¼  a  [        U 5      $ S[	        U 5      -
  $ )z
erfc of a real number.
r,   g      Àrk   g      "@rJ   )r   Úerfcr¯   r¬   r¨   r/   s    r   r³   r³   ¸  si   € ô 	ˆa‹€AØƒvØˆØˆ3ƒwØˆt‹8ØØ”4˜˜“8‰|ÐØˆ3ƒwÜ˜1‹~ÐØˆCƒxÜ˜‹|ÐØ”˜Q“ÑÐr   )*)gtÖüÁãòï?çd—›sÑp?)gtÖüÁãòï¿r´   )gè+K »ï?çÁr–Ò¹‡ƒ?)gè+K »ï¿rµ   )gÞìvÆVï?ç5ž@}¨�Ž?)gÞìvÆVï¿r¶   )gLú—“»Æî?ç=Y�¶”?)gLú—“»Æî¿r·   )g²‰j©î?ç{Âíîyš?)g²‰j©î¿r¸   )gÛöM•&í?çuý8ó5Ÿ?)gÛöM•&í¿r¹   )gÇß˜¾ì?ç�àNå¢?)gÇß˜¾ì¿rº   )gÒ±,nŸãê?çT!•ü~ƒ¤?)gÒ±,nŸãê¿r»   )gqÍE·çˆé?çÇ
ŽsjÎ¦?)gqÍE·çˆé¿r¼   )gu¬=8|
è?çÉöíÎrù¨?)gu¬=8|
è¿r½   )gYÖ®¡sjæ?çÊ´þ�«?)gYÖ®¡sjæ¿r¾   )gc#¤«ä?ç1¦Æêã¬?)gc#¤«ä¿r¿   )gá,ÂÍÎâ?çN@¹à�®?)gá,ÂÍÎâ¿rÀ   )gÔ¦Væ;Øà?çÛK‘òƒ°?)gÔ¦Væ;Øà¿rÁ   )gW9Þ9”Ý?ç)™Ç°?)gW9Þ9”Ý¿rÂ   )g’N%ØŸNÙ?çÛ¿¼78a±?)g’N%ØŸNÙ¿rÃ   )g×„ã£åÔ?ç­ÕôƒŠâ±?)g×„ã£åÔ¿rÄ   )g§�Q'p_Ð?ç”÷é,ÛJ²?)g§�Q'p_Ð¿rÅ   )g^>¢°„Ç?çK†X˜™²?)g^>¢°„Ç¿rÆ   )g\S…;S¼?çžøQïSÎ²?)g\S…;S¼¿rÇ   )g|­Íøê¢?çM_6Äè²?)g|­Íøê¢¿rÈ   g      D@c                 ó¶  • SU -  nS=p4Sn XEU-  -  nX4-  n[        U5      S:  a  OUS-  nM"  U[        U 5      -  U -  nU(       aI  [        U 5      [        L a  U R                  nU R
                  nOU nSnUS:X  a  US:”  a  U[        S-  -  nU$ [        U 5      [        L a8  U R
                  S:”  a  U[        S-  -  nU R
                  S:  a  U[        S-  -  nU$ )NrJ   rS   g¼‰Ø—²Òœ<r,   rb   r   )r‹   rx   r   r	   r9   r:   rX   )	r<   Ú_e1rM   r—   r{   Úkr‘   ÚzrealÚzimags	            r   Úei_asymprÎ   ÷  sê   € Ø
ˆ1‰€AØ€K€AØ	€AØ
Ø	ˆq‰S‰ˆØ	‰ˆÜˆq‹6�E‹>ØØ	ˆQ‰ˆñ ð 	
Œ#ˆa‹&‰�‰
€AÞ
Ü�‹7”gÒØ—F‘FˆEØ—F‘F‰EàˆEØˆEØ�C‹<˜E C›KØ”�B‘‰JˆAð €Hô �‹7”gÒØ�v‰v˜‹zØ”R˜‘U‘
�Ø�v‰v˜‹zØ”R˜‘U‘
�Ø€Hr   c                 óH  • U =p#Sn X0-  U-  nX4-  n[        U5      S:  a  OX%-  nUS-  nM&  U[        -  nU(       a  U[        U * 5      -  nU$ [        U 5      [        L d  U R
                  S:X  a  U[        [        U 5      5      -  nU$ U[        R                  " U 5      -  nU$ )NrT   rS   r§   r,   )r‹   Úeulerr.   r   r   r:   r0   rC   )r<   rÊ   r—   r{   rË   Úterms         r   Ú	ei_taylorrÒ     s¶   € Ø€I€AØ	€AØ
Ø‰C�‰EˆØ‰sˆÜˆt‹9�uÓØØ	‰	ˆØ	ˆQ‰ˆñ ð Œ�J€AÞ
Ø	ŒS�!�‹W‰ˆð €Hô	 �‹7”eÒ˜qŸv™v¨›}Ø”œ#˜a›&Ó!Ñ!ˆAð €Hð ”—’˜1“ÑˆAØ€Hr   c                 ó¬  • [        U 5      nU[        [        4;  a   [        U 5      n [        nU (       d  [
