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 CoshIntegral

 http://functions.wolfram.com/06.40.21.0014.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", "z"]]], " ", RowBox[List["CoshIntegral", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "b"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["n", "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]], "+", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "z"]], "]"]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "n"], RowBox[List[FractionBox["1", "m"], " ", SuperscriptBox[RowBox[List["(", FractionBox["b", RowBox[List["b", "-", "a"]]], ")"]], "m"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "m"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "k"], " ", SuperscriptBox["z", "k"]]], RowBox[List["k", "!"]]]]]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "n"], RowBox[List[FractionBox["1", "m"], SuperscriptBox[RowBox[List["(", FractionBox["b", RowBox[List["a", "+", "b"]]], ")"]], "m"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "m"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", "b"]], ")"]], "k"], " ", SuperscriptBox["z", "k"]]], RowBox[List["k", "!"]]]]]]]]]]], "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", "z"]]], " ", RowBox[List["CoshIntegral", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], " ", "z"]], ")"]], "k"], RowBox[List["k", "!"]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n b z Chi ( a z ) z 1 2 ( - b ) - n - 1 n ! ( Ei ( ( b - a ) z ) + Ei ( ( a + b ) z ) - 2 b z Chi ( a z ) k = 0 n ( - b z ) k k ! - ( a + b ) z m = 1 n 1 m ( b a + b ) m k = 0 m - 1 ( - a - b ) k z k k ! - ( b - a ) z m = 1 n 1 m ( b b - a ) m k = 0 m - 1 ( a - b ) k z k k ! ) /; n Condition z z n b z CoshIntegral a z 1 2 -1 b -1 n -1 n ExpIntegralEi b -1 a z ExpIntegralEi a b z -1 2 b z CoshIntegral a z k 0 n -1 b z k k -1 -1 a b z m 1 n 1 m -1 b a b -1 m k 0 m -1 -1 a -1 b k z k k -1 -1 b -1 a z m 1 n 1 m -1 b b -1 a -1 m k 0 m -1 a -1 b k z k k -1 n [/itex]

 Rule Form

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

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

 2001-10-29