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/01.20.21.0127.01

 Input Form

 Integrate[z^n Cosh[a Sqrt[z] + b], z] == (-a^(-2 (1 + n))) (E^b (-(ExpIntegralEi[a Sqrt[z]]/(-2 (1 + n))!) + E^(a Sqrt[z]) Sum[((-a) Sqrt[z])^k/Pochhammer[2 (1 + n), 1 + k - 2 (1 + n)], {k, 0, -1 + 2 (1 + n)}] - E^(a Sqrt[z]) Sum[((-a) Sqrt[z])^k/Pochhammer[2 (1 + n), 1 + k - 2 (1 + n)], {k, 2 (1 + n), -1}]) + (-(ExpIntegralEi[(-a) Sqrt[z]]/(-2 (1 + n))!) + Sum[(a Sqrt[z])^k/Pochhammer[2 (1 + n), 1 + k - 2 (1 + n)], {k, 0, -1 + 2 (1 + n)}]/E^(a Sqrt[z]) - Sum[(a Sqrt[z])^k/Pochhammer[2 (1 + n), 1 + k - 2 (1 + n)], {k, 2 (1 + n), -1}]/E^(a Sqrt[z]))/E^b) /; Element[n, Integers]

 Standard Form

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 MathML Form

 z n cosh ( z a + b ) z - a - 2 ( n + 1 ) ( b ( - Ei ( a z ) ( - 2 ( n + 1 ) ) ! + a z k = 0 2 ( n + 1 ) - 1 ( - a z ) k ( 2 ( n + 1 ) ) k - 2 ( n + 1 ) + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], ")"]], RowBox[List["k", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], "+", "1"]]], Pochhammer] - a z k = 2 ( n + 1 ) - 1 ( - a z ) k ( 2 ( n + 1 ) ) k - 2 ( n + 1 ) + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], ")"]], RowBox[List["k", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], "+", "1"]]], Pochhammer] ) + - b ( - Ei ( - a z ) ( - 2 ( n + 1 ) ) ! + - a z k = 0 2 ( n + 1 ) - 1 ( a z ) k ( 2 ( n + 1 ) ) k - 2 ( n + 1 ) + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], ")"]], RowBox[List["k", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], "+", "1"]]], Pochhammer] - - a z k = 2 ( n + 1 ) - 1 ( a z ) k ( 2 ( n + 1 ) ) k - 2 ( n + 1 ) + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], ")"]], RowBox[List["k", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], "+", "1"]]], Pochhammer] ) ) /; n TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition z z n z 1 2 a b -1 a -2 n 1 b -1 ExpIntegralEi a z 1 2 -2 n 1 -1 a z 1 2 k 0 2 n 1 -1 -1 a z 1 2 k Pochhammer 2 n 1 k -1 2 n 1 1 -1 -1 a z 1 2 k 2 n 1 -1 -1 a z 1 2 k Pochhammer 2 n 1 k -1 2 n 1 1 -1 -1 b -1 ExpIntegralEi -1 a z 1 2 -2 n 1 -1 -1 a z 1 2 k 0 2 n 1 -1 a z 1 2 k Pochhammer 2 n 1 k -1 2 n 1 1 -1 -1 -1 a z 1 2 k 2 n 1 -1 a z 1 2 k Pochhammer 2 n 1 k -1 2 n 1 1 -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["a_", " ", SqrtBox["z_"]]], "+", "b_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["a", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "b"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["ExpIntegralEi", "[", RowBox[List["a", " ", SqrtBox["z"]]], "]"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ")"]], "!"]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", SqrtBox["z"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", SqrtBox["z"]]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List["1", "+", "k", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]]]]], "]"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", SqrtBox["z"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", SqrtBox["z"]]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List["1", "+", "k", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]]]]], "]"]]]]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "b"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["-", "a"]], " ", SqrtBox["z"]]], "]"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ")"]], "!"]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", SqrtBox["z"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SqrtBox["z"]]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List["1", "+", "k", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]]]]], "]"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", SqrtBox["z"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SqrtBox["z"]]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List["1", "+", "k", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]]]]], "]"]]]]]]]]], ")"]]]]]], ")"]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]]]

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

 2002-12-18