        * $ [        U 5      nU[        :”  a  [        X5      $ US::  d  U[        L a  U S:”  a  [        X5      $ U[        L a   U R                  S:”  a  X-  n[        XA5      nO[        U -  U-  n[        XA5      nX@-
  S-  nX@-   S-  nSn[        U 5      [        L a  [        R                  n	O[        R                  n	[         H  u  p«Xj-  U-   nX‹U	" U5      -  U-  -  nM     XVU-  -  nU$ ! [        [        4 a    [        U 5      n [        n GN$f = f)Nrk   r,   rQ   )r   r   r	   r
   r   ÚINFr‹   ÚEI_ASYMP_CONVERGENCE_RADIUSrÎ   rÒ   r9   rC   rx   r-   Úgauss42)r<   rÊ   ÚtypezÚabszÚzrefÚrefÚCÚDr—   Ú_expr   Úwr{   s                r   Úeirß   '  sJ  € Ü�‹G€EØ”UœGÐ$Ó$ð	Ü�a“ˆAÜˆEö ÜˆtˆÜˆq‹6€DØÔ)Ó)Ü˜ÓÐØ	�‹˜¤%š¨A°«GÜ˜Ó Ð ð ”Ò˜AŸF™F S›LØ‰xˆÜ˜Ó"‰ä*¨QÑ.°Ñ5ˆÜ�tÓ!ˆØ	‰�‰€AØ	‰�‰€AØ€AÜˆAƒw”'ÒÜ�y‰y‰ä�x‰xˆß‰ˆØ‰C�‰EˆØ	‰t�A‹w‰Y�q‰[ÑŠñ ð ˆQ‰3�J€CØ€Jøô; œ:Ð&ó 	Ü˜“
ˆAÜ‹Eð	ús   �D. Ä.!EÅEc                 ó4  • [        U 5      n[        U 5      [        [        4;  a   [        U 5      n [        nU[        L a'  U R
                  (       d  [        U R                  S5      n [        U * SS9* $ ! [        [        4 a    [        U 5      n [        n N_f = f)Nr,   T)rÊ   )r   r   r	   r
   r   r:   r9   rß   )r<   r×   s     r   Úe1rá   L  s‡   € ô �‹G€EÜˆAƒw”uœgÐ&Ó&ð	Ü�a“ˆAÜˆEð ”Ò §§Ü�A—F‘F˜CÓ ˆä�ˆr�tÑÐÐøô œ:Ð&ó 	Ü˜“
ˆAÜŠEð	ús   ¦A3 Á3!BÂB)g      à¿r,   gÓSb¦Qú?r   gHØÇ*2Qñ?gÎ|ÊŽA—ð?gE’À„	Gð?g9ÏÚ2"ð?g—cøj³ð?gµØó9ð?g¹[:ãð?g³H¾1ð?gÓ,[
ð?g¼�¬€ ð?gÔÊ+9@ ð?gâ—÷  ð?g²ÞL ð?g´9 ð?gNe²  ð?ga;  ð?g”Å  ð?gÖ•€  ð?g›1@  ð?g»    ð?gZ>   ð?gÇ   ð?gí   ð?)	gA   ÀgîáêÖpæ¿gp »Ï6±¿gçì|› Up¿gµïÞ ‹%¿gE3†Ô+�Ó¾g’«Áf{¾g'(ø
Þ¾gÑ²GÐ]´½)	rJ   g/"®=øí¿g{õ‰Èû%®¿gh{sr¿g’å9’DÌ"¿glXx‹‚kÕ¾gê%ñÂé¼y¾g†Õ®*·¾geuyÚ-´½)g]u:õßô=g h#%ß.J¾g`5'>‹>gø^õ±Ò¹¾gU3 ßì¡¾gPFÓ; ?g„á2¾­M¿g=Æ§_Îšc¿gviT{¼�µ?g†ž\€ãÏÚ?gH$  à?)g��ü…È÷½gI„™’™;¾gÑ.kš^­‡>g�P�N«¯¸¾g„/†wðo¥¾gSŒ' ?gº7zƒ2¯M¿g)ÐŽšc¿gËú�»�µ?gˆ|É�ãÏÚ?rQ   c                 óH  • [        U [        [        45      (       d   [        U 5      n U S:X  a  [        S5      eU S:¼  a  SSU * -  -   SU * -  -   $ [        U 5      nX:X  a  US:¼  a	  [        U   $ US-  (       d  g	U S	::  a?  SU -  [        U S-
  -  -  [        S
U -  5      -  [        SU -
