<s> """<STR_LIT>""" <EOL> from __future__ import print_function , division <EOL> from sympy import ( Add , expand , Eq , Expr , Mul , Piecewise , Pow , sqrt , Sum , <EOL> symbols , sympify , Wild ) <EOL> from sympy . core . compatibility import range <EOL> from sympy . printing . pretty . stringpict import prettyForm , stringPict <EOL> from sympy . functions . special . tensor_functions import KroneckerDelta <EOL> from sympy . physics . wigner import clebsch_gordan , wigner_3j , wigner_6j , wigner_9j <EOL> __all__ = [ <EOL> '<STR_LIT>' , <EOL> '<STR_LIT>' , <EOL> '<STR_LIT>' , <EOL> '<STR_LIT>' , <EOL> '<STR_LIT>' <EOL> ] <EOL> class Wigner3j ( Expr ) : <EOL> """<STR_LIT>""" <EOL> is_commutative = True <EOL> def __new__ ( cls , j1 , m1 , j2 , m2 , j3 , m3 ) : <EOL> args = map ( sympify , ( j1 , m1 , j2 , m2 , j3 , m3 ) ) <EOL> return Expr . __new__ ( cls , * args ) <EOL> @ property <EOL> def j1 ( self ) : <EOL> return self . args [ <NUM_LIT:0> ] <EOL> @ property <EOL> def m1 ( self ) : <EOL> return self . args [ <NUM_LIT:1> ] <EOL> @ property <EOL> def j2 ( self ) : <EOL> return self . args [ <NUM_LIT:2> ] <EOL> @ property <EOL> def m2 ( self ) : <EOL> return self . args [ <NUM_LIT:3> ] <EOL> @ property <EOL> def j3 ( self ) : <EOL> return self . args [ <NUM_LIT:4> ] <EOL> @ property <EOL> def m3 ( self ) : <EOL> return self . args [ <NUM_LIT:5> ] <EOL> @ property <EOL> def is_symbolic ( self ) : <EOL> return not all ( [ arg . is_number for arg in self . args ] ) <EOL> def _pretty ( self , printer , * args ) : <EOL> m = ( ( printer . _print ( self . j1 ) , printer . _print ( self . m1 ) ) , <EOL> ( printer . _print ( self . j2 ) , printer . _print ( self . m2 ) ) , <EOL> ( printer . _print ( self . j3 ) , printer . _print ( self . m3 ) ) ) <EOL> hsep = <NUM_LIT:2> <EOL> vsep = <NUM_LIT:1> <EOL> maxw = [ - <NUM_LIT:1> ] * <NUM_LIT:3> <EOL> for j in range ( <NUM_LIT:3> ) : <EOL> maxw [ j ] = max ( [ m [ j ] [ i ] . width ( ) for i in range ( <NUM_LIT:2> ) ] ) <EOL> D = None <EOL> for i in range ( <NUM_LIT:2> ) : <EOL> D_row = None <EOL> for j in range ( <NUM_LIT:3> ) : <EOL> s = m [ j ] [ i ] <EOL> wdelta = maxw [ j ] - s . width ( ) <EOL> wleft = wdelta // <NUM_LIT:2> <EOL> wright = wdelta - wleft <EOL> s = prettyForm ( * s . right ( '<STR_LIT:U+0020>' * wright ) ) <EOL> s = prettyForm ( * s . left ( '<STR_LIT:U+0020>' * wleft ) ) <EOL> if D_row is None : <EOL> D_row = s <EOL> continue <EOL> D_row = prettyForm ( * D_row . right ( '<STR_LIT:U+0020>' * hsep ) ) <EOL> D_row = prettyForm ( * D_row . right ( s ) ) <EOL> if D is None : <EOL> D = D_row <EOL> continue <EOL> for _ in range ( vsep ) : <EOL> D = prettyForm ( * D . below ( '<STR_LIT:U+0020>' ) ) <EOL> D = prettyForm ( * D . below ( D_row ) ) <EOL> D = prettyForm ( * D . parens ( ) ) <EOL> return D <EOL> def _latex ( self , printer , * args ) : <EOL> label = map ( printer . _print , ( self . j1 , self . j2 , self . j3 , <EOL> self . m1 , self . m2 , self . m3 ) ) <EOL> return r'<STR_LIT>' % tuple ( label ) <EOL> def doit ( self , ** hints ) : <EOL> if self . is_symbolic : <EOL> raise ValueError ( "<STR_LIT>" ) <EOL> return wigner_3j ( self . j1 , self . j2 , self . j3 , self . m1 , self . m2 , self . m3 ) <EOL> class CG ( Wigner3j ) : <EOL> """<STR_LIT>""" <EOL> def doit ( self , ** hints ) : <EOL> if self . is_symbolic : <EOL> raise ValueError ( "<STR_LIT>" ) <EOL> return clebsch_gordan ( self . j1 , self . j2 , self . j3 , self . m1 , self . m2 , self . m3 ) <EOL> def _pretty ( self , printer , * args ) : <EOL> bot = printer . _print_seq ( <EOL> ( self . j1 , self . m1 , self . j2 , self . m2 ) , delimiter = '<STR_LIT:U+002C>' ) <EOL> top = printer . _print_seq ( ( self . j3 , self . m3 ) , delimiter = '<STR_LIT:U+002C>' ) <EOL> pad = max ( top . width ( ) , bot . width ( ) ) <EOL> bot = prettyForm ( * bot . left ( '<STR_LIT:U+0020>' ) ) <EOL> top = prettyForm ( * top . left ( '<STR_LIT:U+0020>' ) ) <EOL> if not pad == bot . width ( ) : <EOL> bot = prettyForm ( * bot . right ( '<STR_LIT:U+0020>' * ( pad - bot . width ( ) ) ) ) <EOL> if not pad == top . width ( ) : <EOL> top = prettyForm ( * top . right ( '<STR_LIT:U+0020>' * ( pad - top . width ( ) ) ) ) <EOL> s = stringPict ( '<STR_LIT:C>' + '<STR_LIT:U+0020>' * pad ) <EOL> s = prettyForm ( * s . below ( bot ) ) <EOL> s = prettyForm ( * s . above ( top ) ) <EOL> return s <EOL> def _latex ( self , printer , * args ) : <EOL> label = map ( printer . _print , ( self . j3 , self . m3 , self . j1 , <EOL> self . m1 , self . j2 , self . m2 ) ) <EOL> return r'<STR_LIT>' % tuple ( label ) <EOL> class Wigner6j ( Expr ) : <EOL> """<STR_LIT>""" <EOL> def __new__ ( cls , j1 , j2 , j12 , j3 , j , j23 ) : <EOL> args = map ( sympify , ( j1 , j2 , j12 , j3 , j , j23 ) ) <EOL> return Expr . __new__ ( cls , * args ) <EOL> @ property <EOL> def j1 ( self ) : <EOL> return self . args [ <NUM_LIT:0> ] <EOL> @ property <EOL> def j2 ( self ) : <EOL> return self . args [ <NUM_LIT:1> ] <EOL> @ property <EOL> def j12 ( self ) : <EOL> return self . args [ <NUM_LIT:2> ] <EOL> @ property <EOL> def j3 ( self ) : <EOL> return self . args [ <NUM_LIT:3> ] <EOL> @ property <EOL> def j ( self ) : <EOL> return self . args [ <NUM_LIT:4> ] <EOL> @ property <EOL> def j23 ( self ) : <EOL> return self . args [ <NUM_LIT:5> ] <EOL> @ property <EOL> def is_symbolic ( self ) : <EOL> return not all ( [ arg . is_number for arg in self . args ] ) <EOL> def _pretty ( self , printer , * args ) : <EOL> m = ( ( printer . _print ( self . j1 ) , printer . _print ( self . j3 ) ) , <EOL> ( printer . _print ( self . j2 ) , printer . _print ( self . j ) ) , <EOL> ( printer . _print ( self . j12 ) , printer . _print ( self . j23 ) ) ) <EOL> hsep = <NUM_LIT:2> <EOL> vsep = <NUM_LIT:1> <EOL> maxw = [ - <NUM_LIT:1> ] * <NUM_LIT:3> <EOL> for j in range ( <NUM_LIT:3> ) : <EOL> maxw [ j ] = max ( [ m [ j ] [ i ] . width ( ) for i in range ( <NUM_LIT:2> ) ] ) <EOL> D = None <EOL> for i in range ( <NUM_LIT:2> ) : <EOL> D_row = None <EOL> for j in range ( <NUM_LIT:3> ) : <EOL> s = m [ j ] [ i ] <EOL> wdelta = maxw [ j ] - s . width ( ) <EOL> wleft = wdelta // <NUM_LIT:2> <EOL> wright = wdelta - wleft <EOL> s = prettyForm ( * s . right ( '<STR_LIT:U+0020>' * wright ) ) <EOL> s = prettyForm ( * s . left ( '<STR_LIT:U+0020>' * wleft ) ) <EOL> if D_row is None : <EOL> D_row = s <EOL> continue <EOL> D_row = prettyForm ( * D_row . right ( '<STR_LIT:U+0020>' * hsep ) ) <EOL> D_row = prettyForm ( * D_row . right ( s ) ) <EOL> if D is None : <EOL> D = D_row <EOL> continue <EOL> for _ in range ( vsep ) : <EOL> D = prettyForm ( * D . below ( '<STR_LIT:U+0020>' ) ) <EOL> D = prettyForm ( * D . below ( D_row ) ) <EOL> D = prettyForm ( * D . parens ( left = '<STR_LIT:{>' , right = '<STR_LIT:}>' ) ) <EOL> return D <EOL> def _latex ( self , printer , * args ) : <EOL> label = map ( printer . _print , ( self . j1 , self . j2 , self . j12 , <EOL> self . j3 , self . j , self . j23 ) ) <EOL> return r'<STR_LIT>' % tuple ( label ) <EOL> def doit ( self , ** hints ) : <EOL> if self . is_symbolic : <EOL> raise ValueError ( "<STR_LIT>" ) <EOL> return wigner_6j ( self . j1 , self . j2 , self . j12 , self . j3 , self . j , self . j23 ) <EOL> class Wigner9j ( Expr ) : <EOL> """<STR_LIT>""" <EOL> def __new__ ( cls , j1 , j2 , j12 , j3 , j4 , j34 , j13 , j24 , j ) : <EOL> args = map ( sympify , ( j1 , j2 , j12 , j3 , j4 , j34 , j13 , j24 , j ) ) <EOL> return Expr . __new__ ( cls , * args ) <EOL> @ property <EOL> def j1 ( self ) : <EOL> return self . args [ <NUM_LIT:0> ] <EOL> @ property <EOL> def j2 ( self ) : <EOL> return self . args [ <NUM_LIT:1> ] <EOL> @ property <EOL> def j12 ( self ) : <EOL> return self . args [ <NUM_LIT:2> ] <EOL> @ property <EOL> def j3 ( self ) : <EOL> return self . args [ <NUM_LIT:3> ] <EOL> @ property <EOL> def j4 ( self ) : <EOL> return self . args [ <NUM_LIT:4> ] <EOL> @ property <EOL> def j34 ( self ) : <EOL> return self . args [ <NUM_LIT:5> ] <EOL> @ property <EOL> def j13 ( self ) : <EOL> return self . args [ <NUM_LIT:6> ] <EOL> @ property <EOL> def j24 ( self ) : <EOL> return self . args [ <NUM_LIT:7> ] <EOL> @ property <EOL> def j ( self ) : <EOL> return self . args [ <NUM_LIT:8> ] <EOL> @ property <EOL> def is_symbolic ( self ) : <EOL> return not all ( [ arg . is_number for arg in self . args ] ) <EOL> def _pretty ( self , printer , * args ) : <EOL> m = ( <EOL> ( printer . _print ( <EOL> self . j1 ) , printer . _print ( self . j3 ) , printer . _print ( self . j13 ) ) , <EOL> ( printer . _print ( <EOL> self . j2 ) , printer . _print ( self . j4 ) , printer . _print ( self . j24 ) ) , <EOL> ( printer . _print ( self . j12 ) , printer . _print ( self . j34 ) , printer . _print ( self . j ) ) ) <EOL> hsep = <NUM_LIT:2> <EOL> vsep = <NUM_LIT:1> <EOL> maxw = [ - <NUM_LIT:1> ] * <NUM_LIT:3> <EOL> for j in range ( <NUM_LIT:3> ) : <EOL> maxw [ j ] = max ( [ m [ j ] [ i ] . width ( ) for i in range ( <NUM_LIT:3> ) ] ) <EOL> D = None <EOL> for i in range ( <NUM_LIT:3> ) : <EOL> D_row = None <EOL> for j in range ( <NUM_LIT:3> ) : <EOL> s = m [ j ] [ i ] <EOL> wdelta = maxw [ j ] - s . width ( ) <EOL> wleft = wdelta // <NUM_LIT:2> <EOL> wright = wdelta - wleft <EOL> s = prettyForm ( * s . right ( '<STR_LIT:U+0020>' * wright ) ) <EOL> s = prettyForm ( * s . left ( '<STR_LIT:U+0020>' * wleft ) ) <EOL> if D_row is None : <EOL> D_row = s <EOL> continue <EOL> D_row = prettyForm ( * D_row . right ( '<STR_LIT:U+0020>' * hsep ) ) <EOL> D_row = prettyForm ( * D_row . right ( s ) ) <EOL> if D is None : <EOL> D = D_row <EOL> continue <EOL> for _ in range ( vsep ) : <EOL> D = prettyForm ( * D . below ( '<STR_LIT:U+0020>' ) ) <EOL> D = prettyForm ( * D . below ( D_row ) ) <EOL> D = prettyForm ( * D . parens ( left = '<STR_LIT:{>' , right = '<STR_LIT:}>' ) ) <EOL> return D <EOL> def _latex ( self , printer , * args ) : <EOL> label = map ( printer . _print , ( self . j1 , self . j2 , self . j12 , self . j3 , <EOL> self . j4 , self . j34 , self . j13 , self . j24 , self . j ) ) <EOL> return r'<STR_LIT>' % tuple ( label ) <EOL> def doit ( self , ** hints ) : <EOL> if self . is_symbolic : <EOL> raise ValueError ( "<STR_LIT>" ) <EOL> return wigner_9j ( self . j1 , self . j2 , self . j12 , self . j3 , self . j4 , self . j34 , self . j13 , self . j24 , self . j ) <EOL> def cg_simp ( e ) : <EOL> """<STR_LIT>""" <EOL> if isinstance ( e , Add ) : <EOL> return _cg_simp_add ( e ) <EOL> elif isinstance ( e , Sum ) : <EOL> return _cg_simp_sum ( e ) <EOL> elif isinstance ( e , Mul ) : <EOL> return Mul ( * [ cg_simp ( arg ) for arg in e . args ] ) <EOL> elif isinstance ( e , Pow ) : <EOL> return Pow ( cg_simp ( e . base ) , e . exp ) <EOL> else : <EOL> return e <EOL> def _cg_simp_add ( e ) : <EOL> """<STR_LIT>""" <EOL> cg_part = [ ] <EOL> other_part = [ ] <EOL> e = expand ( e ) <EOL> for arg in e . args : <EOL> if arg . has ( CG ) : <EOL> if isinstance ( arg , Sum ) : <EOL> other_part . append ( _cg_simp_sum ( arg ) ) <EOL> elif isinstance ( arg , Mul ) : <EOL> terms = <NUM_LIT:1> <EOL> for term in arg . args : <EOL> if isinstance ( term , Sum ) : <EOL> terms *= _cg_simp_sum ( term ) <EOL> else : <EOL> terms *= term <EOL> if terms . has ( CG ) : <EOL> cg_part . append ( terms ) <EOL> else : <EOL> other_part . append ( terms ) <EOL> else : <EOL> cg_part . append ( arg ) <EOL> else : <EOL> other_part . append ( arg ) <EOL> cg_part , other = _check_varsh_871_1 ( cg_part ) <EOL> other_part . append ( other ) <EOL> cg_part , other = _check_varsh_871_2 ( cg_part ) <EOL> other_part . append ( other ) <EOL> cg_part , other = _check_varsh_872_9 ( cg_part ) <EOL> other_part . append ( other ) <EOL> return Add ( * cg_part ) + Add ( * other_part ) <EOL> def _check_varsh_871_1 ( term_list ) : <EOL> a , alpha , b , lt = map ( Wild , ( '<STR_LIT:a>' , '<STR_LIT>' , '<STR_LIT:b>' , '<STR_LIT>' ) ) <EOL> expr = lt * CG ( a , alpha , b , <NUM_LIT:0> , a , alpha ) <EOL> simp = ( <NUM_LIT:2> * a + <NUM_LIT:1> ) * KroneckerDelta ( b , <NUM_LIT:0> ) <EOL> sign = lt / abs ( lt ) <EOL> build_expr = <NUM_LIT:2> * a + <NUM_LIT:1> <EOL> index_expr = a + alpha <EOL> return _check_cg_simp ( expr , simp , sign , lt , term_list , ( a , alpha , b , lt ) , ( a , b ) , build_expr , index_expr ) <EOL> def _check_varsh_871_2 ( term_list ) : <EOL> a , alpha , c , lt = map ( Wild , ( '<STR_LIT:a>' , '<STR_LIT>' , '<STR_LIT:c>' , '<STR_LIT>' ) ) <EOL> expr = lt * CG ( a , alpha , a , - alpha , c , <NUM_LIT:0> ) <EOL> simp = sqrt ( <NUM_LIT:2> * a + <NUM_LIT:1> ) * KroneckerDelta ( c , <NUM_LIT:0> ) <EOL> sign = ( - <NUM_LIT:1> ) ** ( a - alpha ) * lt / abs ( lt ) <EOL> build_expr = <NUM_LIT:2> * a + <NUM_LIT:1> <EOL> index_expr = a + alpha <EOL> return _check_cg_simp ( expr , simp , sign , lt , term_list , ( a , alpha , c , lt ) , ( a , c ) , build_expr , index_expr ) <EOL> def _check_varsh_872_9 ( term_list ) : <EOL> a , alpha , alphap , b , beta , betap , c , gamma , lt = map ( Wild , ( <EOL> '<STR_LIT:a>' , '<STR_LIT>' , '<STR_LIT>' , '<STR_LIT:b>' , '<STR_LIT>' , '<STR_LIT>' , '<STR_LIT:c>' , '<STR_LIT>' , '<STR_LIT>' ) ) <EOL> expr = lt * CG ( a , alpha , b , beta , c , gamma ) ** <NUM_LIT:2> <EOL> simp = <NUM_LIT:1> <EOL> sign = lt / abs ( lt ) <EOL> x = abs ( a - b ) <EOL> y = abs ( alpha + beta ) <EOL> build_expr = a + b + <NUM_LIT:1> - Piecewise ( ( x , x > y ) , ( <NUM_LIT:0> , Eq ( x , y ) ) , ( y , y > x ) ) <EOL> index_expr = a + b - c <EOL> term_list , other1 = _check_cg_simp ( expr , simp , sign , lt , term_list , ( a , alpha , b , beta , c , gamma , lt ) , ( a , alpha , b , beta ) , build_expr , index_expr ) <EOL> x = abs ( a - b ) <EOL> y = a + b <EOL> build_expr = ( y + <NUM_LIT:1> - x ) * ( x + y + <NUM_LIT:1> ) <EOL> index_expr = ( c - x ) * ( x + c ) + c + gamma <EOL> term_list , other2 = _check_cg_simp ( expr , simp , sign , lt , term_list , ( a , alpha , b , beta , c , gamma , lt ) , ( a , alpha , b , beta ) , build_expr , index_expr ) <EOL> expr = CG ( a , alpha , b , beta , c , gamma ) * CG ( a , alphap , b , betap , c , gamma ) <EOL> simp = KroneckerDelta ( alpha , alphap ) * KroneckerDelta ( beta , betap ) <EOL> sign = sympify ( <NUM_LIT:1> ) <EOL> x = abs ( a - b ) <EOL> y = abs ( alpha + beta ) <EOL> build_expr = a + b + <NUM_LIT:1> - Piecewise ( ( x , x > y ) , ( <NUM_LIT:0> , Eq ( x , y ) ) , ( y , y > x ) ) <EOL> index_expr = a + b - c <EOL> term_list , other3 = _check_cg_simp ( expr , simp , sign , sympify ( <NUM_LIT:1> ) , term_list , ( a , alpha , alphap , b , beta , betap , c , gamma ) , ( a , alpha , alphap , b , beta , betap ) , build_expr , index_expr ) <EOL> x = abs ( a - b ) <EOL> y = a + b <EOL> build_expr = ( y + <NUM_LIT:1> - x ) * ( x + y + <NUM_LIT:1> ) <EOL> index_expr = ( c - x ) * ( x + c ) + c + gamma <EOL> term_list , other4 = _check_cg_simp ( expr , simp , sign , sympify ( <NUM_LIT:1> ) , term_list , ( a , alpha , alphap , b , beta , betap , c , gamma ) , ( a , alpha , alphap , b , beta , betap ) , build_expr , index_expr ) <EOL> return term_list , other1 + other2 + other4 <EOL> def _check_cg_simp ( expr , simp , sign , lt , term_list , variables , dep_variables , build_index_expr , index_expr ) : <EOL> """<STR_LIT>""" <EOL> other_part = <NUM_LIT:0> <EOL> i = <NUM_LIT:0> <EOL> while i < len ( term_list ) : <EOL> sub_1 = _check_cg ( term_list [ i ] , expr , len ( variables ) ) <EOL> if sub_1 is None : <EOL> i += <NUM_LIT:1> <EOL> continue <EOL> if not sympify ( build_index_expr . subs ( sub_1 ) ) . is_number : <EOL> i += <NUM_LIT:1> <EOL> continue <EOL> sub_dep = [ ( x , sub_1 [ x ] ) for x in dep_variables ] <EOL> cg_index = [ None ] * build_index_expr . subs ( sub_1 ) <EOL> for j in range ( i , len ( term_list ) ) : <EOL> sub_2 = _check_cg ( term_list [ j ] , expr . subs ( sub_dep ) , len ( variables ) - len ( dep_variables ) , sign = ( sign . subs ( sub_1 ) , sign . subs ( sub_dep ) ) ) <EOL> if sub_2 is None : <EOL> continue <EOL> if not sympify ( index_expr . subs ( sub_dep ) . subs ( sub_2 ) ) . is_number : <EOL> continue <EOL> cg_index [ index_expr . subs ( sub_dep ) . subs ( sub_2 ) ] = j , expr . subs ( lt , <NUM_LIT:1> ) . subs ( sub_dep ) . subs ( sub_2 ) , lt . subs ( sub_2 ) , sign . subs ( sub_dep ) . subs ( sub_2 ) <EOL> if all ( i is not None for i in cg_index ) : <EOL> min_lt = min ( * [ abs ( term [ <NUM_LIT:2> ] ) for term in cg_index ] ) <EOL> indicies = [ term [ <NUM_LIT:0> ] for term in cg_index ] <EOL> indicies . sort ( ) <EOL> indicies . reverse ( ) <EOL> [ term_list . pop ( i ) for i in indicies ] <EOL> for term in cg_index : <EOL> if abs ( term [ <NUM_LIT:2> ] ) > min_lt : <EOL> term_list . append ( ( term [ <NUM_LIT:2> ] - min_lt * term [ <NUM_LIT:3> ] ) * term [ <NUM_LIT:1> ] ) <EOL> other_part += min_lt * ( sign * simp ) . subs ( sub_1 ) <EOL> else : <EOL> i += <NUM_LIT:1> <EOL> return term_list , other_part <EOL> def _check_cg ( cg_term , expr , length , sign = None ) : <EOL> """<STR_LIT>""" <EOL> matches = cg_term . match ( expr ) <EOL> if matches is None : <EOL> return <EOL> if sign is not None : <EOL> if not isinstance ( sign , tuple ) : <EOL> raise TypeError ( '<STR_LIT>' ) <EOL> if not sign [ <NUM_LIT:0> ] == ( sign [ <NUM_LIT:1> ] ) . subs ( matches ) : <EOL> return <EOL> if len ( matches ) == length : <EOL> return matches <EOL> def _cg_simp_sum ( e ) : <EOL> e = _check_varsh_sum_871_1 ( e ) <EOL> e = _check_varsh_sum_871_2 ( e ) <EOL> e = _check_varsh_sum_872_4 ( e ) <EOL> return e <EOL> def _check_varsh_sum_871_1 ( e ) : <EOL> a = Wild ( '<STR_LIT:a>' ) <EOL> alpha = symbols ( '<STR_LIT>' ) <EOL> b = Wild ( '<STR_LIT:b>' ) <EOL> match = e . match ( Sum ( CG ( a , alpha , b , <NUM_LIT:0> , a , alpha ) , ( alpha , - a , a ) ) ) <EOL> if match is not None and len ( match ) == <NUM_LIT:2> : <EOL> return ( ( <NUM_LIT:2> * a + <NUM_LIT:1> ) * KroneckerDelta ( b , <NUM_LIT:0> ) ) . subs ( match ) <EOL> return e <EOL> def _check_varsh_sum_871_2 ( e ) : <EOL> a = Wild ( '<STR_LIT:a>' ) <EOL> alpha = symbols ( '<STR_LIT>' ) <EOL> c = Wild ( '<STR_LIT:c>' ) <EOL> match = e . match ( <EOL> Sum ( ( - <NUM_LIT:1> ) ** ( a - alpha ) * CG ( a , alpha , a , - alpha , c , <NUM_LIT:0> ) , ( alpha , - a , a ) ) ) <EOL> if match is not None and len ( match ) == <NUM_LIT:2> : <EOL> return ( sqrt ( <NUM_LIT:2> * a + <NUM_LIT:1> ) * KroneckerDelta ( c , <NUM_LIT:0> ) ) . subs ( match ) <EOL> return e <EOL> def _check_varsh_sum_872_4 ( e ) : <EOL> a = Wild ( '<STR_LIT:a>' ) <EOL> alpha = Wild ( '<STR_LIT>' ) <EOL> b = Wild ( '<STR_LIT:b>' ) <EOL> beta = Wild ( '<STR_LIT>' ) <EOL> c = Wild ( '<STR_LIT:c>' ) <EOL> cp = Wild ( '<STR_LIT>' ) <EOL> gamma = Wild ( '<STR_LIT>' ) <EOL> gammap = Wild ( '<STR_LIT>' ) <EOL> match1 = e . match ( Sum ( CG ( a , alpha , b , beta , c , gamma ) * CG ( <EOL> a , alpha , b , beta , cp , gammap ) , ( alpha , - a , a ) , ( beta , - b , b ) ) ) <EOL> if match1 is not None and len ( match1 ) == <NUM_LIT:8> : <EOL> return ( KroneckerDelta ( c , cp ) * KroneckerDelta ( gamma , gammap ) ) . subs ( match1 ) <EOL> match2 = e . match ( Sum ( <EOL> CG ( a , alpha , b , beta , c , gamma ) ** <NUM_LIT:2> , ( alpha , - a , a ) , ( beta , - b , b ) ) ) <EOL> if match2 is not None and len ( match2 ) == <NUM_LIT:6> : <EOL> return <NUM_LIT:1> <EOL> return e <EOL> def _cg_list ( term ) : <EOL> if isinstance ( term , CG ) : <EOL> return ( term , ) , <NUM_LIT:1> , <NUM_LIT:1> <EOL> cg = [ ] <EOL> coeff = <NUM_LIT:1> <EOL> if not ( isinstance ( term , Mul ) or isinstance ( term , Pow ) ) : <EOL> raise NotImplementedError ( '<STR_LIT>' ) <EOL> if isinstance ( term , Pow ) and sympify ( term . exp ) . is_number : <EOL> if sympify ( term . exp ) . is_number : <EOL> [ cg . append ( term . base ) for _ in range ( term . exp ) ] <EOL> else : <EOL> return ( term , ) , <NUM_LIT:1> , <NUM_LIT:1> <EOL> if isinstance ( term , Mul ) : <EOL> for arg in term . args : <EOL> if isinstance ( arg , CG ) : <EOL> cg . append ( arg ) <EOL> else : <EOL> coeff *= arg <EOL> return cg , coeff , coeff / abs ( coeff ) </s>
