html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Cosh

 http://functions.wolfram.com/01.20.21.3991.01

 Input Form

 Integrate[z^n E^(b Sqrt[z] + d z + e) Cos[a Sqrt[z] + p z + q] Cosh[c Sqrt[z] + f z + g]^v, z] == 2^(-2 - 2 n - v) Binomial[v, v/2] (1 - Mod[v, 2]) (E^(e - ((-I) a + b)^2/(4 (d - I p)) - I q) (d - I p)^(-2 - 2 n) Sum[(-1)^(-h + i) 4^i ((-I) a + b)^(-h - i + 2 n) ((-I) a + b + 2 (d - I p) Sqrt[z])^(h + i) (-(((-I) a + b + 2 (d - I p) Sqrt[z])^2/(d - I p)))^ ((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] (((-I) a + b) ((-I) a + b + 2 (d - I p) Sqrt[z]) Gamma[(1/2) (1 + h + i), -(((-I) a + b + 2 (d - I p) Sqrt[z])^2/ (4 (d - I p)))] + 2 (d - I p) Sqrt[-(((-I) a + b + 2 (d - I p) Sqrt[z])^2/(d - I p))] Gamma[(1/2) (2 + h + i), -(((-I) a + b + 2 (d - I p) Sqrt[z])^2/ (4 (d - I p)))]), {i, 0, n}, {h, 0, i}] + E^(e - (I a + b)^2/(4 (d + I p)) + I q) (d + I p)^(-2 - 2 n) Sum[(-1)^(-h + i) 4^i (I a + b)^(-h - i + 2 n) (I a + b + 2 (d + I p) Sqrt[z])^(h + i) (-((I a + b + 2 (d + I p) Sqrt[z])^2/(d + I p)))^((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((I a + b) (I a + b + 2 (d + I p) Sqrt[z]) Gamma[(1/2) (1 + h + i), -((I a + b + 2 (d + I p) Sqrt[z])^2/(4 (d + I p)))] + 2 (d + I p) Sqrt[-((I a + b + 2 (d + I p) Sqrt[z])^2/(d + I p))] Gamma[(1/2) (2 + h + i), -((I a + b + 2 (d + I p) Sqrt[z])^2/ (4 (d + I p)))]), {i, 0, n}, {h, 0, i}]) + 2^(-2 - 2 n - v) Sum[Binomial[v, s] (E^(e - I q - ((-I) a + b + c (2 s - v))^2/ (4 (d - I p + f (2 s - v))) + g (2 s - v)) (d - I p + f (2 s - v))^ (-2 - 2 n) Sum[(-1)^(-h + i) 4^i ((-I) a + b + c (2 s - v))^ (-h - i + 2 n) ((-I) a + b + c (2 s - v) + 2 (d - I p + f (2 s - v)) Sqrt[z])^(h + i) (-(((-I) a + b + c (2 s - v) + 2 (d - I p + f (2 s - v)) Sqrt[z])^2/ (d - I p + f (2 s - v))))^((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] (((-I) a + b + c (2 s - v)) ((-I) a + b + c (2 s - v) + 2 (d - I p + f (2 s - v)) Sqrt[z]) Gamma[(1/2) (1 + h + i), -(((-I) a + b + c (2 s - v) + 2 (d - I p + f (2 s - v)) Sqrt[z])^2/(4 (d - I p + f (2 s - v))))] + 2 (d - I p + f (2 s - v)) Sqrt[-(((-I) a + b + c (2 s - v) + 2 (d - I p + f (2 s - v)) Sqrt[z])^2/(d - I p + f (2 s - v)))] Gamma[(1/2) (2 + h + i), -(((-I) a + b + c (2 s - v) + 2 (d - I p + f (2 s - v)) Sqrt[z])^2/(4 (d - I p + f (2 s - v))))]), {i, 0, n}, {h, 0, i}] + E^(e + I q - (I a + b + c (2 s - v))^2/(4 (d + I p + f (2 s - v))) + g (2 s - v)) (d + I p + f (2 s - v))^(-2 - 2 n) Sum[(-1)^(-h + i) 4^i (I a + b + c (2 s - v))^(-h - i + 2 n) (I a + b + c (2 s - v) + 2 (d + I p + f (2 s - v)) Sqrt[z])^(h + i) (-((I a + b + c (2 s - v) + 2 (d + I p + f (2 s - v)) Sqrt[z])^2/ (d + I p + f (2 s - v))))^((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((I a + b + c (2 s - v)) (I a + b + c (2 s - v) + 2 (d + I p + f (2 s - v)) Sqrt[z]) Gamma[(1/2) (1 + h + i), -((I a + b + c (2 s - v) + 2 (d + I p + f (2 s - v)) Sqrt[z])^2/( 4 (d + I p + f (2 s - v))))] + 2 (d + I p + f (2 s - v)) Sqrt[-((I a + b + c (2 s - v) + 2 (d + I p + f (2 s - v)) Sqrt[z])^2/(d + I p + f (2 s - v)))] Gamma[(1/2) (2 + h + i), -((I a + b + c (2 s - v) + 2 (d + I p + f (2 s - v)) Sqrt[z])^2/(4 (d + I p + f (2 s - v))))]), {i, 0, n}, {h, 0, i}] + E^(e - I q + g (-2 s + v) - ((-I) a + b + c (-2 s + v))^2/ (4 (d - I p + f (-2 s + v)))) (d - I p + f (-2 s + v))^(-2 - 2 n) Sum[(-1)^(-h + i) 4^i ((-I) a + b + c (-2 s + v))^(-h - i + 2 n) ((-I) a + b + c (-2 s + v) + 2 (d - I p + f (-2 s + v)) Sqrt[z])^ (h + i) (-(((-I) a + b + c (-2 s + v) + 2 (d - I p + f (-2 s + v)) Sqrt[z])^2/(d - I p + f (-2 s + v))))^((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] (((-I) a + b + c (-2 s + v)) ((-I) a + b + c (-2 s + v) + 2 (d - I p + f (-2 s + v)) Sqrt[z]) Gamma[(1/2) (1 + h + i), -(((-I) a + b + c (-2 s + v) + 2 (d - I p + f (-2 s + v)) Sqrt[z])^2/(4 (d - I p + f (-2 s + v))))] + 2 (d - I p + f (-2 s + v)) Sqrt[-(((-I) a + b + c (-2 s + v) + 2 (d - I p + f (-2 s + v)) Sqrt[z])^2/(d - I p + f (-2 s + v)))] Gamma[(1/2) (2 + h + i), -(((-I) a + b + c (-2 s + v) + 2 (d - I p + f (-2 s + v)) Sqrt[z])^2/(4 (d - I p + f (-2 s + v))))]), {i, 0, n}, {h, 0, i}] + E^(e + I q + g (-2 s + v) - (I a + b + c (-2 s + v))^2/ (4 (d + I p + f (-2 s + v)))) (d + I p + f (-2 s + v))^(-2 - 2 n) Sum[(-1)^(-h + i) 4^i (I a + b + c (-2 s + v))^(-h - i + 2 n) (I a + b + c (-2 s + v) + 2 (d + I p + f (-2 s + v)) Sqrt[z])^ (h + i) (-((I a + b + c (-2 s + v) + 2 (d + I p + f (-2 s + v)) Sqrt[z])^2/(d + I p + f (-2 s + v))))^((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((I a + b + c (-2 s + v)) (I a + b + c (-2 s + v) + 2 (d + I p + f (-2 s + v)) Sqrt[z]) Gamma[(1/2) (1 + h + i), -((I a + b + c (-2 s + v) + 2 (d + I p + f (-2 s + v)) Sqrt[z])^2/(4 (d + I p + f (-2 s + v))))] + 2 (d + I p + f (-2 s + v)) Sqrt[-((I a + b + c (-2 s + v) + 2 (d + I p + f (-2 s + v)) Sqrt[z])^2/(d + I p + f (-2 s + v)))] Gamma[(1/2) (2 + h + i), -((I a + b + c (-2 s + v) + 2 (d + I p + f (-2 s + v)) Sqrt[z])^2/(4 (d + I p + f (-2 s + v))))]), {i, 