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; }

 Sech

 http://functions.wolfram.com/01.24.21.0209.01

 Input Form

 Integrate[Sin[a z] Tanh[c z] Sech[c z], z] == E^(2 c z) (-((1/((-I) a + c)) (E^((I Pi)/2 + ((-I) a - c) z) HypergeometricPFQ[{2, 1/2 - (I a)/(2 c)}, {3/2 - (I a)/(2 c)}, -E^(2 c z)])) + (1/((-I) a + 3 c)) (E^((I Pi)/2 + ((-I) a + c) z) HypergeometricPFQ[{2, 3/2 - (I a)/(2 c)}, {5/2 - (I a)/(2 c)}, -E^(2 c z)]) - (1/(I a + c)) (E^(-((I Pi)/2) + (I a - c) z) HypergeometricPFQ[{2, 1/2 + (I a)/(2 c)}, {3/2 + (I a)/(2 c)}, -E^(2 c z)]) + (1/(I a + 3 c)) (E^(-((I Pi)/2) + (I a + c) z) HypergeometricPFQ[{2, 3/2 + (I a)/(2 c)}, {5/2 + (I a)/(2 c)}, -E^(2 c z)]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Sin", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[FractionBox["1", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "c"]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "-", "c"]], ")"]], " ", "z"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]]]]]], "+", RowBox[List[FractionBox["1", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", RowBox[List["3", " ", "c"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "c"]], ")"]], " ", "z"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List[FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "c"]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "c"]], ")"]], " ", "z"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", RowBox[List["3", " ", "c"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "c"]], ")"]], " ", "z"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 sin ( a z ) tanh ( c z ) sech ( c z ) z 2 c z ( - - 1 2 ( π ) + ( a - c ) z 2 F 1 ( 2 , 1 2 + a 2 c ; 3 2 + a 2 c ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["2", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox["1", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["3", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] c + a + - 1 2 ( π ) + ( c + a ) z 2 F 1 ( 2 , 3 2 + a 2 c ; 5 2 + a 2 c ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["2", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox["3", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["5", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] 3 c + a - π 2 + ( - c - a ) z 2 F 1 ( 2 , 1 2 - a 2 c ; 3 2 - a 2 c ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["2", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] c - a + π 2 + ( c - a ) z 2 F 1 ( 2 , 3 2 - a 2 c ; 5 2 - a 2 c ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["2", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["5", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] 3 c - a ) z a z c z c z 2 c z -1 -1 1 2 a -1 c z HypergeometricPFQ 2 1 2 a 2 c -1 3 2 a 2 c -1 -1 2 c z c a -1 -1 1 2 c a z HypergeometricPFQ 2 3 2 a 2 c -1 5 2 a 2 c -1 -1 2 c z 3 c a -1 -1 2 -1 -1 c -1 a z HypergeometricPFQ 2 1 2 -1 a 2 c -1 3 2 -1 a 2 c -1 -1 2 c z c -1 a -1 2 -1 c -1 a z HypergeometricPFQ 2 3 2 -1 a 2 c -1 5 2 -1 a 2 c -1 -1 2 c z 3 c -1 a -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Sin", "[", RowBox[List["a_", " ", "z_"]], "]"]], " ", RowBox[List["Tanh", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", RowBox[List["Sech", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "-", "c"]], ")"]], " ", "z"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "c"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "c"]], ")"]], " ", "z"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List[FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", RowBox[List["3", " ", "c"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "c"]], ")"]], " ", "z"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "c"]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "c"]], ")"]], " ", "z"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", RowBox[List["3", " ", "c"]]]]]]], ")"]]]]]]]]

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

 2002-12-18