
    [h|-                        S SK JrJr  \S 5       r\S 5       r\S 5       r\S 5       r\SS j5       r\S 5       r\S 5       r	\SS	 j5       r
\SS
 j5       r\S S j5       r\S!S j5       r\S"S j5       r\S"S j5       r\S"S j5       r\S 5       r\S"S j5       r\S 5       r\S 5       r\S 5       r\S 5       r\S 5       r\S 5       r\S 5       r\S 5       r\S 5       r\S 5       r\S 5       rg)#   )defundefun_wrappedc                     U R                  US5      nSU R                  U R                  5      -  U-  U R                  SSU5      -  nU R	                  U5      (       d  U R                  U5      U R                  -  nU$ )N   r   r      r   square_exp_argsqrtpihyp1f1_re_imjctxzz2vs       U/var/www/auris/envauris/lib/python3.13/site-packages/mpmath/functions/expintegrals.py_erf_complexr      sj    			Ar	"B	
388CFF	QE%!<<A771::GGAJsuuH    c                    U R                  U5      S:  ac  U R                  U5      nU R                  USS9nU R                  U5      U R	                  U R
                  5      -  U R                  SSU5      -  nOSU R                  U5      -
  nU R                  U5      (       d!  SU R                  U5      U R                  -  -   nU$ )Nr   Texactr   r   )rer   fnegexpr   r   hyperur   r   r   r   )r   r   r   nz2r   s        r   _erfc_complexr#      s    
vvay1}"hhrh&GGCL#&&))CJJuUB,GG  ##771::cggajHr   c                 z   U R                  U5      nU R                  U5      (       a   U R                  U5      $ U R	                  U5      (       a<  UR
                  (       d+   [        U5      " U R                  UR                  5      5      $ U R                  U5      $ ! [         a     Nof = f! [         a     N-f = fN)	convert_is_real_type_erfNotImplementedError_is_complex_typeimagtyperealr   r   r   s     r   erfr/      s    AA
	88A; Aqvv	7388AFF+,, A # 		
 # 		#   B ")B- 
B*)B*-
B:9B:c                 z   U R                  U5      nU R                  U5      (       a   U R                  U5      $ U R	                  U5      (       a<  UR
                  (       d+   [        U5      " U R                  UR                  5      5      $ U R                  U5      $ ! [         a     Nof = f! [         a     N-f = fr%   )	r&   r'   _erfcr)   r*   r+   r,   r-   r#   r.   s     r   erfcr3   '   s    AA
	99Q< Aqvv	7399QVV,-- Q # 		
 # 		r0   c                     U R                   S-  S-   nU(       a+  U R                  XUS9nU R                  U R                  XTS9nOU R                  XUS9nUS:w  a  U R                  XRSS9nU$ )N      )precr   Tr   )r7   fmulfdivone)r   r   mult
reciprocalr7   r   s         r   r   r   6   sn    88A:b=DXXaX&XXcggrX-XXaX&qyXXbdX+Ir   c                    U(       d  U$ U R                  U5      nSU R                  U R                  5      -  U-  U R                  SSU5      -  nU R	                  U5      (       d  U R                  U5      U R                  -  nU$ )Nr   r   r	   r   r   s       r   erfir>   B   sp    			A	B	
388CFF	A	E5"!==A771::GGAJsuuHr   c                 N  ^ ^ T R                  T5      nUT:w  d  US:  d  US:  a  T R                  S5      $ UmT(       d  T$ TS:X  a  T R                  $ TS:X  a  T R                  $ [	        T5      S:  a  STS-  -  ST-  -   nOwT R                  ST R                  -  [	        T5      S-
  S-  -  5      nT R                  T5      T R                  UT R                  U5      -
  5      -  T R                  S5      -  nT =R                  S	-  sl	        T R                  U U4S
 jU5      $ )Nr   r   z*erfinv(x) is defined only for -1 <= x <= 1g?goe1?r
   gϷ?r   
   c                 ,   > TR                  U 5      T-
  $ r%   )r/   )tr   xs    r   <lambda>erfinv.<locals>.<lambda>]   s    #''!*Q,r   )r   
