
    \h*                         S SK JrJrJrJrJrJrJr  S SKJ	r	J
r
  S SKJrJrJrJr  S SKJr  S SKJrJrJr  S SKJr  S SKJr  S SKJrJr  S S	KJr  S S
KJ r J!r!  S SK"J#r#J$r$J%r%  S SK&J'r'J(r(J)r)J*r*  S SK+J,r,  S SK-J.r.  S SK/J0r0J1r1J2r2  S SK3J4r4  S SK5J6r6J7r7J8r8  S SK9J:r:J;r;  S SK<J=r=  S SK>J?r?  S SK@JArAJBrBJCrCJDrDJErEJFrFJGrGJHrH  S SKIJJrJJKrKJLrLJMrM  S SKNJOrOJPrP  S SKQJRrRJSrS  S SKTJUrU  \OS 5       rV\OS 5       rW\OS 5       rX\OS 5       rYg)    )laplace_transforminverse_laplace_transformLaplaceTransformInverseLaplaceTransform_laplace_deep_collectlaplace_correspondencelaplace_initial_conds)Function
expand_mul)
EulerGammaSubs
Derivativediff)factor_terms)Ioopi)Eq)S)Symbolsymbolssimplify)Absre)explog	exp_polar)coshsinhcothasinh)sqrt)	Piecewise)atancossin)And)
lowergammagamma
uppergamma)
DiracDelta	Heaviside)SingularityFunction)lerchphi)fresnelcfresnelserferfcEiCiexpintE1)besselibesseljbesselkbessely)slowwarns_deprecated_sympy)Matrixeye)sc                  @u   [         n [        n[        SSS9u  p#n[        SSSS9n[        S5      u  pgn[        S5      n	[        S5      n
[        S	5      n[        S
5      n[        S5      n[	        [        Xb-   Xc-   -  5      [        S[        Xb-   Xc-   -  US-  -
  5      5      -   U5      [        X#-  US-  -   XbU-   -  -   5      [        S[        X#-  XbU-   -  -   5      5      -   :X  d   e[        [        U" U5      US5      U[        SS9n[        XU05      n[        XU/ SQ05      nU[        S-  U" [        5      -  S[        S-  -  -
  S[        -  -
  S-
  :X  d   eS[        [        U5      U[        5      -  S-
  R                  5       SS[        S-
  -  -   :X  d   eU " X&-  US-  -   U[        S5      S-  -  -   U[        5      U[        S-  -  S[        S-  -  -   S[        [        5      -  S[        [        S5      S-  -  -  -  -   SS4:X  d   eU " X6U-   -  U[        5      U* [        U* [        -  5      -  [!        U* [        -  5      -  SS4:X  d   eU " S[        Xb-   5      -  U[        5      [        [        5      [        S[        -  5      -  [        U[        -  5      -  [#        [        U5      [        [        5      -  5      -  SS4:X  d   eU " [        U5      Xb-   -  U[        5      [        * [        U5      -  [        U[        -  5      -  [#        [        U5      [        [        5      -  5      -  [        [        5      [        S[        -  5      -  -   SS4:X  d   eU " Xb-   [        S5      * S-  -  U[        5      S[        [        5      -  [        [        5      -  [        U[        -  5      -  [#        [        U5      [        [        5      -  5      -  S[        U5      -  -   SS4:X  d   eU " U[        S5      S-  -  Xb-   S-  -  U[        5      [        * [        U5      -  [        U[        -  5      -  [#        [        U5      [        [        5      -  5      -  [        [        5      [        S[        -  5      -  -   SS4:X  d   eU " SU[        U5      -  US-  -   -  U[        5      [        [        U5      -  [        U[        -  5      -  [#        [        U5      [        [        5      -  5      -  SS4:X  d   eU " Xb-   U-  U[        5      [        U* S-
  -  [        U* [        -  5      -  [%        US-   U[        -  5      -  SS4:X  d   eU " US-  Xb-   -  U[        5      SUS-  -  [%        SU[        -  5      -  SS4:X  d   eU " [        U5      U[        5      S[        S-
  -  SS4:X  d   eU " [        SU-  5      U[        5      S[        S-
  -  SS4:X  d   eU " [        X&-  5      U[        5      S[        U-
  -  US4:X  d   eU " [        X&U-
  -  5      U[        5      [        U* U-  5      U* [        -   -  US4:X  d   eU " U[        U* U-  5      -  U[        5      U[        -   S-  U* S4:X  d   eU " U[        U* Xc-
  -  5      -  U[        5      [        X#-  5      U[        -   S-  -  U* S4:X  d   eU " X6-  [        U* U-  5      -  U[        5      X2[        -   S-  -  U* S4:X  d   eU " [        U* [        U* 5      -  5      U[        5      ['        [        U5      U[        -  -  SS4:X  d   eU " [        U* [        U5      -  5      U[        5      U[        -  [%        [        * U5      -  SS4:X  d   eU " U[        S5      S-  -  [        SU-  5      -  [)        [        S5      S-  5      -  U[        5      [        S-   [        S5      * S-  -  SS4:X  d   eU " U[        S5      S-  -  [        SU-  5      -  U[        5      S[        [        5      -  S[        S-   [        S5      S-  -  -  -  SS4:X  d   eU " Xb-  [        U* U-  5      -  U[        5      U[        -   U* S-
  -  [)        US-   5      -  U* S4:X  d   eU " U[        U* US-  -  5      -  U[        5      [        [        5      U-  [        [        S-  SU-  -  5      -  [#        [        S[        U5      -  -  5      -  S[        U5      -  -  SS4:X  d   eU " [        SUS-  -  5      U[        5      [        S5      [        [        5      -  [        [        S-  S-  5      -  [#        [        S5      [        -  S-  5      -  S-  SS4:X  d   eU " U[        SUS-  -  5      -  U[        5      U[        [        SUS-  -  5      U[        5      -  [*        * S4:X  d   eU " U[        U* US-  -  5      -  U[        5      SSU-  -  [        [#        [        S[        U5      -  -  5      -  S[        [        5      -  U[        S5      S-  -  -  -  -
  SS4:X  d   eU " [        U* U-  5      U[        5      S[        U5      -  [        S[        -  5      -  [-        SS[        U5      -  [        [        5      -  5      -  SS4:X  d   eU " [        U5      [        U* U-  5      -  U[        SS9[        [        5      [        U5      [        [        5      -  S[        S5      -  -   -  [        [        S-  5      -  [        S[        U5      -  [        [        5      -  5      -  SS4:X  d   eU " [        U* U-  5      [        U5      -  U[        5      [        [        5      [        S[        -  5      -  [        S[        U5      -  [        [        5      -  5      -  SS4:X  d   eU " [        U* U-  5      U[        U5      -  -  U[        5      [        [        5      [        SU-  5      -  [        S[        U5      -  [        [        5      -  5      -  SS4:X  d   eU " US-  [        SU-  5      -  U[        5      S[        S5      -  S[        -  [        S5      S-  -  -  [-        SS[        S5      -  [        [        5      -  5      -  SS4:X  d   eU " [/        X&-  5      U[        5      X"S-  * [        S-  -   -  US4:X  d   eU " U[/        X&-  5      S-  -  U[        5      SUS-  -  U-  S US-  -  [        -  [        S-  -   -  SU-  S4:X  d   eU " U[/        X&-  5      S-  -  U[        SS9SUS-  -  U-  [        S US-  -  [        S-  -   -  -  SU-  S4:X  d   eU " [1        SU-  5      U[        5      [        S [        S-  -   -  SS4:X  d   eU " [1        X&-  5      U[        5      [        US-  * [        S-  -   -  US4:X  d   eU " [1        X&-  5      S-  U[        SS9SUS-  -  [        S-  -
  [        SUS-  -  [        S-  -
