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/01.24.21.0067.01

 Input Form

 Integrate[E^(p z) Cos[b z]^m Sech[c z], z] == Binomial[m, m/2] (1 - Mod[m, 2]) ((2^(1 - m) E^((c + p) z) Hypergeometric2F1[1, (c + p)/(2 c), (1/2) (3 + p/c), -E^(2 c z)])/ (c + p)) + 2^(1 - m) Sum[Binomial[m, s] ((E^((c + p - I b (m - 2 s)) z) Hypergeometric2F1[1, (c + p - I b (m - 2 s))/(2 c), (3 c + p - I b (m - 2 s))/(2 c), -E^(2 c z)])/(c + p - I b (m - 2 s)) + (E^((c + p + I b (m - 2 s)) z) Hypergeometric2F1[1, (c + p + I b (m - 2 s))/(2 c), (3 c + p + I b (m - 2 s))/(2 c), -E^(2 c z)])/(c + p + I b (m - 2 s))), {s, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0

 Standard Form

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 MathML Form

 p z cos m ( b z ) sech ( c z ) z 2 1 - m ( c + p ) z ( 1 - m mod 2 \$CellContext`m 2 ) c + p ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] 2 F 1 ( 1 , c + p 2 c ; 1 2 ( p c + 3 ) ; - 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["1", Hypergeometric2F1], ",", TagBox[FractionBox[RowBox[List["c", "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox["p", "c"], "+", "3"]], ")"]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] + 2 1 - m s = 0 m - 1 2 ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( c + p + b ( m - 2 s ) ) z c + p + b ( m - 2 s ) 2 F 1 ( 1 , c + p + b ( m - 2 s ) 2 c ; 3 c + p + b ( m - 2 s ) 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["1", Hypergeometric2F1], ",", TagBox[FractionBox[RowBox[List["c", "+", "p", "+", RowBox[List["b", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "+", "p", "+", RowBox[List["b", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] + ( c + p - b ( m - 2 s ) ) z c + p - b ( m - 2 s ) 2 F 1 ( 1 , c + p - b ( m - 2 s ) 2 c ; 3 c + p - b ( m - 2 s ) 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["1", Hypergeometric2F1], ",", TagBox[FractionBox[RowBox[List["c", "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) /; m + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] Condition z p z b z m c z 2 1 -1 m c p z 1 -1 \$CellContext`m 2 c p -1 Binomial m m 2 -1 Hypergeometric2F1 1 c p 2 c -1 1 2 p c -1 3 -1 2 c z 2 1 -1 m s 0 m -1 2 -1 Binomial m s c p b m -1 2 s z c p b m -1 2 s -1 Hypergeometric2F1 1 c p b m -1 2 s 2 c -1 3 c p b m -1 2 s 2 c -1 -1 2 c z c p -1 b m -1 2 s z c p -1 b m -1 2 s -1 Hypergeometric2F1 1 c p -1 b m -1 2 s 2 c -1 3 c p -1 b m -1 2 s 2 c -1 -1 2 c z m [/itex]

 Rule Form

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

 2002-12-18