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; }

 Csch

 http://functions.wolfram.com/01.23.21.0203.01

 Input Form

 Integrate[Sin[a z]^m Sinh[c z]^\[Mu] Csch[c z], z] == ((-(1/(c (1 - \[Mu])))) 2^(1 - m) E^(c z) Sinh[c z]^\[Mu] Binomial[m, m/2] Hypergeometric2F1[(c - c \[Mu])/(2 c), 1 - \[Mu], (3 - \[Mu])/2, E^(2 c z)] (1 - Mod[m, 2]))/(1 - E^(2 c z))^\[Mu] - (2^(1 - m) E^(c z) Sinh[c z]^\[Mu] Sum[(-1)^k Binomial[m, k] ((E^((I m Pi)/2 + I a (2 k - m) z) Hypergeometric2F1[ (c + I a (2 k - m) - c \[Mu])/(2 c), 1 - \[Mu], (I a (2 k - m) + c (3 - \[Mu]))/(2 c), E^(2 c z)])/ (I a (2 k - m) + c (1 - \[Mu])) + (E^((-(1/2)) I m Pi + I a (-2 k + m) z) Hypergeometric2F1[ (c + I a (-2 k + m) - c \[Mu])/(2 c), 1 - \[Mu], (I a (-2 k + m) + c (3 - \[Mu]))/(2 c), E^(2 c z)])/ (I a (-2 k + m) + c (1 - \[Mu]))), {k, 0, Floor[(1/2) (-1 + m)]}])/ (1 - E^(2 c z))^\[Mu] /; Element[m, Integers] && m > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["a", " ", "z"]], "]"]], "m"], SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]"]], ")"]]]]]]], SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["1", "-", "\[Mu]"]], ",", FractionBox[RowBox[List["3", "-", "\[Mu]"]], "2"], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]]]], " ", "-", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "2"], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", "z"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["1", "-", "\[Mu]"]], ",", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["3", "-", "\[Mu]"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]"]], ")"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", "z"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["1", "-", "\[Mu]"]], ",", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["3", "-", "\[Mu]"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]"]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 sin m ( a z ) sinh μ ( c z ) csch ( c z ) z - 2 1 - m c z ( 1 - m mod 2 \$CellContext`m 2 ) sinh μ ( c z ) ( 1 - 2 c z ) - μ c ( 1 - μ ) ( 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 F 1 ( c - c μ 2 c , 1 - μ ; 3 - μ 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[FractionBox[RowBox[List["c", "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Mu]"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["3", "-", "\[Mu]"]], "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] - 2 1 - m c z sinh μ ( c z ) ( 1 - 2 c z ) - μ 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 + a ( 2 k - m ) z a ( 2 k - m ) + c ( 1 - μ ) 2 F 1 ( - μ c + c + a ( 2 k - m ) 2 c , 1 - μ ; a ( 2 k - m ) + c ( 3 - μ ) 2 c ; 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[RowBox[List["-", "\[Mu]"]], " ", "c"]], "+", "c", "+", RowBox[List["a", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Mu]"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["a", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["3", "-", "\[Mu]"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], 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] + a ( m - 2 k ) z - m π 2 a ( m - 2 k ) + c ( 1 - μ ) 2 F 1 ( - μ c + c + a ( m - 2 k ) 2 c , 1 - μ ; a ( m - 2 k ) + c ( 3 - μ ) 2 c ; 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[RowBox[List["-", "\[Mu]"]], " ", "c"]], "+", "c", "+", RowBox[List["a", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Mu]"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["a", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["3", "-", "\[Mu]"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], 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] ) z a z m c z μ c z -1 2 1 -1 m c z 1 -1 \$CellContext`m 2 c z μ 1 -1 2 c z -1 μ c 1 -1 μ -1 Binomial m m 2 -1 Hypergeometric2F1 c -1 c μ 2 c -1 1 -1 μ 3 -1 μ 2 -1 2 c z -1 2 1 -1 m c z c z μ 1 -1 2 c z -1 μ k 0 m -1 2 -1 -1 k Binomial m k m 2 -1 a 2 k -1 m z a 2 k -1 m c 1 -1 μ -1 Hypergeometric2F1 -1 μ c c a 2 k -1 m 2 c -1 1 -1 μ a 2 k -1 m c 3 -1 μ 2 c -1 2 c z a m -1 2 k z -1 m 2 -1 a m -1 2 k c 1 -1 μ -1 Hypergeometric2F1 -1 μ c c a m -1 2 k 2 c -1 1 -1 μ a m -1 2 k c 3 -1 μ 2 c -1 2 c z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["a_", " ", "z_"]], "]"]], "m_"], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "\[Mu]_"], " ", RowBox[List["Csch", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["1", "-", "\[Mu]"]], ",", FractionBox[RowBox[List["3", "-", "\[Mu]"]], "2"], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]"]], ")"]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "2"], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", "z"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["1", "-", "\[Mu]"]], ",", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["3", "-", "\[Mu]"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]"]], ")"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", "z"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["1", "-", "\[Mu]"]], ",", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["3", "-", "\[Mu]"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]"]], ")"]]]]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

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

 2002-12-18