  -  -  SU-  S4:X  d   eU " [/        US-   5      U[        SS9[        [/        S5      -  [1        S5      -   [        S-  S-
  -  SS4:X  d   eU " S![3        Xv-  U-   5      S-  -  U[        5      u  n  nUS"[        S-  [        [        * [5        SU-  5      -  SU-  [3        SU-  5      -  -   -  -   SUS-  -  -   -  [        [        S-  SUS-  -  -   -  -  -
  R7                  5       S:X  d   eU " [/        X&-  5      [1        X&-  5      -  U[        SS9US US-  -  [        S-  -   -  SU-  S4:X  d   eU " [/        X&-  5      U-  U[        5      [9        U[        -   U* [        -   -  5      S-  US4:X  d   eU " U[        S5      * S-  -  [/        X&-  5      -  U[        5      [        [        5      * [        U* [        -   5      [        U[        -   5      -
  -  US4:X  d   eU " U[        S5      S-  -  [1        X&-  5      -  U[        5      U[        -   [        S#5      * S-  -  U* [        -   [        S#5      * S-  -  -   [)        [        S#5      S-  5      -  S-  US4:X  d   eU " [9        U5      U[        SS9[9        [        5      * [:        -
  [        -  SS4:X  d   eU " [9        Xb-  5      * U[        SS9[9        U5      [9        [        5      -   [:        -   [        -  SS4:X  d   eU " [9        SX&-  -   5      U[        5      [        [        U-  5      * [!        [        * U-  5      -  [        -  SS4:X  d   eU " [9        Xb-   5      U[        SS9[        [9        U5      -  [        [        U-  5      [!        [        * U-  5      -  -
  [        S-  -  SS4:X  d   eU " [9        U5      [        U5      -  U[        SS9[        [        5      [9        [        5      * [9        S5      -
  [:        -
  -  [        [        5      -  SS4:X  d   eU " U[        S5      S-  -  [9        U5      -  U[        SS9[        [        5      S$[9        [        5      -  [9        S%5      -
  S[:        -  -
  S&-   -  S[        [        S5      S-  -  -  -  SS4:X  d   eU " US-  [9        U5      -  U[        SSS'9S([9        [        5      * [        R:                  -
  [        S5      S(-  -   -  [        S-  -  -
  R7                  5       [        R<                  :X  d   eU " [9        U5      S-  U[        SS9[9        [        5      [:        -   S-  [        S-  S(-  -   [        -  SS4:X  d   eU " [        U* U-  5      [9        U5      -  U[        SS9[9        U[        -   5      * [:        -
  U[        -   -  U* S4:X  d   eU " [3        X&-  5      U[        5      X"S-  [        S-  -   -  SS4:X  d   eU " [?        [3        X&-  5      5      U[        5      U[A        [        [        -  SU-  -  5      -  US-  [        S-  -   -  SS4:X  d   eU " [3        X&-  5      U-  U[        5      [C        U[        -  5      SS4:X  d   eU " [3        X&-  5      S-  U-  U[        5      [9        SUS-  -  [        S-  -  S-   5      S-  SS4:X  d   eU " [3        X&-  5      S-  US-  -  U[        5      U[C        SU-  [        -  5      -  [        [9        SUS-  -  [        S-  -  S-   5      -  S-  -
  SS4:X  d   eU " [5        X&-  5      U[        5      [        US-  [        S-  -   -  SS4:X  d   eU " [5        X&-  5      S-  U[        5      SUS-  -  [        S-  -   [        SUS-  -  [        S-  -   -  -  SS4:X  d   eU " [3        X&-  5      [3        X6-  5      -  U[        5      SU-  U-  [        -  [        S-  X#-
  S-  -   [        S-  X#-   S-  -   -  -  SS4:X  d   eU " [5        X&-  5      [3        X6-  5      -  U[        5      X2S-  * US-  -   [        S-  -   -  [        S-  X#-
  S-  -   [        S-  X#-   S-  -   -  -  SS4:X  d   eU " [5        X&-  5      [5        X6-  5      -  U[        5      [        US-  US-  -   [        S-  -   -  [        S-  X#-
  S-  -   [        S-  X#-   S-  -   -  -  SS4:X  d   eU " U* U-  [5        X&-  5      -  [3        X&-  5      -   U[        SS9SUS-  -  US-  SUS-  -  [        S-  -  -   [        S-  -   -  SS4:X  d   eU " U[        U* U-  5      -  [3        X&-  5      -  U[        5      UU-  US-  U[        -   S-  -   -  U* S4:X  d   eU " U[        U* U-  5      -  [5        X&-  5      -  U[        5      XC[        -   -  US-  U[        -   S-  -   -  U* S4:X  d   eU " [5        US-   5      U[        SS9u  nnnUS:X  d   eU[        [5        S5      -  [3        S5      -
  [        S-  S-   -  -
  S:X  d   eU " [E        X&-  5      U[        5      [        [        S-  SUS-  -  -  5      [#        [        SU-  -  5      -  [        -  SS4:X  d   eU " [        SX&-  5      U[        5      S[        US-  [        S-  -   5      -  SS4:X  d   eU " [        SX&-  5      U[        SS9X"S-  [        S-  -   [        [        US-  [        S-  -   5      -  -   -  SS4:X  d   eU " [        SX&-  5      U[        SS9US-  [        US-  [        S-  -   5      [        [        US-  [        S-  -   5      -   S-  -  -  SS4:X  d   eU " U[        SX&-  5      -  U[        5      [        US-  [        S-  -   [        S5      S-  -  -  SS4:X  d   eU " U[        SX&-  5      -  U[        5      X"S-  [        S-  -   [        S5      S-  -  -  SS4:X  d   eU " US-  [        SX&-  5      -  U[        5      SUS-  -  US-  [        S-  -   [        S5      S-  -  -  SS4:X  d   eU " [        SU[        US-  X6-  -   5      -  5      U[        SS9[        U[        [        US-  [        S-  -   5      -
  -  5      [        US-  [        S-  -   5      -  SS4:X  d   eU " [G        SX&-  5      U[        5      S[        US-  * [        S-  -   5      -  US4:X  d   eU " [G        SX&-  5      U[        SS9X"S-  * [        S-  -   [        [        US-  * [        S-  -   5      -  -   -  US4:X  d   eU " [G        SX&-  5      U[        SS9US-  [        US-  * [        S-  -   5      [        [        US-  * [        S-  -   5      -   S-  -  -  US4:X  d   eU " U[G        SX&-  5      -  U[        5      [        US-  * [        S-  -   [        S5      S-  -  -  US4:X  d   eU " U[G        SX&-  5      -  U[        5      X"S-  * [        S-  -   [        S5      S-  -  -  US4:X  d   eU " US-  [G        SX&-  5      -  U[        5      SUS-  -  US-  * [        S-  -   [        S5      S-  -  -  US4:X  d   eU " [I        SX&-  5      U[        5      S[K        [        U-  5      -  [        [        US-  [        S-  -   5      -  -  SS4:X  d   eU " [-        SX&-  5      U[        5      [9        [        [        US-  * [        S-  -   5      -   U-  5      [        US-  * [        S-  -   5      -  U* S4:X  d   eU " [3        X&-  5      S-  U[        SS9S)US-  -  [        S*US-  -  S+US-  -  [        S-  -  -   [        S-  -   -  -  SS4:X  d   eU " [M        US-
  5      [5        US-
  5      -  U[        5      [        [        [        * 5      -  [        S-  S-   -  SS4:X  d   eU " X)" U5      -  Xg5      U[        U	" U5      Xg5      -  [*        * S4:X  d   eU " U[M        US-   5      -  U	" US-   5      -  U[        5      U[        U	" US-   5      U[        5      -  [*        * S4:X  d   eU " U[M        US-