0, n}, {h, 0, i}]), {s, 0, Floor[(1/2) (-1 + v)]}] /; Element[n, Integers] && n >= 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 z n z b + d z + e cos ( z a + p z + q ) cosh v ( z c + f z + g ) z 2 - 2 n - v - 2 ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - v mod 2 \$CellContext`v 2 ) ( - ( b + a ) 2 4 ( d + p ) + e + q ( d + p ) - 2 n - 2 i = 0 n h = 0 i ( - 1 ) i - h 4 i ( b + a ) - h - i + 2 n ( b + a + 2 ( d + p ) z ) h + i ( - ( b + a + 2 ( d + p ) z ) 2 d + p ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b + a ) ( b + a + 2 ( d + p ) z ) Γ ( 1 2 ( h + i + 1 ) , - ( b + a + 2 ( d + p ) z ) 2 4 ( d + p ) ) + 2 - ( b + a + 2 ( d + p ) z ) 2 d + p ( d + p ) Γ ( 1 2 ( h + i + 2 ) , - ( b + a + 2 ( d + p ) z ) 2 4 ( d + p ) ) ) + - ( b - a ) 2 4 ( d - p ) + e - q ( d - p ) - 2 n - 2 i = 0 n h = 0 i ( - 1 ) i - h 4 i ( b - a ) - h - i + 2 n ( b - a + 2 ( d - p ) z ) h + i ( - ( b - a + 2 ( d - p ) z ) 2 d - p ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b - a ) ( b - a + 2 ( d - p ) z ) Γ ( 1 2 ( h + i + 1 ) , - ( b - a + 2 ( d - p ) z ) 2 4 ( d - p ) ) + 2 - ( b - a + 2 ( d - p ) z ) 2 d - p ( d - p ) Γ ( 1 2 ( h + i + 2 ) , - ( b - a + 2 ( d - p ) z ) 2 4 ( d - p ) ) ) ) + 2 - 2 n - v - 2 s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - 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v ) ) z ) 2 4 ( d + p + f ( 2 s - v ) ) ) - ( b + a + c ( 2 s - v ) + 2 ( d + p + f ( 2 s - v ) ) z ) 2 d + p + f ( 2 s - v ) ) + - ( b - a + c ( 2 s - v ) ) 2 4 ( d - p + f ( 2 s - v ) ) + e - q + g ( 2 s - v ) ( d - p + f ( 2 s - v ) ) - 2 n - 2 i = 0 n h = 0 i ( - 1 ) i - h 4 i ( b - a + c ( 2 s - v ) ) - h - i + 2 n ( b - a + c ( 2 s - v ) + 2 ( d - p + f ( 2 s - v ) ) z ) h + i ( - ( b - a + c ( 2 s - v ) + 2 ( d - p + f ( 2 s - v ) ) z ) 2 d - p + f ( 2 s - v ) ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b - a + c ( 2 s - v ) ) ( b - a + c ( 2 s - v ) + 2 ( d - p + f ( 2 s - v ) ) z ) Γ ( 1 2 ( h + i + 1 ) , - ( b - a + c ( 2 s - v ) + 2 ( d - p + f ( 2 s - v ) ) z ) 2 4 ( d - p + f ( 2 s - v ) ) ) + 2 ( d - p + f ( 2 s - v ) ) Γ ( 1 2 ( h + i + 2 ) , - ( b - a + c ( 2 s - v ) + 2 ( d - p + f ( 2 s - v ) ) z ) 2 4 ( d - p + f ( 2 s - v ) ) ) - ( b - a + c ( 2 s - v ) + 2 ( d - p + f ( 2 s - v ) ) z ) 2 d - p + f ( 2 s - v ) ) + - ( b + a + c ( v - 2 s ) ) 2 4 ( d + p + f ( v - 2 s ) ) + e + q + g ( v - 2 s ) ( d + p + f ( v - 2 s ) ) - 2 n - 2 i = 0 n h = 0 i ( - 1 ) i - h 4 i ( b + a + c ( v - 2 s ) ) - h - i + 