bad_domaininfninfabslnr   signr   r7   findroot)r   rC   xreaus   ``   r   erfinvrP   L   s    
''!*CqcBhC!G~~JKKAQhAvcgg~Bwsxx
1v|AqDL8A:% FF1SVV8SVAXM)*HHQK#((1svvay=11#((1+=HHNH<<.22r   c                     U R                  U5      nU R                  X-
  S-  * SUS-  -  -  5      X0R                  SU R                  -  5      -  -  $ )Nr   )r&   r    r   r   )r   rC   musigmas       r   npdfrT   _   sM    KKE77QTAI:qz*+uXXah5G/GHHr   c                     X-
  X0R                  S5      -  -  nUS:  a  U R                  U* 5      S-  $ SU R                  U5      -   S-  $ )Nr       r   )r   r3   r/   )r   rC   rR   rS   rN   s        r   ncdfrW   d   sK    	
hhqk!"A1uxx|A~#''!*ar   c                 :   X4:X  a  SnOU(       dA  US:X  a  US:X  a  U R                  X5      nOXA-  U R                  USU-
  US-   U5      -  U-  nOU R                  U5      u  pxUS::  aR  XR                  * :  a'  U R                  7n	U =R                  S-  sl        X-  nOUS:  a  U =R                  U-  sl        XA-  U R                  USU-
  US-   U5      -  n
X1-  U R                  USU-
  US-   U5      -  nX-
  U-  nU(       a  X`R                  X5      -  nU$ )NrV   r   r   )betahyp2f1nint_distancer7   eps)r   rN   bx1x2regularizedr   mdhs1s2s               r   betaincrg   l   s   	x7rQwA

1ac1Q333a7A  #6HH9}WWHARAUSZZ!A#ac"--USZZ!A#ac"--WM	XXa]Hr   Nc                    [        U5      nU R                  U5      nUc  U R                  nSnOU R                  U5      nX R                  :g  nUc  U R                  nSnOU R                  U5      nX0R                  :g  nU(       dm  U(       df  U(       aN  U R	                  U5      S:  a  U R                  $ U R	                  U5      S:  a  U R
                  $ U R                  $ U R                  U5      $ X#:X  a  U R                  $ U R	                  U5      U R	                  U5      :  a  U R                  XX$5      * $ U(       a  U(       a  U R                  XX45      7$ U(       a  U R                  XU5      $ U(       a  U R                  XU5      $ g )NFrV   )boolr&   zerorG   r   r:   nangammagammainc_gamma3_upper_gamma_lower_gamma)r   r   rN   r^   ra   lower_modifiedupper_modifieds          r   rm   rm      sB   {#KAAyHHKKNhhyGGKKNggnvvay1}wwQwwwwyy|vxx
vvay366!9Q1222.A!111	k22	k22 
r   c                    ^ ^^^ T R                  U5      (       a  [        U5      " T R                  5      $ U/U-  mT R                  TSS9mUUU U4S jnT R	                  XA/5      $ )NTr   c                 P   > TR                  T5      TU /SU S// TS/SU -   /T4nU4$ )Nr   r   r    )r   T1Gr^   r   negbs     r   rd   _lower_gamma.<locals>.h   s;    ggdmQ"Q2JAsQqSE1Dur   )isnpintr,   rG   r   	hypercomb)r   r   r^   ra   rd   rw   rx   s   ` `  @@r   rp   rp      s`     {{1~~Awsww	
kA88AT8"D  ==C  r   c                   ^ ^^^ T R                  U5      (       a   U(       a~  T R                  U5      (       a  [        U5      " T R                  5      $ T R                  n T =R                  S-  sl        T R                  UT5      T R                  U5      -  UT l        $ T R                  UT5      $ US:X  a  TS:X  a  UT-   S-  $ US:X  a  TS:X  d  TS:X  a  UT-   S-  $ T R                  TSS	9mU/U-  m UUU U4S
 jnT R                  XQ/SS9$ ! UT l        f = f! [         a     Nwf = f! T R                   a    UUU U4S jnT R                  XQ/5      s $ f = f)Nr@   r   r   rV   r
   y            y            ?Tr   c                 T   > U S-
  nTR                  T5      T/SU// TSU* // ST-  4/$ Nr   ru   )r   rrw   rN   r   negas     r   rd   _upper_gamma.<locals>.h   s?    !AggdmQ'!QQQBQtVLMMr   )force_seriesc                 p   > / SU S-