  5      -  [        SU -
  5      -  $ U S::  a2  U S::  a  [        [        U 5      U S-
  -  $ [        [         U 5      U S-
  -  $ [        ["        U 5      [        [$        U 5      -  nSSU * -  -   SU * -  -   SU * -  U-  -   $ ! [        [        4 a`     [        U 5      n U R                  (       d   [        [        U R                  5      5      s $  [        e! [        [        4 a     [        ef = ff = f)z&
Riemann zeta function, real argument
rS   zzeta(1) poleé   rJ   rk   g      @r   rT   r,   rQ   g      @)Ú
isinstancer   rp   r   r
   r	   r:   Úzetar9   ÚNotImplementedErrorÚ	_zeta_intrX   rV   rt   r¥   Ú_zeta_0Ú_zeta_1Ú_zeta_PÚ_zeta_Q)r—   rL   r<   s      r   rå   rå   €  s®  € ô �aœ%¤˜×&Ñ&ð		&Ü�a“ˆAð 	ˆAƒvÜ˜Ó(Ð(ØˆBƒwØ�S˜A˜2‘Y‰ ¨ r¡Ñ*Ð*ÜˆA‹€AØƒvØ�‹6Ü˜Q‘<ÐØ�A—ØØˆCƒxØ�1‰u”R˜!˜A™#‘Y‰œ{¨3¨q©5Ó1Ñ1´+¸aÀ¹cÓ2BÑBÄ4ÈÈ!ÉÃ9ÑLÐLØˆCƒxØ�‹8ÜœG AÓ&¨¨!©Ñ,Ð,Üœ Ó" A a¡CÑ(Ð(Ü”˜Óœh¤w¨qÓ1Ñ1€AØ�˜�r‘‰?˜S A 2™YÑ&¨°¨r©°1©Ñ4Ð4øô3 œIÐ&ó 	&ðÜ˜A“J�Ø—v—vÜ"¤4¨¯©£<Ó0Ò0ð ô &Ð%øô ¤	Ð*ó ØÜ%Ð%ðúð	&ús5   �D1 Ä1F!Å9FÅ;F!Å>F!ÆFÆF!ÆFÆF!)F)YÚ__doc__Úoperatorr-   rC   rX   ÚeÚsqrt2Úsqrt5ÚphiÚln2Úln10rÐ   ÚcatalanÚkhinchinÚaperyr�   r   r   r)   r.   r0   r3   r   r
   r2   Úpowrx   r@   rA   ÚtanÚacosÚasinÚatanÚcoshÚsinhÚtanhr8   r>   Úcos_sinÚcbrtrN   rV   rZ   r\   r_   re   rd   rg   rj   rÔ   ÚNINFÚNANÚEPSrs   Úlenrr   rw   ru   rt   r}   r   r€   r‚   r…   rŽ   r�   r    r¢   Údigammarª   r«   r¥   r¨   r¬   r¯   r±   r³   rÖ   rÕ   rÎ   rÒ   rß   rá   rç   rê   rë   ré   rè   rå   rF   r   r   Ú<module>r     s¡  ðñó Û Û ð €Ø€Ø€Ø€Ø€Ø€Ø€Ø€Ø
 €Ø €Ø€à€òò	òðØ‡H‚HˆT„Nòòñ �—‘Ñ9Ó:€Ù�˜5Ÿ9™9Ó%€Ù�	˜5Ÿ:™:Ó&€Ù�D—H‘H˜eŸi™iÓ(€á�D—H‘H˜eŸi™iÓ(€Ù�D—H‘H˜eŸi™iÓ(€Ù�D—H‘H˜eŸi™iÓ(€á�—	‘	˜5Ÿ:™:Ó&€Ù�—	‘	˜5Ÿ:™:Ó&€Ù�T—Y‘Y §
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Ó+€á�T—Y‘Y §
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Ó+€Ù�T—Y‘Y §
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Ó+€Ù�T—Y‘Y §
¡
Ó+€á�d—j‘jÙ=ó	?€á�T—Y‘YÙ;ó=€ñ Ñ<Ù>ó@€ñ Ñ#Ñ%8Ó9€òò	#ò	"ò	$ò	#ñ 	�k >Ó2€Ù�k >Ó2€òòð €Ø€t€Ø	ˆ#�g€Ø€à�S˜#˜s C¨¨u°e¸VÀWØ
ˆI�z ;°¸mØÐ#Ð%6Ð8JØÐ-ð/€ñ
 �|Ó$ QÑ&Ð ð €
ðI€
òGò&Hñ 	�k >Ó2€òòò%ò5òn
€
ò#ò,$ñ( ˜Ð'7Ó
8€ò	ñ  $ ˜tñ	%€ò
ñ  $ ˜tñ
%€òò#òIòòò ò"*/€ðX #Ð ôô8ô(#òJò €	òñ ˜B˜$ñ	 €òñ ˜B˜$ñ	 €òB€òC€ó
 5øðm 	�IÐó Ø�x‰x€HØ—	‘	ƒIðús   ±K Ë K1Ë0K1