  5      -  U	" US-
  5      -  U[        5      U[        U	" U5      U[        5      -  [        [        * 5      -  [*        * S4:X  d   eU " X9" Xb-  5      -  U[        5      X#-  [        U	" U5      Xb[        -  5      -  [*        * S4:X  d   eU " [        U	" U5      * U-  5      U[        5      S[        U	" U5      -   -  [O        U	" U5      5      * S4:X  d   eU " [        U* U-  5      U	" U5      -  U[        5      [        U	" U5      Xb[        -   5      [*        * S4:X  d   eU " [/        X&-  5      U	" U5      -  U[        5      [        U	" U5      Xb* [        -   5      S-  [        U	" U5      Xb[        -   5      S-  -
  [*        * S4:X  d   eU " [/        X&-  5      U-  U[        SS9SU-  [        -  US-  SUS-  -  [        S-  -  -
  [        S-  -   -  US4:X  d   eU " [1        X&-  5      U	" U5      -  U[        5      [        U	" U5      Xb* [        -   5      S-  [        U	" U5      Xb[        -   5      S-  -   [*        * S4:X  d   eU " [1        X&-  5      U-  U[        SS9SSU[        -   S-  -  -  SSU[        -
  S-  -  -  -   US4:X  d   eU " [3        X&-  5      U	" U5      -  U[        SS9[P        [        U	" U5      U[P        * U-  [        -   5      * [        U	" U5      U[P        U-  [        -   5      -   -  S-  [*        * S4:X  d   eU " [3        U	" U5      5      U[        5      [        [3        U	" U5      5      U[        5      [*        * S4:X  d   eU " [3        X&-  5      U-  U[        SS9SU-  [        -  US-  SUS-  -  [        S-  -  -   [        S-  -   -  SS4:X  d   eU " [5        X&-  5      U	" U5      -  U[        5      [        U	" U5      U[P        * U-  [        -   5      S-  [        U	" U5      U[P        U-  [        -   5      S-  -   [*        * S4:X  d   eU " [5        X&-  5      U-  U[        SS9US-  * [        S-  -   US-  SUS-  -  [        S-  -  -   [        S-  -   -  SS4:X  d   eU " [3        X&-  U-   5      S-  U	" U5      -  U[        5      u  nnnU[*        * :X  d   eU* [        U	" U5      U[        5      S-  [        U	" U5      US[P        -  U-  [        -   5      [        S[P        -  U-  5      -  S-  -
  [        U	" U5      US[P        -  U-  [        -   5      [        S[P        -  U-  5      -  S-  -
  -   S:X  d   eU " [3        X&-  U-   5      S-  U	" U5      -  U[        SS9n[        XU
05      U
" [        5      S-  U
" S[P        -  U-  [        -   5      [        S[P        -  U-  5      -  S-  -
  U
" S[P        -  U-  [        -   5      [        S[P        -  U-  5      -  S-  -
  :X  d   eU " [3        X&-  5      S-  [1        X6-  5      -  U[        5      u  nnnUU:X  d   eU* SU-  SS,US-  -  US-  -   SU-  [        -  -   [        S-  -   -  -  -
  SU-  SS,US-  -  US-  -   SU-  [        -  -
  [        S-  -   -  -  -
  SU-  SUS-  US-  -   SU-  [        -  -   [        S-  -   -  -  -   SU-  SUS-  US-  -   SU-  [        -  -
  [        S-  -   -  -  -   R7                  5       S:X  d   eU " US-  [        US-  * 5      -  U[        5      [        [        5      [        S-  -  [        [        S-  S-  5      -  [#        [        S-  5      -  S-  [        S-  -
  [        [        5      [        [        S-  S-  5      -  [#        [        S-  5      -  S-  -   SS4:X  d   eU " X&S-  -  X6-  -   U-   U	" U5      -  U[        5      U[S        [        U	" U5      U[        5      [        S45      -  U[S        [        U	" U5      U[        5      [        5      -  -
  U[        U	" U5      U[        5      -  -   [*        * S4:X  d   eU " Xe-  U" U5      -  U[        5      SU-  [S        [        U" U5      U[        5      [        U45      -  [*        * S4:X  d   e[U        SUS:*  4SUS:*  4S-5      nU " UU[        5      S   nUS[        -  [        [        * 5      [        -  -
  :X  d   eU" U[        U5      nU[M        U5      [M        US-
  5      -
  :X  d   e[U        SUS:*  4XfS:*  4SU-
  US:*  4S-5      nU " UU[        5      S   R7                  5       nU[        S[        -  5      S[        [        5      -  -
  S-   [        S[        -  5      -  [        S-  -  :X  d   eU" U[        U5      nU* U[M        U5      -  -   US-
  [M        US-
  5      -  -   SUS-
  -  [M        US-
  5      -  -
  R7                  5       S:X  d   e[U        [        U5      US:*  4SUS:*  4[        U* 5      S45      nU " UU[        5      S   nU[        S5      [        [        * 5      -  [        S-   -  S[        -  -   [        [        * 5      [        -  -
  :X  d   eU" U[        U5      nU[        S5      [        SU-
  5      -  [M        US-
  5      -  [M        U5      -   [M        US-
  5      -
  :X  d   e[U        SUS:*  4SUS:*  4S-5      nU " UU[        5      S   nU[U        SUS:*  4SUS:*  4S-5      [        -  :X  d   eU[U        S[W        US:  US:*  5      4S.5      -  U[U        S[W        US:  US:*  5      4S.5      -  U[U        S[W        US:  US:  5      4S.5      -  U[U        S[W        US:  US:  5      4S.5      -  /nU Hj  nU " UU[        5      S   RY                  5       SU-  [        -  U[        [        * 5      -  [        -  -
  U[        S[        -  5      -  [        -  -   :X  a  Mj   e   U " [U        S[[        US5      4S.5      U[        5      S   [        [U        S[[        US5      4S.5      U[        5      :X  d   eS[        U[        S-  -  U-
  5      -  nU" U[        U5      nU " UU[        5      S   nUS-  US-  -  R7                  5       nUS:X  d   eU " [        U	" U5      US45      U[        SS9[        [        U	" U5      U[        5      -  U	" S5      -
  :X  d   eU " [        U	" U5      US45      U[        SS9[        S-  [        U	" U5      U[        5      -  [        S-  U	" S5      -  -
  [        []        [S        U	" U5      U5      US5      -  -
  []        [S        U	" U5      US45      US5      -
  :X  d   eU " [        U	" XU5      US5      U[        5      [        S-  [        U	" XU5      U[        5      -  [        U	" USU5      -  -
  []        [S        U	" XU5      U5      US5      -
  [*        * S4:X  d   eU " S#[        U	" U5      US45      -  U[        SS9S#[        -  [        U	" U5      U[        5      -  S#U	" S5      -  -
  :X  d   eU " X)" X6-  5      -  U" XF-  5      -   U[        SS9U[        U	" U5      U[        U-  5      -  U-  [        U" U5      U[        U-  5      U-  -   :X  d   e[        U	" U5      XvSS/9[_        U	" U5      XvS5      :X  d   eU " U	" U5      U" U5      -  U[        SS9[        U	" U5      U" U5      -  U[        5      :X  d   eU " X9" X&-  5      -  U[        SS9U[        U	" U5      U[        U-  5      -  U-  :X  d   eU " S[        U5      -  [M        U5      -  U[        5      S[        S-
  -  SS4:X  d   eU " S[3        U5      -  [M        U5      -  U[        SS9S[        S-  S-   -  SS4:X  d   eU " SUS-
  -  [3        S[        -  US-
  -  5      -  [a        US-
  5      -  [M        US0-