2 n ( b + a + c ( v - 2 s ) + 2 ( d + p + f ( v - 2 s ) ) z ) h + i ( - ( b + a + c ( v - 2 s ) + 2 ( d + p + f ( v - 2 s ) ) z ) 2 d + p + f ( v - 2 s ) ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b + a + c ( v - 2 s ) ) ( b + a + c ( v - 2 s ) + 2 ( d + p + f ( v - 2 s ) ) z ) Γ ( 1 2 ( h + i + 1 ) , - ( b + a + c ( v - 2 s ) + 2 ( d + p + f ( v - 2 s ) ) z ) 2 4 ( d + p + f ( v - 2 s ) ) ) + 2 ( d + p + f ( v - 2 s ) ) Γ ( 1 2 ( h + i + 2 ) , - ( b + a + c ( v - 2 s ) + 2 ( d + p + f ( v - 2 s ) ) z ) 2 4 ( d + p + f ( v - 2 s ) ) ) - ( b + a + c ( v - 2 s ) + 2 ( d + p + f ( v - 2 s ) ) z ) 2 d + p + f ( v - 2 s ) ) + - ( b - a + c ( v - 2 s ) ) 2 4 ( d - p + f ( v - 2 s ) ) + e - q + g ( v - 2 s ) ( d - p + f ( v - 2 s ) ) - 2 n - 2 i = 0 n h = 0 i ( - 1 ) i - h 4 i ( b - a + c ( v - 2 s ) ) - h - i + 2 n ( b - a + c ( v - 2 s ) + 2 ( d - p + f ( v - 2 s ) ) z ) h + i ( - ( b - a + c ( v - 2 s ) + 2 ( d - p + f ( v - 2 s ) ) z ) 2 d - p + f ( v - 2 s ) ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b - a + c ( v - 2 s ) ) ( b - a + c ( v - 2 s ) + 2 ( d - p + f ( v - 2 s ) ) z ) Γ ( 1 2 ( h + i + 1 ) , - ( b - a + c ( v - 2 s ) + 2 ( d - p + f ( v - 2 s ) ) z ) 2 4 ( d - p + f ( v - 2 s ) ) ) + 2 ( d - p + f ( v - 2 s ) ) Γ ( 1 2 ( h + i + 2 ) , - ( b - a + c ( v - 2 s ) + 2 ( d - p + f ( v - 2 s ) ) z ) 2 4 ( d - p + f ( v - 2 s ) ) ) - ( b - a + c ( v - 2 s ) + 2 ( d - p + f ( v - 2 s ) ) z ) 2 d - p + f ( v - 2 s ) ) ) /; n v + Condition z z n z 1 2 b d z e z 1 2 a p z q z 1 2 c f z g v 2 -2 n -1 v -2 Binomial v v 2 -1 1 -1 \$CellContext`v 2 -1 b a 2 4 d p -1 e q d p -2 n -2 h 0 i i 0 n -1 i -1 h 4 i b a -1 h -1 i 2 n b a 2 d p z 1 2 h i -1 b a 2 d p z 1 2 2 d p -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i b a b a 2 d p z 1 2 Gamma 1 2 h i 1 -1 b a 2 d p z 1 2 2 4 d p -1 2 -1 b a 2 d p z 1 2 2 d p -1 1 2 d p Gamma 1 2 h i 2 -1 b a 2 d p z 1 2 2 4 d p -1 -1 b -1 a 2 4 d -1 p -1 e -1 q d -1 p -2 n -2 h 0 i i 0 n -1 i -1 h 4 i b -1 a -1 h -1 i 2 n b -1 a 2 d -1 p z 1 2 h i -1 b -1 a 2 d -1 p z 1 2 2 d -1 p -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i b -1 a b -1 a 2 d -1 p z 1 2 Gamma 1 2 h i 1 -1 b -1 a 2 d -1 p z 1 2 2 4 d -1 p -1 2 -1 b -1 a 2 d -1 p z 1 2 2 d -1 p -1 1 2 d -1 p Gamma 1 2 h i 2 -1 b -1 a 2 d -1 p z 1 2 2 4 d -1 p -1 2 -2 n -1 v -2 s 0 v -1 2 -1 Binomial v s -1 b a c 2 s -1 v 2 4 d p f 2 s -1 v -1 e q g 2 s -1 v d p f 2 s -1 v -2 n -2 h 0 i i 0 n -1 i -1 h 4 i b a c 2 s -1 v -1 h -1 i 2 