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; }

 Coth

 http://functions.wolfram.com/01.22.21.0052.01

 Input Form

 Integrate[E^(p z) Sin[b z]^m Coth[c z], z] == ((-(1/(p (2 c + p)))) E^(p z) Binomial[m, m/2] ((2 c + p) Hypergeometric2F1[p/(2 c), 1, 1 + p/(2 c), E^(2 c z)] + E^(2 c z) p Hypergeometric2F1[1 + p/(2 c), 1, 2 + p/(2 c), E^(2 c z)]) (1 - Mod[m, 2]))/2^m - Sum[(-1)^k Binomial[m, k] ((E^((I m Pi)/2 + ((-I) b (-2 k + m) + p) z) ((2 c - I b (-2 k + m) + p) Hypergeometric2F1[ ((-I) b (-2 k + m) + p)/(2 c), 1, 1 + ((-I) b (-2 k + m) + p)/ (2 c), E^(2 c z)] + E^(2 c z) ((-I) b (-2 k + m) + p) Hypergeometric2F1[1 + ((-I) b (-2 k + m) + p)/(2 c), 1, 2 + ((-I) b (-2 k + m) + p)/(2 c), E^(2 c z)]))/ (((-I) b (-2 k + m) + p) (2 c - I b (-2 k + m) + p)) + (E^((-(1/2)) I m Pi + (I b (-2 k + m) + p) z) ((2 c + I b (-2 k + m) + p) Hypergeometric2F1[(I b (-2 k + m) + p)/ (2 c), 1, 1 + (I b (-2 k + m) + p)/(2 c), E^(2 c z)] + E^(2 c z) (I b (-2 k + m) + p) Hypergeometric2F1[ 1 + (I b (-2 k + m) + p)/(2 c), 1, 2 + (I b (-2 k + m) + p)/ (2 c), E^(2 c z)]))/((I b (-2 k + m) + p) (2 c + I b (-2 k + m) + p))), {k, 0, Floor[(1/2) (-1 + m)]}]/2^m /; Element[m, Integers] && m > 0

 Standard Form

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 MathML Form

 p z sin m ( b z ) coth ( c z ) z 1 p ( 2 c + p ) ( 2 - m p z ( - 1 ) ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( 2 c + p ) 2 F 1 ( p 2 c , 1 ; p 2 c + 1 ; 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["p", RowBox[List["2", " ", "c"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["p", RowBox[List["2", " ", "c"]]], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + 2 c z p 2 F 1 ( p 2 c + 1 , 1 ; p 2 c + 2 ; 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["p", RowBox[List["2", " ", "c"]]], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["p", RowBox[List["2", " ", "c"]]], "+", "2"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) ( 1 - m mod 2 \$CellContext`m 2 ) ) - 2 - m k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( π m 2 + ( p - b ( m - 2 k ) ) z ( ( 2 c - b ( m - 2 k ) + p ) 2 F 1 ( p - b ( m - 2 k ) 2 c , 1 ; p - b ( m - 2 k ) 2 c + 1 ; 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + 2 c z ( p - b ( m - 2 k ) ) 2 F 1 ( p - b ( m - 2 k ) 2 c + 1 , 1 ; p - b ( m - 2 k ) 2 c + 2 ; 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "2"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) ) / ( ( p - b ( m - 2 k ) ) ( 2 c - b ( m - 2 k ) + p ) ) + ( ( b ( m - 2 k ) + p ) z - m π 2 ( ( 2 c + b ( m - 2 k ) + p ) 2 F 1 ( b ( m - 2 k ) + p 2 c , 1 ; b ( m - 2 k ) + p 2 c + 1 ; 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List["b", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["b", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + 2 c z ( b ( m - 2 k ) + p ) 2 F 1 ( b ( m - 2 k ) + p 2 c + 1 , 1 ; b ( m - 2 k ) + p 2 c + 2 ; 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["b", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["b", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "2"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) ) / ( ( b ( m - 2 k ) + p ) ( 2 c + b ( m - 2 k ) + p ) ) ) /; m + Condition z p z b z m c z 1 p 2 c p -1 2 -1 m p z -1 Binomial m m 2 -1 2 c p Hypergeometric2F1 p 2 c -1 1 p 2 c -1 1 2 c z 2 c z p Hypergeometric2F1 p 2 c -1 1 1 p 2 c -1 2 2 c z 1 -1 \$CellContext`m 2 -1 2 -1 m k 0 m -1 2 -1 -1 k Binomial m k m 2 -1 p -1 b m -1 2 k z 2 c -1 b m -1 2 k p Hypergeometric2F1 p -1 b m -1 2 k 2 c -1 1 p -1 b m -1 2 k 2 c -1 1 2 c z 2 c z p -1 b m -1 2 k Hypergeometric2F1 p -1 b m -1 2 k 2 c -1 1 1 p -1 b m -1 2 k 2 c -1 2 2 c z p -1 b m -1 2 k 2 c -1 b m -1 2 k p -1 b m -1 2 k p z -1 m 2 -1 2 c b m -1 2 k p Hypergeometric2F1 b m -1 2 k p 2 c -1 1 b m -1 2 k p 2 c -1 1 2 c z 2 c z b m -1 2 k p Hypergeometric2F1 b m -1 2 k p 2 c -1 1 1 b m -1 2 k p 2 c -1 2 2 c z b m -1 2 k p 2 c b m -1 2 k p -1 m SuperPlus [/itex]

 Rule Form

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

 2002-12-18