  5      [M        US-
  5      -
  -  U[        5      S   S[        -  [        [        * 5      -  :X  d   eU " Xb-
  U-  [        U* Xb-
  -  5      -  [M        Xb-
  5      -  U[        5      U[        -   U* S-
  -  [        U* [        -  5      -  [)        US-   5      -  U* S4:X  d   eU " [        SU-  5      S-
  [        U* U-
  5      -  [M        U5      -  S-  U[        SSS'9[        U* 5      [        S-  S-
  -  :X  d   eU " [a        U5      U[        5      S[*        * S4:X  d   eU " [a        X&-  5      U[        5      SU-  [*        * S4:X  d   eU " [a        US!-  5      U[        5      S![*        * S4:X  d   eU " [a        US!-   5      U[        5      S[*        * S4:X  d   eU " [a        U5      [a        US!-
  5      -   U[        5      S[        S1[        -  5      -   [*        * S4:X  d   eU " [a        U5      U[        U* U-  5      -  -
  U[        SS9[        U[        -   -  U* S4:X  d   eU " [        U* 5      [a        U5      [a        US!-
  5      -   -  U[        SS9[        S1[        -  S!-
  5      S-   [*        * S4:X  d   eU " U	" U5      [a        US!-
  5      -  U[        5      U	" S!5      [        S1[        -  5      -  [*        * S4:X  d   eU " U	" U5      [a        X6-  U-
  5      -  U[        5      U	" X#-  5      [        U* [        -  U-  5      -  U-  [*        * S4:X  d   eU " U	" U5      [a        X6-  U-   5      -  U[        5      S[*        * S4:X  d   eU " [c        XbS5      U[        5      S   [        U* [        -  5      :X  d   eU " [c        XbS5      U[        5      S   [        U* [        -  5      [        S-  -  :X  d   eU " [c        XbU5      U[        5      S   [        [c        XbU5      U[        5      :X  d   eU " [a        US-  5      U[        SS9[        [a        US-  5      U[        5      :X  d   eU " [a        US-  S-
  5      U[        5      [        [        * 5      S-  [*        * S4:X  d   eU " [a        USU-
  -  5      U[        5      S[        [        * 5      -
  [*        * S4:X  d   eU " [a        U5      S-   [a        US-
  5      S-   -  U[        5      [        [a        U5      [a        US-
  5      -  U[        5      S-   [        [        * 5      -   S[        -  -   SS4:X  d   eU " [a        SU-  S[        U5      -  -
  5      U[        5      [        [        * [        U5      -  5      S-  [*        * S4:X  d   eU " [a        SU-  S[        U5      -  -   5      U[        5      [        [        * [        U5      -  5      S-  [*        * S4:X  d   eU " [M        U5      U[        5      S[        -  SS4:X  d   eU " [M        Xb-
  5      U[        5      [        U* [        -  5      [        -  SS4:X  d   eU " [M        US-
  5      U[        5      [        [        * 5      [        -  SS4:X  d   eU " [M        SU-  S-
  5      U[        5      [        S[        -  5      [        -  SS4:X  d   eU " [M        SU-  S-   5      U[        5      S[        -  SS4:X  d   eU " [M        SU-  S-   5      U[        SS9S[        -  [        S[        -  5      [        -  -
  SS4:X  d   eU " U" U5      [M        Xg-
  5      -  U[        5      [        U" U5      [M        Xg-
  5      -  U[        5      [*        * S4:X  d   eU " [M        Xb-
  5      U" U5      -  U[        5      [        U" X&-   5      U[        5      [        U* [        -  5      -  [*        * S4:X  d   eU " [M        Xb-   5      U" U5      -  U[        5      [        U" U5      U[        5      [*        * S4:X  d   eU " [M        U* U-   5      U" U5      -  U[        5      [        U" U5      U[        5      [        U" X&-   5      U[        5      [        U* [        -  5      -  -
  [*        * S4:X  d   eU " [M        U* U-
  5      U" U5      -  U[        5      S2:X  d   e[        [e        U5      U[        SS9[3        [        S-  S[        -  -  5      * [e        [        [        -  5      -  [        S5      [3        [        S-  S[        -  -  [        S-  -   5      -  S-  -   [5        [        S-  S[        -  -  5      [g        [        [        -  5      -  -
  [        -  SS4:X  d   e[        [g        U5      U[        SS9[3        [        S-  S[        -  -  5      [g        [        [        -  5      -  [5        [        S-  S[        -  -  5      [e        [        [        -  5      -  -
  [        S5      [5        [        S-  S[        -  -  [        S-  -   5      -  S-  -   [        -  SS4:X  d   e[i        [        U5      U[        U* 5      -  /U[        U* 5      -  [        U5      //5      n[i        S[        S-
  -  [        S-   S-  /[        S-   S-  S[        S-
  -  //5      n[k        5          [i        S[        S-
  -  SS4[        S-   S-  SS4/[        S-   S-  SS4S[        S-
  -  SS4//5      nS S S 5        [k        5          U " UU[        5      W:X  d   e S S S 5        U " UU[        S3S49USS4:X  d   eU " UU[        SS9U:X  d   eU " UU[        S3SS59U:X  d   eg ! , (       d  f       Nt= f! , (       d  f       N]= f)6Nza, b, cTpositivenp)integerrC   zt, w, xfFgyY      noconds)rK                rO   rP               r   g      ?x   i   r   *      
   ii   @.   )rN   r         @      	   )r   T)rK   T)planeg      ?i)r   r   TF)legacy_matrix)rh   rN   )6r   r   r   r
   r   r   r9   r   r@   r   r	   r   doitr   r#   r   r4   r3   r+   r)   r*   r   r:   r    r   r'   r&   r   r   r   Zeror   r!   r%   r2   r8   r;   r"   r-   r   r   r   r$   r(   expandr   r   r   r,   r.   r1   r0   r>   r=   )LTILTabcrD   twxrF   rG   rH   rI   rJ   L_rg   condx1X1y1x2Y1Z1MtMsMs_condss                              Z/var/www/auris/envauris/lib/python3.13/site-packages/sympy/integrals/tests/test_laplace.pytest_laplace_transformr   "   s;   	B
#Ci$/GA!	td	3Bi GA!AAAAA 	c13+.%aacAC[A-=)>?@AB	DAC!Q$JE"#gaQS1!e9_1E&FF	GH	G 	$qtQ*Aq$?Aqa&)AaQ(9$:;A1QqT	AadF"QqS(1,,,,s1vq!,,q0668BAENJJJqs1a4xAaDF#Q*q!tVa1f_r$r({Aa!A$q&kM::AtDE F Ea1gq!!C1Ib!Ah!64 @@@@qac{Aq!"Xd1Q3iAaC(d1gd1go)>>4HI J ItAw}a#Sa[QqS!$tAwtAw"77$r(4!9:LL   !uQwA&R[a QqS)$tAwtAw*??!DG)K   q1Q46{AC2;&1-Sa[QqS!$tAwtAw"77$r(4!9:LL   q!DG)a#h&'A.QZAaC d1gd1go!664@A B AaxA!a[aRT":a!eQqS#991dCD E DadACj!QC1HZAaC-@$@!T#JJJJc!fa1q5	1d3333c!A#h1!QU)Q!5555c!#h1!QU)Q!5555c!qS'lAq!c1"Q$i!a&8!T%BBBBaQBFmQ"A}qb$&????aQBHoq!$QS1q51*)<qb$(GGGGac#qbd)mQ"qa%!|aR&>>>>c1"S!W*oq!$Aq)9!Q$)>4(HHHHc1"SV)na#1ZA->(>4'HHHHq1Q46{3r!t9$U1R57^3Q:!e"vax "d+, - ,q1Q46{3r!t9$a+tBxZAEQqT!V,,-r489 : 9ad3r!t9na#11eAaCj(@1"d'KKKKqaR1W~q!$"XaZAqD!A#J'Q$q'	](;;QtAwYG   s2ad7|Q"!WT"Xc!Q$q&k)$tAwqy{*;;A=q$GH I HqQq!tV}a#AadFQ22RC>? @ ?qaR1W~q!$!Wqa47m,,aRjQqT!V.DEE   sA2a4y!QtAwYtAaCy Ad1gId1g,=!>>4HI J Id1gc1"Q$iA5R$q'$q'/AadF*+DRM9BtAwJtAw	 !"D:* * * * sA2a4ya !Q'"Xd1Q3iBtAwJtAw$6 77DAB C BsA2a4y!DG)$a+"Xd1Q3iBtAwJtAw$6 77DAB C B q!tC1I~q!$tAwY!qtAv&wq!DG)DG2C'DD   d13iA1!teadl#3Q"====qacA~q!$q!tVAXr!Q$wqy1a4'(!A#t45 6 5qacA~q!d3q!tVAXq"QT'AqD.)*AaC67 8 7 d1Q3iA1b1a4i=!T"::::d13iA1q!teadl#3Q"====tACy!|QD11fq!tma1a4!Q$/0!A#t<= > =tAaCy!Q.Qi$q'!AqD1H-q$78 9 8CAJM!1a(GAq!	