  /U /T/ / S4nTR                  T5      * TU /SU S// TS/SU -   /T4nX4$ )Nr   rV   r   ru   )r   rv   T2rw   rN   r   r   s      r   rd   r      s[    a1XsAr2q0B774=.!Q'!QRQC!A#IB6Mr   )isintrz   r,   rj   r7   _gamma_upper_intrl   r)   r   r{   NoConvergence)r   r   rN   ra   origrd   rw   r   s   ` `   @@r   ro   ro      s_    yy||	;;q>>7388,,xx$HHNH//15		!D#CH++Aq11 	Av!r'!QwAv1:e!Qw88AT8"D	
kA
%	N 	N }}Q$}77#  $CH # 		  %	 	 }}Q$$%sG   7D D !8D D !D 8D. 	DD 
D+*D+.+EEc                 n   U R                  U5      nU(       a  U(       a  U R                  $  U =R                  S-  sl        U R                  XUS9nU R                  XUS9nXg-
  nU R	                  U5      [        U R	                  U5      U R	                  U5      5      -
  S:  a  UU =R                  S-  sl        $ U(       d>  U R                  USX4S9nU R                  USX$S9nXg-
  n UU =R                  S-  sl        $ U =R                  S-  sl        [        e! U =R                  S-  sl        f = f)N   )ra   irV   )rz   rj   r7   rm   magmaxr)   )	r   r   rN   r^   ra   polerv   r   Rs	            r   rn   rn      s   ;;q>DtxxB \\!K\8\\!K\8G771:CGGBK55; 	B aA?BaA?BA BB
 	Bs   A:D =/D D4c                 n   U R                  U5      (       a(  U R                  U5      (       a   U R                  X5      $ U R	                  U5      (       d  U R	                  U5      (       a  X!-  $ X R
                  :X  a  SU-  $ US:X  aB  U R                  U5      S::  a  [        U5      " U R
                  5      $ U R                  US-
  -  $ US:X  a  U R                  U* 5      U-  $ US:X  a  U R                  U* 5      US-   -  US-  -  $ X!S-
  -  U R                  SU-
  U5      -  $ ! [         a     Nf = f)Nr   rV   r   r   )r   r'   _expint_intr)   isnanrG   r   r,   r:   r    rm   )r   nr   s      r   expintr      s   
yy||))!,,	??1(( yy||syy||s
GG|s
Av66!9>7377##77AaC= Avwwr{1}Bwwwr{AaC A%%s8cll1Q3***# # 		s   D' '
D43D4c                 "   U(       aN  US:X  a  U R                   $ U R                  U R                  U5      5      U R                  U R                  5      -
  $ U(       d  U$ US:X  a  U R                  $ U R                  U R                  U5      5      $ )Nr   r   )rj   eirJ   ln2rH   )r   r   offsets      r   lir     sl    688OvvcffQi 366#''?22Avxx66#&&)r   c                 h     U R                  U5      $ ! [         a    U R                  U5      s $ f = fr%   )_eir)   _ei_genericr.   s     r   r   r     3    "wwqz "q!!"    11c           	         XR                   :X  a  U$ XR                  :X  a  U R                  $ U R                  U5      S:  a   U R                  U-  nU R                  U5      U R                  SS// UU R                  SS9-  U-  nU R                  U5      nUS:  a  X0R                  U R                  -  -  nUS:  a  X0R                  U R                  -  -  nU$ XR                  SSSSU5      -  U R                  -   nU R                  U5      (       a8  USU R                  U5      U R                  U R                  U-  5      -
  -  -  nU$ X0R                  [        U5      5      -  nU$ ! U R                   a     Nf = f)Nr   T)maxtermsr   rV   r         ?)rG   rH   rj   r   r:   r    hyperr7   r   r   r   r   hyp2f2eulerlogrI   )r   r   r   r   ims        r   r   r   &  sd    	GG|HH}xx
wwqzA~		A
399aU2a % 6 6678ABAvVVCEE\!AvVVCEE\!H 	
**Qq1Q
#))+A
wwqzz	S#''!*swwswwqy1122 H 	
