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/01.20.21.4447.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Sinh[c z] Cosh[b + a z]^v, z] == (2^(-1 - v) z^\[Alpha] Binomial[v, v/2] ((-(c z)^\[Alpha]) Gamma[\[Alpha], (-c) z] + ((-c) z)^\[Alpha] Gamma[\[Alpha], c z]) (1 - Mod[v, 2]))/((-c^2) z^2)^\[Alpha] + 2^(-1 - v) z^\[Alpha] Sum[(Binomial[v, s] ((-E^(-2 b s + b v)) ((c + a (-2 s + v)) z)^\[Alpha] ((-(-c - 2 a s + a v)^2) z^2)^\[Alpha] Gamma[\[Alpha], (-c + 2 a s - a v) z] + ((-c + 2 a s - a v) z)^\[Alpha] (E^(-2 b s + b v) ((-c - 2 a s + a v) z)^\[Alpha] ((c + a (-2 s + v)) z)^\[Alpha] Gamma[\[Alpha], (c + 2 a s - a v) z] - E^(b (2 s - v)) ((c + 2 a s - a v) z)^\[Alpha] ((c + a (-2 s + v)) z)^\[Alpha] Gamma[\[Alpha], (-c - 2 a s + a v) z] + E^(b (2 s - v)) ((-(-c - 2 a s + a v)^2) z^2)^\[Alpha] Gamma[\[Alpha], (c + a (-2 s + v)) z])))/ (((-c + 2 a s - a v) z)^\[Alpha] ((c + a (-2 s + v)) z)^\[Alpha] ((-(-c - 2 a s + a v)^2) z^2)^\[Alpha]), {s, 0, Floor[(1/2) (-1 + v)]}] /; Element[v, Integers] && v > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", " ", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "v"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["c", "2"]]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", "z"]], ")"]], "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["-", "c"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], " ", "z"]], ")"]], "\[Alpha]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["c", " ", "z"]]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "v"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]]]], ")"]], " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "a", " ", "s"]], "+", RowBox[List["a", " ", "v"]]]], ")"]], "2"]]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "s"]], "+", RowBox[List["b", " ", "v"]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]], ")"]], "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "a", " ", "s"]], "+", RowBox[List["a", " ", "v"]]]], ")"]], "2"]]], " ", SuperscriptBox["z", "2"]]], ")"]], "\[Alpha]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]]]], ")"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]]]], ")"]], " ", "z"]], ")"]], "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "s"]], "+", RowBox[List["b", " ", "v"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "a", " ", "s"]], "+", RowBox[List["a", " ", "v"]]]], ")"]], " ", "z"]], ")"]], "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]], ")"]], "\[Alpha]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]]]], ")"]], " ", "z"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]]]], ")"]], " ", "z"]], ")"]], "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]], ")"]], "\[Alpha]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "a", " ", "s"]], "+", RowBox[List["a", " ", "v"]]]], ")"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "a", " ", "s"]], "+", RowBox[List["a", " ", "v"]]]], ")"]], "2"]]], " ", SuperscriptBox["z", "2"]]], ")"]], "\[Alpha]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 z α - 1 sinh ( c z ) cosh v ( b + a z ) z 2 - v - 1 z α ( 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]]]]] ( ( - c z ) α Γ ( α , c z ) - ( c z ) α Γ ( α , - c z ) ) ( 1 - v mod 2 \$CellContext`v 2 ) ( - c 2 z 2 ) - α + 2 - v - 1 z α s = 0 v - 1 2 ( ( - c + 2 a s - a v ) z ) - α ( ( c + a ( v - 2 s ) ) z ) - α ( - ( - c - 2 a s + a v ) 2 z 2 ) - α ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( ( - c + 2 a s - a v ) z ) α ( - b ( 2 s - v ) ( ( c + 2 a s - a v ) z ) α Γ ( α , ( - c - 2 a s + a v ) z ) ( ( c + a ( v - 2 s ) ) z ) α + b v - 2 b s ( ( - c - 2 a s + a v ) z ) α Γ ( α , ( c + 2 a s - a v ) z ) ( ( c + a ( v - 2 s ) ) z ) α + b ( 2 s - v ) ( - ( - c - 2 a s + a v ) 2 z 2 ) α Γ ( α , ( c + a ( v - 2 s ) ) z ) ) - b v - 2 b s ( ( c + a ( v - 2 s ) ) z ) α ( - ( - c - 2 a s + a v ) 2 z 2 ) α Γ ( α , ( - c + 2 a s - a v ) z ) ) /; v + Condition z z α -1 c z b a z v 2 -1 v -1 z α Binomial v v 2 -1 -1 c z α Gamma α c z -1 c z α Gamma α -1 c z 1 -1 \$CellContext`v 2 -1 c 2 z 2 -1 α 2 -1 v -1 z α s 0 v -1 2 -1 -1 c 2 a s -1 a v z -1 α c a v -1 2 s z -1 α -1 -1 c -1 2 a s a v 2 z 2 -1 α Binomial v s -1 c 2 a s -1 a v z α -1 b 2 s -1 v c 2 a s -1 a v z α Gamma α -1 c -1 2 a s a v z c a v -1 2 s z α b v -1 2 b s -1 c -1 2 a s a v z α Gamma α c 2 a s -1 a v z c a v -1 2 s z α b 2 s -1 v -1 -1 c -1 2 a s a v 2 z 2 α Gamma α c a v -1 2 s z -1 b v -1 2 b s c a v -1 2 s z α -1 -1 c -1 2 a s a v 2 z 2 α Gamma α -1 c 2 a s -1 a v z v SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["Sinh", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["b_", "+", RowBox[List["a_", " ", "z_"]]]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "v"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["c", "2"]]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", "z"]], ")"]], "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["-", "c"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], " ", "z"]], ")"]], "\[Alpha]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["c", " ", "z"]]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "v"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]]]], ")"]], " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "a", " ", "s"]], "+", RowBox[List["a", " ", "v"]]]], ")"]], "2"]]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "s"]], "+", RowBox[List["b", " ", "v"]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]], ")"]], "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "a", " ", "s"]], "+", RowBox[List["a", " ", "v"]]]], ")"]], "2"]]], " ", SuperscriptBox["z", "2"]]], ")"]], "\[Alpha]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]]]], ")"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]]]], ")"]], " ", "z"]], ")"]], "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "s"]], "+", RowBox[List["b", " ", "v"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "a", " ", "s"]], "+", RowBox[List["a", " ", "v"]]]], ")"]], " ", "z"]], ")"]], "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]], ")"]], "\[Alpha]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]]]], ")"]], " ", "z"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]]]], ")"]], " ", "z"]], ")"]], "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]], ")"]], "\[Alpha]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "a", " ", "s"]], "+", RowBox[List["a", " ", "v"]]]], ")"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "a", " ", "s"]], "+", RowBox[List["a", " ", "v"]]]], ")"]], "2"]]], " ", SuperscriptBox["z", "2"]]], ")"]], "\[Alpha]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]]]]]]]]]]

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

 2002-12-18