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; }

 SinhIntegral

 http://functions.wolfram.com/06.39.21.0039.01

 Input Form

 Integrate[z^n Cosh[a z] SinhIntegral[a z], z] == (1/((-a)^n (4 a))) (2 (Gamma[1 + n, (-a) z] - (-1)^n Gamma[1 + n, a z]) SinhIntegral[a z] - n! ((-1)^n ExpIntegralEi[-2 a z] + ExpIntegralEi[2 a z] - (1 + (-1)^n) Log[z] - 2 Sum[(((-a) z)^k/(2 k) + 2^(-1 - k) Gamma[k, -2 a z])/k!, {k, 1, n}] - 2 (-1)^n Sum[((a z)^k/(2 k) + 2^(-1 - k) Gamma[k, 2 a z])/k!, {k, 1, n}])) /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], " ", RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "a"]], ")"]], RowBox[List["-", "n"]]], RowBox[List["4", "a"]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "a"]], " ", "z"]]]], "]"]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["a", " ", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", "z"]], "]"]]]], "+", RowBox[List["ExpIntegralEi", "[", RowBox[List["2", " ", "a", " ", "z"]], "]"]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], " ", ")"]], RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", "z"]], ")"]], "k"], RowBox[List["2", " ", "k"]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", "z"]]]], "]"]]]]]], RowBox[List["k", "!"]]]]]]], "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], "k"], RowBox[List["2", " ", "k"]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", ",", RowBox[List["2", " ", "a", " ", "z"]]]], "]"]]]]]], RowBox[List["k", "!"]]]]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n cosh ( a z ) Shi ( a z ) z ( - a ) - n 4 a ( 2 ( Γ ( n + 1 , - a z ) - ( - 1 ) n Γ ( n + 1 , a z ) ) Shi ( a z ) - n ! ( ( - 1 ) n Ei ( - 2 a z ) + Ei ( 2 a z ) - ( 1 + ( - 1 ) n ) log ( z ) - 2 k = 1 n 1 k ! ( ( - a z ) k 2 k + 2 - k - 1 Γ ( k , - 2 a z ) ) - 2 ( - 1 ) n k = 1 n 1 k ! ( ( a z ) k 2 k + 2 - k - 1 Γ ( k , 2 a z ) ) ) ) /; n Condition z z n a z SinhIntegral a z -1 a -1 n 4 a -1 2 Gamma n 1 -1 a z -1 -1 n Gamma n 1 a z SinhIntegral a z -1 n -1 n ExpIntegralEi -2 a z ExpIntegralEi 2 a z -1 1 -1 n z -1 2 k 1 n 1 k -1 -1 a z k 2 k -1 2 -1 k -1 Gamma k -2 a z -1 2 -1 n k 1 n 1 k -1 a z k 2 k -1 2 -1 k -1 Gamma k 2 a z 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["a_", " ", "z_"]], "]"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "a"]], ")"]], RowBox[List["-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "a"]], " ", "z"]]]], "]"]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["a", " ", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", "z"]], "]"]]]], "+", RowBox[List["ExpIntegralEi", "[", RowBox[List["2", " ", "a", " ", "z"]], "]"]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", "z"]], ")"]], "k"], RowBox[List["2", " ", "k"]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", "z"]]]], "]"]]]]]], RowBox[List["k", "!"]]]]]]], "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], "k"], RowBox[List["2", " ", "k"]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", ",", RowBox[List["2", " ", "a", " ", "z"]]]], "]"]]]]]], RowBox[List["k", "!"]]]]]]]]], ")"]]]]]], ")"]]]], RowBox[List["4", " ", "a"]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29