WWSV_H    		s   BE+ +E>=E>c                 j     U R                  U5      $ ! [         a    U R                  SU5      s $ f = fr~   )_e1r)   r   r.   s     r   e1r   B  s5     wwqz  zz!Q s    22c                 h     U R                  U5      $ ! [         a    U R                  U5      s $ f = fr%   )_cir)   _ci_genericr.   s     r   cir   I  r   r   c                    U R                  U5      (       a9  XR                  :X  a  U R                  $ XR                  :X  a  U R                  S-  $ U R                  U R                  USS9nU R                  USS9nSU R                  U5      U R                  U5      -   -  nU R                  U5      nU R                  U5      nUS:X  a.  US:  a  X@R                  S-  -  nUS:  a  X@R                  S-  -  nUS:  a.  US:  a  X@R                  S-  -  nUS:  a  X@R                  S-  -  nU R                  U5      (       a  US:  a  U R                  U5      nU$ )N              ?Tr   r   rV                 ?)isinfrG   rj   rH   r   r8   r   r   r   r   r   r'   )r   r   jznjzr   zrealzimags          r   r   r   P  s&   
yy||<=*	#%%	%B
((2D(
!CSVVBZ#&&+%&AGGAJEGGAJEz19a66$;&a19a66$;&aqyA:qFF2I~qA:qFF2I~q
	GGAJHr   c                 h     U R                  U5      $ ! [         a    U R                  U5      s $ f = fr%   )_sir)   _si_genericr.   s     r   sir   d  r   r   c                 ~   U R                  U5      (       a<  XR                  :X  a  SU R                  -  $ XR                  :X  a  SU R                  -  $ U R	                  U5      S:  a  U R                  U R                  USS9nU R                  USS9nSU R                  U5      U R                  U5      -
  -  nU R                  U5      nUS:  a  USU R                  -  -  nUS:  a  USU R                  -  -  nU R                  U5      (       a  U R                  U5      nU$ XR                  SS	S	S
U-  U-  5      -  $ )Nr         r   Tr   y             rV   r   r	   g      п)r   rG   r   rH   r   r8   r   r   r   r   r'   hyp1f2)r   r   r   r   r   r   s         r   r   r   k  s   
yy||<CFF
*=cff,
wwqzRXXceeADX)hhrh%SVVBZ#&&+-.
19SVVOA19SVVOAQ
AE%eAgai888r   c                 H   U R                  USS9nSU R                  U5      U R                  U5      -   -  nU R                  U5      nU R                  U5      nUS:  a  X0R                  S-  -  nU$ US:  a  X0R                  S-  -  nU$ US:  a  X0R                  S-  -  nU$ )NTr   r   rV   r   r   )r   r   r   r   r   )r   r   nzr   r   r   s         r   chir     s    	!4	 BSVVAY#$AGGAJEGGAJEqy	VVD[
 H	 
	VVD[ H 
	VVBYHr   c                 V   U R                  U5      S:  ay  U R                  USS9nSU R                  U5      U R                  U5      -
  -  nU R                  U5      nUS:  a  USU R                  -  -  nUS:  a  USU R                  -  -  nU$ XR                  SSSS	U-  U-  5      -  $ )
Nr   Tr   r   rV   r   r   r	   g      ?)r   r   r   r   r   r   )r   r   r   r   r   s        r   shir     s     wwqzRXXatX$SVVBZ'(
19a4;&a19a4;&a::eE%Qq999r   c           	         XR                   :X  a  U R                  S5      $ XR                  :X  a  U R                  S5      $ U R                  US-  -  S-  U R	                  SSSU R                  S-  * US	-  -  S
-  5      -  $ )Nr   r   r
      )r
   r5   r	   )   r5   r   r5      )rG   mpfrH   r   r   r.   s     r   fresnelsr     sq    GG|wws|HH}wwt}66!Q$;q=E%svvqyjAob6HIIIr   c           	          XR                   :X  a  U R                  S5      $ XR                  :X  a  U R                  S5      $ XR                  SSSU R                  S-  * US-  -  S-  5      -  $ )	Nr   r   )r   r5   r   )   r5   r   r5   r   )rG   r   rH   r   r   r.   s     r   fresnelcr     s_    GG|wws|HH}wwt}ZZeE36619*QT/"*<===r   )r   F)rV   r   )rV   r   F)rV   NF)F)	functionsr   r   r   r#   r/   r3   r   r>   rP   rT   rW   rg   rm   rp   ro   rn   r   r   r   r   r   r   r   r   r   r   r   r   r    r   r   <module>r      s(   +  	 	       	 	   3 3$ I I      0 &3 &3P 	! 	! $% $%L  2 + +. 	 	 " "  6     " "  & " " 9 9(   
: 
: J J > >r   