AqD1qbQqSkAaCAaCL011adF 	1qAv&	(	()1q9 9 9
 d13iQS	!1a$7Ar!Q$wA~<N<=aC<G G G Gd13ik1a S!a%1"q&)9%:1%<a$FFFFqAaD57|DI%q!,2hYaR!VtAE{23Q=> ? > q1Q46{49$a+1u2q!aRTaeVAX$66aeAgFqH   c!faT*
a&:	q !T/+ + + +ACy!Q.!fs1vo
*A-q$78 9 8c!AC%j!QS1XIb!Ah$6q$8!T#BBBBs13xA-AhQqS"aRT(**AqD0!T:; < ;s1vd1g~q!d3"XAwQ'*45d1g=q$GH I Hq1Q46{3q6!1a$7"Xs3q6zC
O3bmCbHIAaDFm'' ( ' q!tCF{Aq$>Aw%"a/0A567?xzQVVL L Ls1vqy!Q.1v
"Q&Qq0!3Q=> ? >sA2a4yQA51q5zkJ&Q/!T:; < ;c!#h1!TAqD[/1d!;;;;s3qs8}a#tBqD!A#JA1-q$78 9 8c!#hqj!QD1Iq$#7777c!#hk!mQ"s1QT6!Q$;?';A'=q$&GGGGs13x{1a4A&tAaCE{]Qs1QT6!Q$;?33A55q$?@ A @
 c!#h1!QTAqD[/1d!;;;;s13x{Aq!1fq!tma1a4!Q$/0!T:; < ; s13xAC !Q'qSU1Wq!tquqj(1a4151*+<=>4HI J Is13xAC !Q'A1q!t#$q!tquqj'81a4151*;L&MN   s13xAC !Q'1q!tad"#adaeaZ&7!Q$!%!:K%LM   r!tCH}s13x'A=q!tVQTAadF1a4K'!Q$./D9: ; :aQBqD	k#ac("Aq)a.//012AQ
1B/DFGR.O O O OaQBqD	k#ac("Aq)aQiAQ
9J.K/0b$.8 8 8 8AE
Aq48NAudA::#a&3q6!AqD1H--222c!#h1#adAadFm"4T!QqS']"B1"Da!NNNN gaoq!$41q!t+<)<a(FFFFwq!#1t41q!taQTAqD[ 1112At<= > =wq!#1t4T41q!t$a$q!tad{*;&;a%??@!TJK L KqAC !Q'1q!tqtAv&&401 2 1qAC !Q'1q!tqtAv&&401 2 1q!tGAqsO#Q*q!tVQTAqD[AaDF++Q56 7 6
 wq!DAacN*+QDAAQTAqD[))*+DA1,==q$GH I Hgaoq!$4A1+=)=q$(GGGGwq!#1t4A1qq!teadl!3334a>? @ ?wq!#1t4T4A1%q4A1+='=&AABAtLM N Ma13A&1q!teadlad1f-E+Eq$*OOOOa13A&1!teadlad1f-E+Eq$*OOOOq!tGAqsO#Q*q!tVadUQT\QqT!V,,a67 8 7 wq!#1%ac
]BtAqD1a4K0011d;< = <wq!#1%!dAqD51a4<((!+,T1a4%!Q$,-??!TJK L Ks13x{Aq401WaAqD2ad71a4</!Q$678!TBC D C i!nS1X%q!,3r7AqD1H1Eq$0OOOOa!fa"21Q4"> >TJJJJq1Q3!A#&1-!a%!Q//"d;< = <q1Q3!A#&1-!a++CG3bS$?@ A @q13xAS!!A$Q3//"d;< = <c1Q4%'lAq!aQqTlR!XIt%DDDDsA2a4y1~q!$adA1u-sD9: ; : tACy1~q!$adArAv.q0adA1u-a/024d<< = < tACy{Aq40qSUAqD1QT6!Q$;&A-.489 : 9tACy1~q!$adArAv.q0adA1u-a/024d<< = < tACy{Aq401q51*1a!eaZ< 00!T:; < ;s13x!}aT2!!A$A2a4!844 1q!A#'23 44568:sDBB C B s1Q4y!Qc!A$iA.T:; < ;s13xz1a$/qSUAqD1QT6!Q$;&A-.489 : 9s13x!}a#adAr!tax02adAqsQw/12463>> ? > s13xz1a$/!teadlQTAadF1a4K/!Q$67DAB C BSQZ]1Q4'A.KAuaRC<<	
QqT1a(*QqT1bd1fqj1#ac!e*<Q>?QqT1ac!eai0RT!V<Q>?	@ EF	F F F
 	3qs1u:q=1q!T2Aqa&)	!Q2a46A:s1Q3q5z)!++	!A#a%!)SAa[ "	#	#$	# SXq[ac*Aq1KAuaA::	
QqS!Qq!tVad]QqSU*QT1233	!Q!Q$A!A%1,-.	/	!Q1q!tac!e#ad*+,	- 	
!Q1q!tac!e#ad*+,	- /7hjA	> > >
 q!tCAJ1%"Xad]3q!tAv;&tAaCy02QqS8"Xc!Q$q&k!$qs)+A-./0$88 9 8 Q$q !A$&1-z*1Q4A6A??z*1Q4A6::;qtQ**+-/C77 8 7 quQqTz1a 2Xj!1!A$1!=2wGGS$   
AqAv;AFY	7B	B1aB1sA2wqy   	RAB1	!a% 00000	AqAv;Fac16]I	FB	B1a	!	!	#B#ac(Qs1vX%)3r!t94QT9999	RAB	a	!nAyQ'777	1q5	)AE""	#$,HJ!4 4 4 
CFAF#aa[3t9d2C	DB	B1aBRaR!a%(1Q3.aR::::	RABBAE
9QU++il:Yq1u=MMO O O	AqAv;AFY	7B	B1aBAqAv;AFY?AAAA	)QAE16*+Y
77	)QAFAF+,i
88	)QAFAE*+Y
77	)QAE1q5)*I
66	
8B
 "a|A%%'1Q3q51S!W9Q;+>3r!t9Q+NNNN  	9aAq]I.15a8Ar!Qx=)<aC	DE	D
 
4!Q$q>	B	RAB	B1aB
a%A+			!B7N7tAaD1a&!1a6qtQ**QqT12 3 2tAaD1a&!1a6qD!!A$1--1QqT	9d:adA&1--.AaD1a&)1a011 2 1
 tAaAJ1%q!,T"1Q1:q!44q1a|C*QqQZ+Q2357C?? @ ? r$qtaV$$aD9qD!!A$1--1Q478 9 8q13x!#1d3qtQ!,,Q.QqT1ac*1,-- . - %	!a! 7!aA FG G GqtAaDy!Q-QqT!A$Y1-. / . q13xAt,qtQ!,,Q./ 0 /aAhy|#Q*q!a%y!T.BBBBqQx	!$aT:1q\1d#$ % $ 	AqsGS2qs_$Z!_4aey1~-/011	667	9<=bDaRL	IJ	I z#qb!%j/))AE*::AqA!erAvsA2a4y(q1u5r4@A B A 	C!HQJQBqD	!)A,.q0!Q	QBA		 jmQ"q2#tn444joq!$1rc4(8888j21%"rc4888j21%!bS$777z!}Z"--q!4SU^bS$'( ) (z!}QsA2a4y[(!Q>AYD!" # " 	3r7JqM*QrT"223QDI	SURZ1	rc4(	)*	)ad:ad##Q*quSQZ/?"d.KKKKad:ac!e$$a+!#sA2a46{0B10D13T0; ; ; ;ad:ac!e$$a+B3~===!!+Q215aRTBBB!!*Aq1!4QBqD	!Q$FFF!!*Aq1!4,Q15q!<> > > z!Q$At4Z1-q!45 6 5jA"Aq)c1"gai"d-CCCCjAE#Q*q3r7{RC.FFFF
1!Jq1u$5$9:AqAjmJq1u,==q!DaRQ3 !4)) * ) j1Qs1vX&1-#qbQi.2BRC1NNNNjAaAh'A.3r#a&y>!3CbS$2OOOOilAq!ac1d^333i1%#qbd)A+q$)????i!na#QB	1d';;;;i!A1%#bd)A+q$)????i!A1%!A#q$777yAa !Q6qS3r!t9Q;4() * )qtIae$$a+ad9QU#33Q:RCFG H G 	9QS>!A$1%	!AE(Aq	)#qbd)	3bS$?	@A	@ 	9QS>!A$1%	!A$1	%sD1	23	2 	9aRT?1Q4A&	!A$1	%	!AE(Aq	)#qbd)	3
468S$	@	@A	@ 	9aRT?1Q4A&,686hqk1a$?1a42;2.1gc!Q$"+1,--a/0!Q$"+x"~-./0123T;; < ; hqk1a$?!Q$"+x"~-!Q$"+x"~-.1gc!Q$"+1,--a/012345t== > =
 
#a&!CG)$qaRy#a&&9:	;B	!QU)a!er]+q5B-QAY/1 
2B 
 	!!a%y!T"a!er]B$=>A}b$'!QU)Q)=>@A 
" 
 	!"a|x''' 
" b!Qe,Q=== b!Q%+++b!QeT:b@@@ 
"	! 