n b a c 2 s -1 v 2 d p f 2 s -1 v z 1 2 h i -1 b a c 2 s -1 v 2 d p f 2 s -1 v z 1 2 2 d p f 2 s -1 v -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i b a c 2 s -1 v b a c 2 s -1 v 2 d p f 2 s -1 v z 1 2 Gamma 1 2 h i 1 -1 b a c 2 s -1 v 2 d p f 2 s -1 v z 1 2 2 4 d p f 2 s -1 v -1 2 d p f 2 s -1 v Gamma 1 2 h i 2 -1 b a c 2 s -1 v 2 d p f 2 s -1 v z 1 2 2 4 d p f 2 s -1 v -1 -1 b a c 2 s -1 v 2 d p f 2 s -1 v z 1 2 2 d p f 2 s -1 v -1 1 2 -1 b -1 a c 2 s -1 v 2 4 d -1 p f 2 s -1 v -1 e -1 q g 2 s -1 v d -1 p f 2 s -1 v -2 n -2 h 0 i i 0 n -1 i -1 h 4 i b -1 a c 2 s -1 v -1 h -1 i 2 n b -1 a c 2 s -1 v 2 d -1 p f 2 s -1 v z 1 2 h i -1 b -1 a c 2 s -1 v 2 d -1 p f 2 s -1 v z 1 2 2 d -1 p f 2 s -1 v -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i b -1 a c 2 s -1 v b -1 a c 2 s -1 v 2 d -1 p f 2 s -1 v z 1 2 Gamma 1 2 h i 1 -1 b -1 a c 2 s -1 v 2 d -1 p f 2 s -1 v z 1 2 2 4 d -1 p f 2 s -1 v -1 2 d -1 p f 2 s -1 v Gamma 1 2 h i 2 -1 b -1 a c 2 s -1 v 2 d -1 p f 2 s -1 v z 1 2 2 4 d -1 p f 2 s -1 v -1 -1 b -1 a c 2 s -1 v 2 d -1 p f 2 s -1 v z 1 2 2 d -1 p f 2 s -1 v -1 1 2 -1 b a c v -1 2 s 2 4 d p f v -1 2 s -1 e q g v -1 2 s d p f v -1 2 s -2 n -2 h 0 i i 0 n -1 i -1 h 4 i b a c v -1 2 s -1 h -1 i 2 n b a c v -1 2 s 2 d p f v -1 2 s z 1 2 h i -1 b a c v -1 2 s 2 d p f v -1 2 s z 1 2 2 d p f v -1 2 s -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i b a c v -1 2 s b a c v -1 2 s 2 d p f v -1 2 s z 1 2 Gamma 1 2 h i 1 -1 b a c v -1 2 s 2 d p f v -1 2 s z 1 2 2 4 d p f v -1 2 s -1 2 d p f v -1 2 s Gamma 1 2 h i 2 -1 b a c v -1 2 s 2 d p f v -1 2 s z 1 2 2 4 d p f v -1 2 s -1 -1 b a c v -1 2 s 2 d p f v -1 2 s z 1 2 2 d p f v -1 2 s -1 1 2 -1 b -1 a c v -1 2 s 2 4 d -1 p f v -1 2 s -1 e -1 q g v -1 2 s d -1 p f v -1 2 s -2 n -2 h 0 i i 0 n -1 i -1 h 4 i b -1 a c v -1 2 s -1 h -1 i 2 n b -1 a c v -1 2 s 2 d -1 p f v -1 2 s z 1 2 h i -1 b -1 a c v -1 2 s 2 d -1 p f v -1 2 s z 1 2 2 d -1 p f v -1 2 s -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i b -1 a c v -1 2 s b -1 a c v -1 2 s 2 d -1 p f v -1 2 s z 1 2 Gamma 1 2 h i 1 -1 b -1 a c v -1 2 s 2 d -1 p f v -1 2 s z 1 2 2 4 d -1 p f v -1 2 s -1 2 d -1 p f v -1 2 s Gamma 1 2 h i 2 -1 b -1 a c v -1 2 s 2 d -1 p f v -1 2 s z 1 2 2 4 d -1 p f v -1 2 s -1 -1 b -1 a c v -1 2 s 2 d -1 p f v -1 2 s z 1 2 2 d -1 p f v -1 2 s -1 1 2 n v SuperPlus [/itex]

 Rule Form

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 Date Added to functions.wolfram.com (modification date)

 2002-12-18