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 Cosh

 http://functions.wolfram.com/01.20.21.0074.01

 Input Form

 Integrate[z^(n + 1/2) Cosh[a z + b], z] == (((-1)^(n - 1) a^(-1 - n) Sqrt[z])/(E^b (2 Sqrt[(-a^2) z^2]))) (Gamma[3/2 + n] ((-E^(2 b)) Sqrt[a z] Erfc[Sqrt[(-a) z]] + (-1)^n Sqrt[(-a) z] Erfc[Sqrt[a z]]) - Sqrt[a z] E^(a z + 2 b) (Sum[((-a) z)^(1/2 + k)/Pochhammer[3/2 + n, k - n], {k, 0, n}] - Sum[((-a) z)^(1/2 + k)/Pochhammer[3/2 + n, k - n], {k, 1 + n, -1}]) + ((-1)^n Sqrt[(-a) z] (Sum[(a z)^(1/2 + k)/Pochhammer[3/2 + n, k - n], {k, 0, n}] - Sum[(a z)^(1/2 + k)/Pochhammer[3/2 + n, k - n], {k, 1 + n, -1}]))/E^(a z)) /; Element[n, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["n", "+", FractionBox["1", "2"]]]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["a", " ", "z"]], "+", "b"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", SuperscriptBox["a", RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "b"]]], SqrtBox["z"], " "]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b"]]]]], " ", SqrtBox[RowBox[List["a", " ", "z"]]], " ", RowBox[List["Erfc", "[", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", "z"]]], "]"]]]], " ", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", "z"]]], " ", RowBox[List["Erfc", "[", SqrtBox[RowBox[List["a", " ", "z"]]], "]"]]]]]], ")"]]]], "-", RowBox[List[SqrtBox[RowBox[List["a", " ", "z"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["a", " ", "z"]], "+", RowBox[List["2", "b"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["k", "-", "n"]]]], "]"]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["1", "+", "n"]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["k", "-", "n"]]]], "]"]]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", "z"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["k", "-", "n"]]]], "]"]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["1", "+", "n"]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["k", "-", "n"]]]], "]"]]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]

 MathML Form

 z n + 1 2 cosh ( b + a z ) z 1 2 - a 2 z 2 ( ( ( - 1 ) n - 1 a - n - 1 - b z ) ( ( ( - 1 ) n - a z erfc ( a z ) - 2 b a z erfc ( - a z ) ) Γ ( n + 3 2 ) - a z 2 b + a z ( k = 0 n ( - a z ) k + 1 2 ( n + 3 2 ) k - n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["3", "2"]]], ")"]], RowBox[List["k", "-", "n"]]], Pochhammer] - k = n + 1 - 1 ( - a z ) k + 1 2 ( n + 3 2 ) k - n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["3", "2"]]], ")"]], RowBox[List["k", "-", "n"]]], Pochhammer] ) + ( - 1 ) n - a z - a z ( k = 0 n ( a z ) k + 1 2 ( n + 3 2 ) k - n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["3", "2"]]], ")"]], RowBox[List["k", "-", "n"]]], Pochhammer] - k = n + 1 - 1 ( a z ) k + 1 2 ( n + 3 2 ) k - n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["3", "2"]]], ")"]], RowBox[List["k", "-", "n"]]], Pochhammer] ) ) ) /; n TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition z z n 1 2 b a z 1 2 -1 a 2 z 2 1 2 -1 -1 n -1 a -1 n -1 -1 b z 1 2 -1 n -1 a z 1 2 Erfc a z 1 2 -1 2 b a z 1 2 Erfc -1 a z 1 2 Gamma n 3 2 -1 a z 1 2 2 b a z k 0 n -1 a z k 1 2 Pochhammer n 3 2 k -1 n -1 -1 k n 1 -1 -1 a z k 1 2 Pochhammer n 3 2 k -1 n -1 -1 n -1 a z -1 a z 1 2 k 0 n a z k 1 2 Pochhammer n 3 2 k -1 n -1 -1 k n 1 -1 a z k 1 2 Pochhammer n 3 2 k -1 n -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["n_", "+", FractionBox["1", "2"]]]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["a_", " ", "z_"]], "+", "b_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", SuperscriptBox["a", RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "b"]]], " ", SqrtBox["z"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b"]]]]], " ", SqrtBox[RowBox[List["a", " ", "z"]]], " ", RowBox[List["Erfc", "[", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", "z"]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", "z"]]], " ", RowBox[List["Erfc", "[", SqrtBox[RowBox[List["a", " ", "z"]]], "]"]]]]]], ")"]]]], "-", RowBox[List[SqrtBox[RowBox[List["a", " ", "z"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["a", " ", "z"]], "+", RowBox[List["2", " ", "b"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["k", "-", "n"]]]], "]"]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["1", "+", "n"]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["k", "-", "n"]]]], "]"]]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", "z"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["k", "-", "n"]]]], "]"]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["1", "+", "n"]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["k", "-", "n"]]]], "]"]]]]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]]]

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

 2002-12-18