"	!s!   g
ACi>h"Cji>
Cjj
Cjc                  |/  ^
^^^ [        S5      m[        SSS9u  pm[        SSS9u  p#pE[        S5      m[        S5      m
UU4S	 jnUU4S
 jnU
UUU4S jnU" [        U* T-  5      T
" T5      -  5      T" U* T-   5      :X  d   eU" U T
" TU-
  5      -  5      U T" T5      -  [        U* T-  5      -  :X  d   eU" [        T
" T5      TS5      5      TS-  * T" T5      -  :X  d   eU" [        T
" T5      TS5      5      TS-  T" T5      -  :X  d   eU" S5      S[	        T5      -  :X  d   eU" ST-  5      [        T5      :X  d   eU" STS-  -  5      T[        T5      -  :X  d   eU" STS-  -  5      TS-  [        T5      -  S-  :X  d   eU" STU-  -  5      TUS-
  -  [        T5      -  [        U5      -  :X  d   eU" X"T-   -  5      U[        U* T-  5      -  [        T5      -  :X  d   eU" TUT-   -  5      U* [        U* T-  5      -  [        T5      -  [	        T5      -   :X  d   eU" UT-  TU-   S-  -  5      X2* T-  [        U* T-  5      -  [        T5      -  [        U* T-  5      [        T5      -  -   -  :X  d   eU" UTU-   TU-   -  -  5      U[        UT-  5      [        UT-  5      -
  -  [        T* X#-   -  5      -  [        T5      -  X#-
  -  :X  d   eU" UT-  TU-   TU-   -  -  5      XB[        UT-  5      -  U[        UT-  5      -  -
  -  [        T* X#-   -  5      -  [        T5      -  X#-
  -  :X  d   eU" TUT-   S-  -  5      TU* T-  S-   -  [        U* T-  5      -  [        T5      -  S-  :X  d   eU" STUT-   S-  -  -  5      US-  * TS-  -  SU-  T-  -
  S[        UT-  5      -  -   S-
  [        U* T-  5      -  [        T5      -  SUS-  -  -  :X  d   eU" STUT-   U-  -  -  5      [        T5      [        XT-  5      -  X!-  [        U5      -  -  :X  d   eU" TU-
  U* -  5      TUS-
  -  [        UT-  5      -  [        T5      -  [        U5      -  :X  d   eU" UT-   S-  5      T[        U* T-  5      -  [        T5      -  :X  d   eU" UT-   S-  5      TS-  [        U* T-  5      -  [        T5      -  S-  :X  d   eU" TS-  TS-  S-   -  5      [        T5      * [        T5      -  [	        T5      -   :X  d   eU" SSTS-  S-   -  -
  5      [        T5      * [        T5      -  [	        T5      -   :X  d   eU" X"S-  TS-  -   -  5      [        UT-  5      [        T5      -  :X  d   eU" TTS-  US-  -   -  5      [        UT-  5      [        T5      -  :X  d   eU" X3S-  UT-   S-  -   -  5      [        U* T-  5      [        UT-  5      -  [        T5      -  :X  d   eU" UT-  US-  UT-   S-  -   -  5      X2* [        U* T-  5      -  [        UT-  5      -  U-  [        U* T-  5      [        UT-  5      -  -   -  [        T5      -  :X  d   eU" STS-  TS-  S-   -  -  5      T[        T5      -  [        T5      [        T5      -  -
  :X  d   eU" UT-  US-  TU-   S-  -  US-  -   -  5      XB* U-  [        UT-  U-  5      -  U[        UT-  U-  5      -  -   -  [        U* T-  5      -  [        T5      -  X5S-  -  -  :X  d   eU" UTS-  -  U-   US-  TS-  -   S-  -  5      SUS-  -  U-  [        UT-  5      -  US-  U-  U-
  UT-  [        UT-  5      -  [        UT-  5      -
  -  -   [        T5      -  SUS-  -  -  :X  d   eU" UTS-  US-  -
  -  5      U[        UT-  5      -  [        T5      -  U-  :X  d   eU" UTS-  US-  -
  -  5      U[        UT-  5      [        T5      -  SU-  -  [        U* T-  5      [        T5      -  SU-  -  -
  -  :X  d   eU" UT-  TS-  US-  -
  -  5      U[        UT-  5      -  [        T5      -  :X  d   eU" UTTU-   -  -  5      U[        T5      U-  [        U* T-  5      [        T5      -  U-  -
  -  :X  d   eU" TS-   TS-   TS-  ST-  -   S-   -  -  5      S[        ST-  5      -  S[        ST-  5      -  -
  [        T5      -  S-   [        ST-  5      -  [        T5      -  S-  :X  d   eU" SUTS-  -  UT-  -   U-   -  5      S[        U* T-  SU-  -  5      -  [        T[        SU-  U-  US-  -
  5      -  SU-  -  5      -  [        T5      -  [        SU-  U-  US-  -
  5      -  :X  d   eU" S[        T5      -  UT-  U-
  -  5      [        UT-  U-  5      [        T5      -  [        [        U5      [        T5      -  [        U5      -  5      -  [        U5      [        U5      -  -  :X  d   eU" S[        U T-  5      -  UT-  U-
  -  T-  5      SU-  [        T5      -  [        [        5      U-  -  U[        S5      S-  -  [        UT-  U-  5      -  [        [        U5      [        T5      -  [        U5      -  5      -  U[        S5      S-  -  -  -   [        T5      -  U[        U 5      -  -  :X  d   eU" S[        UT-  5      U-   -  5      U* [        US-  T-  U-  5      -  [!        U[        T5      -  [        U5      -  5      -  U-  S[        [        5      [        U5      -  [        T5      -  -  -   [        T5      -  :X  d   eU" UT-  U[        T5      -  U-   -  5      [        US-  T-  US-  -  5      * [!        U[        T5      -  U-  5      -  S-   [        T5      -  :X  d   eU" X#-
  [        T5      -  [        T5      [        U5      -   -  TU-
  -  5      [        U5      [        U5      -  [        UT-  5      -  [!        [        U5      [        T5      -  5      -  U[        UT-  5      -  [!        [        U5      [        T5      -  5      -  -   U[        UT-  5      -  -
  [        T5      -  :X  d   eU" S[        T5      -  [        UT-  5      U-   -  5      [        US-  T-  U-  5      [        T5      -  [!        U[        T5      -  [        U5      -  5      -  [        U5      -  :X  d   eU" US-  [        T5      U-   -  T[        S5      S-  -  -  5      SU-  [        T5      -  [        [        5      -  [        US-  T-  5      [!        U[        T5      -  5      -  -   S-
  [        T5      -  :X  d   eU" X#-
  [        U5      -  TU-
  -  [        T5      -  [        T5      [        U5      -   -  5      [        U5      [        UT-  5      -  [        [        U5      [        T5      -  5      -  [        U5      [        UT-  5      -  [!        [        U5      [        T5      -  5      -  -   [        U5      [        UT-  5      -  -
  [        T5      -  :X  d   eU" S[        T5      [        U5      -   S-  -  5      S[        U5      -  [        T5      -  [        [        5      -  SU-  T-  S-   [        [        U5      [        T5      -  5      S-
  -  [        UT-  5      -  -   S-   [        T5      -  :X  d   eU" S[        T5      [        U5      -   S-  -  T-  5      ST-  SU-  -
  [        UT-  5      -  [!        [        U5      [        T5      -  5      -  SU-  -   S[        T5      -  [        [        5      [        U5      -  -  -
  [        T5      -  :X  d   eU" S[        T5      U-   S-  -  [        T5      -  5      SU-  T-  [        US-  T-  5      -  [!        U[        T5      -  5      -  S[        T5      -  [        [        5      -  -   [        T5      -  :X  d   eU" S[        T5      U-   S-  -  5      U* T-  SUS-  -  T-  S-   -  [        US-  T-  5      -  [!        U[        T5      -  5      -  S[        T5      -  US-  T-  S-   -  [        [        5      -  -   [        T5      -  :X  d   eU" [        T5      [        T5      U-   S-  -  5      * S[        [        5      US-  -  T-  S[        [        5      -  [!        U[        T5      -  5      -  S[        US-  * T-  5      -  U[        T5      -  -  -   -  US-  * TS-  -  SUS-  -  T-  S-  -
  [        R"                  -
  -  [        US-  T-  5      -  S-  [        [        5      U-  [        T5      -  US-  T-  S-   -  S-  -   -  [        T5      -  [        US-  -  T-  -  -   R%                  5       n	U	S:X  d   eU" S[        T5      -  [        T5      U-   S-  -  5      * [        T5      [        T5      SUS-  -  T-  S-   [        [        5      U-  [        T5      -  [        US-  T-  5      -  [!        U[        T5      -  5      -  S-
  -  S-   -  [        [        5      U-  -  -  -   R%                  5       n	U	S:X  d   e['        U" S[        T5      U-   S-  -  5      5      SSUS-  -  T[        S5      S-  -  -  SUS-  -  T-  S-   -  S[        [        5      -  -  TSUS-  -  TS-  -  SUS-  -  T-  -   S-   -  [        US-  T-  5      -  [!        U[        T5      -  5      -  S-  -   -  [        T5      -  :X  d   eU" [        T5      U-
  T[        T5      U-   -  -  5      S[        US-  T-  5      -  [!        U[        T5      -  5      -  S-
  [        T5      -  :X  d   eU" [        T5      U-
  S-  T[        T5      U-   S-  -  -  5      SSUS-  -  T-  [        US-  T-  5      -  [!        U[        T5      -  5      -  -   S[        [        5      -  U-  [        T5      -  -
  [        T5      -  :X  d   eU" [        T5      U-
  S-  T[        T5      U-   S-  -  -  5      [        T5      SSUS-  -  TS-  -  SUS-  -  T-  -   S-   -  [        US-  T-  5      -  [!        U[        T5      -  5      -  S[        [        5      -  U-  [        T5      -  SUS-  -  T-  S-   -  -
  S-
  -  :X  d   eU" [        TU-   5      TU-   -  5      [        T5      [        U* T-  5      [        T5      -  [        [        5      -  [        X#-
  5      [        U* T-  5      -  [        [        X#-
  5      [        T5      -  5      -  -   -  :X  d   eU" S[        TU-   5      -  TU-   -  5      [        T5      S[        X2-
  5      -  [        U* T-  5      -  [        [        X2-
  5      [        T5      -  5      -  -  :X  d   eU" S[        TS-  US-  -   5      -  5      [        T5      [)        SUT-  5      -  :X  d   eU" S[        TS-  US-  -
  5      -  5      [        T5      [+        SUT-  5      -  :X  d   eU" TS-  [        ST-  5      -  S[        T5      -  -   S-
  [        ST-  5      -  TTS-  S-   -  -  5      * [        T5      [        T5      -  -   S[        TS-
  5      -  [        TS-
  5      -  -   S[        TS-
  5      -  [        TS-
  5      -  -
  S[        TS-
  5      -  -
  S[        TS-
  5      -  -   R%                  5       S:X  d   eg )Nr@   zk, n, tTrealz
a, b, c, drB   rF   rG   c                    > [        U TT5      $ Nr   rH   r@   rq   s    r   rm   +test_inverse_laplace_transform.<locals>.ILT  s    (Aq11    c                    > [        U TTSS9$ )NTr   r   r   s    r   ILTS,test_inverse_laplace_transform.<locals>.ILTS  s    (Aq4@@r   c                 6   > [        [        U TT5      TT05      $ r   )r   r   )rH   rG   rF   r@   rq   s    r   ILTF,test_inverse_laplace_transform.<locals>.ILTF  s"    %%aA.A8 	8r   rL   rO   rK   rS   rU   rc   rX   rZ   rP   r`   rf   rV   r      )r   r
   r   r   r,   r-   r*   r)   r'   r&   r    r   r#   r2   r   r   r3   Halfr   r   r9   r8   )knrn   ro   rp   drm   r   r   rs   rG   rF   r@   rq   s             @@@@r   test_inverse_laplace_transformr     s9   Aid+GA!5JA!AA2A8 8 QBqD	!A$1aR!V9,,,!AaC&>QqtVC1I----QqT1a !adU1Q4Z///QqT1a !QT!A$Y...
 q6Qz!}_$$$qs8y|###qAv;!IaL.(((qAv;!Q$y|+B....qAv;!a!e*Yq\1%(::::qa%y>QsA2a4y[15555q!a%y>aRQBqD	\)A,6AFFFF!QqS1Hr!tC1I~il*S!AYy|-CCDE F EQqS1Q3K!s1Q3x#ac("#CAE
O3IaL@!%HI J I1qsQqSk"#QqSzAc!A#hJ&'QBJ7	!DaeLM N M1q51*QBqD1Hc1"Q$i!7	!!DQ!FFFF1a!eaZ< !	
Aad
QqSUQs1Q3xZ'!+S!AY&77@|&DEFq!tV&M M M Mq!QUQJ, !ZQ3''eAh79 9 9!r{q1q5z#ac(29Q<?aHHHHA}3r!t9Yq\!9999A}Ac1"Q$i	!!<R!????q!tQTAXCF79Q<#7*Q-#GGGGq1adQh< SVGIaL$8:a=$HHHHqQ$A+3qs8IaL#8888q!Q$A+3qs8IaL#8888qQ$!a%!#$%aRT3qs8);IaL)HHHH!QTQUQJ&'(r#qbd)|C!H$Q&aRT3qs8);;<Yq\IJ K Jq!Q$1q/"#q1~Ay|8K'KKKK1adAaC!8mAqD()*r!tC!AJ3qs1u:-.sA2a4y81EqAvNO P O1a4!adQTkA--.	!Q$qQqSQT!VaZ!A#c!A#h,*-ac(+3 4 	45>q\3BCDQT63K K K K 1a419!D1I+il":1"<<<<1a419s1Q3x	!$ac*S!AYy|-CQqS-IIJK L K!QT!Q$Y Ad1Q3iK	!$<<<<1ac7y|A~QBqD	)A, 6q 889: ; : !a%1q51a4!A#:?345AaCj1S1X:%s1v-13r!t9<Yq\I"LM N M 1QT6!A#:a< !c1"Q$!*oc!D1QA$6"6!"<==aLac!eadl+,, - , 	Ad1gIqs1uAaCE
9Q<DGDGODG$; <<d1gd1goN	OP	O 	Ad1Q3iK1Q!"	Ad1gtBxz	"	
QqT!VS1QZ	DGDGODG$; <	<Q1a[	H
I!	Qi	!	!"	!
 	AtAaCy{O!C1QqM!1$qaya7H2I!I!!K!"DHT!W$4T!W$<!=">?H|!L 	LM	L 	AaC471
ad1fQTk
	4$q'	!,	,q	0)A,>	?@	? d1gtAwtAw/156!WT!W_S1X%d4747?&;;s1Q3xZT!WT!W_--.01#ac(
;<EaLIIJI $q'	4!9Q;'(1QqM)A,&tAd1gId1g,='>>tAwFGHG 1d1gai QqT!V,-qSa[b!C1QKQtAwY$??!CaL d1gqs#DG+T!WT!W_=>!WS1Xc$q'$q'/22!WS1Xd4747?334!WS1X(|,,-,
 4747?Q&&'QZQR(d1fqj3tAwtAw/ !!A#h'')*+,5aL99:9
 4747?Q&&q()cAaCiQqS!$tAwtAw"77!A#=tAwYRa()*+4Q<8898 4719q. a()T!VC1QKQtAwY/tAwYtBx !*1../. 4719q. !RT1QT6!8a< QT!V,T!DG)_<tAwY1Q
#DH,-.7l;;<; d1gtAwqy1n$
%%	48AqD=?BtBxKQtAwY7c1a4%'lNAd1gI67 8TE!Q$J1a4!#aff,.03AqDF<<=> 8A:d1gq!tAvz*1,- 	. 	!		 Agai		!	!
 #+(*  F
aQiaQ&
''!d1g!Q$q1 $R
47 23q!tAv; > $QtAwY!023!4 578 9 : Bxz# 	$	$ &.XZ  FQQ	A~-./r!Q$wq1Q46{"AadF1HqL11T"X:>!AqD&A+1a4	)A-.s1a46{:Ad1gI !! ""+A,//0/
 aQQ	]+,s1a46{]4$q'	?*1,il:;<; aQ4719q. 012AadF1HSAa[(aQi88$r(
1T!W$%&/l73 343 aQ4719q. 012il1QT6!Q$;qAvax')*3q!tAv;6QtAwY  !$r(
1T!W 4a1fQhqj ABBCD7E EFE QqS	1Q3 IaLQBqD	$q'!$r(*QS	#qbd)#CQS	$q'(9$::;%< <=< $qs)QqS!"il$qs)C1I%c$qs)DG*;&<<'> >?> $q!tAqDy/!"il1Q3'! !"! $q!tAqDy/!"il1Q3'! !"! q!tC!H}qQx'!+SAY61a4!8E
FFAy|	AE
l9QU+;;	<	#a!e*Yq1u%%	&())AE*:(:	; 	
)AE
	 %HJ!	, , ,r   c                     SSK Jn   [        n[        SSS9u  p#pE[        SSS9u  pg[        S5      u  p[	        S	5      n
[	        S
5      nS nU" U" [
        5      [
        U5      n[        XU05      U
" U5      :X  d   eU" [        U* [
        -  5      [
        -  [
        U5      [        U* U-   5      :X  d   eU" [        U* [
        -  5      U[
        -   -  [
        U5      [        U* U* U-   -  5      [        U* U-   5      -  :X  d   eU" U[
        -   US-  U[
        -   S-  -   -  [
        U5      [        U* U-  5      [        X(-  5      -  [        U5      -  :X  d   eU" [        U* [
        -  5      [
        U-  -  [
        U5      U* U-   US-
  -  [        U* U-   5      -  [        U5      -  :X  d   eU" [        U* [
        -  5      [        [
        S-  S-   5      -  [
        U5      [        U* U-   5      [        SX(-
  5      -  :X  d   eU" S[
        [        [
        S-   5      -  -  [
        U5      [        U5      [        [        U5      5      -  :X  d   eU" [        U* [
        -  5      [
        U-  -  [
        U5      X-
  US-
  -  [        X-
  5      -  [        U5      -  :X  d   eU" [        U* [
        -  5      [        S[
        S-  -   5      -  [
        U5      [        X-
  5      [        SX(-
  5      -  :X  d   e[        U" X#-  [
        [        [
        S-  US-  -
  5      -   U* -  -  [        [
        S-  US-  -
  5      -  [
        U5      R                  [        5      5      [        U5      [!        X2U-  5      -  :X  d   eU" X#-  [
        [        [
        S-  US-  -   5      -   U* -  -  [        [
        S-  US-  -   5      -  [
        USS9R                  [        5      [        U5      [        X2U-  5      -  :X  d   eU" S[
        [        [
        S-   5      -  -  [
        U5      [        U5      [        [        U5      5      -  :X  d   eU" [
        [#        S5      -  [%        SS/SS//5      -
  R'                  5       [
        U5      [%        [        U5      [        U5      -  S/S[        SU-  5      [        U5      -  //5      :X  d   eU" [
        S-  U
" [
        5      -  [
        U5      [)        [+        U
" [
        5      [
        US 5      US45      :X  d   eU" [        [
        5      [
        U5      [+        [        [
        5      [
        US 5      :X  d   eU" SS[
        -  -   S[
        S-  -  -   S[
        S-  -  -   [
        U5      U " U5      SU " US5      -  -   SU " US5      -  -   SU " US5      -  -   :X  d   eU" S[        S[
        -  5      -  S[        S[
        -  5      -  -
  [
        U5      S[+        [        S[
        -  5      [
        US 5      -  SU " US-
  5      -  -
  :X  d   e[        [-        S5      S-  5      nU" U[        S[
        -  5      -  [
        U5      X " US-
  5      -  :X  d   eU" [        S[
        -  5      [
        U5      [+        [        S[
        -  5      [
        US 5      :X  d   e[/        SSS9nU" [        U[
        -  5      [
        U5      [+        [        U[
        -  5      [
        US 5      :X  d   eU" [
        S-  US-  [
        S-  -   -  [
        U5      U* [-        X(-  5      -  [        U5      -  U " U5      -   :X  d   eU" [
        S-  U
" [
        5      SUS-  [
        S-  -   -  -   -  [
        U5      U* [-        X(-  5      -  [        U5      -  U " U5      -   [)        [+        U
" [
        5      [
        US 5      US45      -   :X  d   eU" [        U[
        -  5      [
        USS9[+        [        U[
        -  5      [
        US 5      S4:X  d   e[/        SSS9[/        SSSS94 HN  n	U" [        U	[
        -  5      [
        USS9n
[1        U
[2        5      (       a  U
S   OU
n
U
R4                  U :w  a  MN   e   g )Nr   )r,   za b c dTrB   zn, rr   zt zrF   rG   c                 P    [        [        U 5      5      R                  [        5      $ r   )r   r   rewriter'   )exprs    r   simp_hyp4test_inverse_laplace_transform_old.<locals>.simp_hyp  s    Jt,-55c::r   rK   rS   r   r^   rL   rU   irW   r\   rFrM   z)extended_real)	imaginaryzero)'sympy.functions.special.delta_functionsr,   r   r   r
   r@   r   r   r-   r&   r*   r#   r9   r2   r   r   r8   r?   r>   invr   r   r'   r   
isinstancetuplefunc)r,   rm   rn   ro   rp   r   r   r   rq   r   rF   rG   r   rt   s                 r   "test_inverse_laplace_transform_oldr     s   B
#CT2JA!6%DA5>DAAA; 	AaD!QA!!V,!444sA2a4y{Aq!YrAv%6666sA2a4y!a% !Q'3rA26{+;Iqb1f<M+MMMMQAQ
*+Q21Ic!#hy|+, - ,QBqD	!Q$1%R!Vq1uiQ//a89 : 9QBqD	$q!tax.(!Q/qb1fga//0 1 0q!DQK- !Q'9Q<DG+DDDD QBqD	!Q$1%Ua!eYqu--eAh67 8 7QBqD	$q1a4x.(!Q/aeWQ../ 0 / 	QT1tAqD1a4K00QB77QTAqD[8II"73<	)!WQ!_$	%&	% 	AD!d1a4!Q$;''A2..tAqD1a4K/@@q4	!!(!WQ!_$	%&	% q!DQK- !Q'9Q<DG+DDDDSVfq!fq!f%566;;=q!DSVIaL(!,q#ac(9Q<2G.HIJK L K 2ad
Aq!.qtQ4@1b'JK L Ks1vq! 7A1d KKKK 	A!Ga1fqAv%q!,
1	*Q
1z!Q//1023Jq!4D2D1E 	EF	E#ac(
Qs2a4y[(!Q/%c!A#h1d;;jQ   !   	CF1HAqRT{Aq!Qz!a%'8%8888s1Q3xA"9#ac(Aq$"OOOOsAs1Q3xA"9#ac(Aq$"OOOOq!tQTAqD[!1a(	
3qs8IaL :a=02 2 2q!tQqTAq!tad{O+,a3	
3qs8IaL :a=0*1Q4At<q!fE	FG G G AaC!Q.$S1Xq!T:DAB C B S.SDu57AaC!Q.q%((AaDavv###	7r   c                  $   [        S5      n [        S5      n[        SSS9n[        [        U 5      U [        5      [	        S[        S-  -   5      * S-  [        -  SS4:X  d   e[        [        X5      U [        SS	9[        [        [        [        [        -  5      -  SU5      S[        U5      [        R                  :  4:X  d   e[        [        SU 5      U [        SS	9[	        [        S-   5      [        -  SS4:X  d   e[        [        SU 5      U [        SS	9[        [	        [        S-   5      -
  [        S-  -  SS4:X  d   e[        [	        S[        S-  -   5      * S-  [        -  [        U5      R                  5       [        U5      [        U5      -  :X  d   e[        [	        [        S-   5      [        -  [        U SS	9R!                  [
        5      [        U 5      [#        U 5      -  :X  d   e[        [        [	        [        S-   5      -
  [        S-  -  [        U SS	9R!                  [
        5      R                  5       [        SU 5      [        U 5      -  R!                  [$        5      R!                  [
        5      R                  5       :X  d   eg )
Nrs   rn   uT)polarrS   rK   r   r   )r   r   r5   r@   r   r6   r/   r   r   r   r   r   rj   r   rk   r-   r   r7   r4   )rs   rn   r   s      r   test_expintr     s5   sAsAs$A RUAq)s1q!t8}nQ.>q.@!T-JJJJfQlAq4@a	!B$'A.2a5166>BC D CfQlAq4@QZ\1d#$ % $fQlAq4@#a!e*nad"At,- . -%s1q!t8}nQ&6q&8!Q?FFHaLA    	"#a!e*Q,1+/	118!RU		 	"QU^QT!1a	"76?668	1il	"++B/77?FFH	IJ	Ir   N)Zsympy.integrals.laplacer   r   r   r   r   r   r	   sympy.core.functionr
   r   
sympy.corer   r   r   r   sympy.core.exprtoolsr   sympy.core.numbersr   r   r   sympy.core.relationalr   sympy.core.singletonr   sympy.core.symbolr   r   sympy.simplify.simplifyr   $sympy.functions.elementary.complexesr   r   &sympy.functions.elementary.exponentialr   r   r   %sympy.functions.elementary.hyperbolicr   r    r!   r"   (sympy.functions.elementary.miscellaneousr#   $sympy.functions.elementary.piecewiser$   (sympy.functions.elementary.trigonometricr%   r&   r'   sympy.logic.boolalgr(   'sympy.functions.special.gamma_functionsr)   r*   r+   r   r,   r-   -sympy.functions.special.singularity_functionsr.   &sympy.functions.special.zeta_functionsr/   'sympy.functions.special.error_functionsr0   r1   r2   r3   r4   r5   r6   r7   sympy.functions.special.besselr8   r9   r:   r;   sympy.testing.pytestr<   r=   sympy.matricesr>   r?   	sympy.abcr@   r   r   r   r    r   r   <module>r      s     
 5 9 9 - ( ( $ " - , 8 F F I I 9 : C C ## # I M ;7 7 7 M M = &  EA EAP s, s,l H$ H$V J Jr   