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 Csch

 http://functions.wolfram.com/01.23.21.0263.01

 Input Form

 Integrate[E^(p z) Sin[a z]^m Coth[c z]^u Csch[c z], z] == 2^(-m - u) (1 - E^(2 c z))^(u + 1) Binomial[m, m/2] Csch[c z]^(u + 1) (1 - Mod[m, 2]) ((1/(p + c (u + 1))) (E^(p z) Binomial[u, u/2] HypergeometricPFQ[{(c (u + 1) + p)/(2 c), u + 1}, {1 + (c (u + 1) + p)/(2 c)}, E^(2 c z)] (1 - Mod[u, 2])) + Sum[Binomial[u, s] ((1/(p + c (2 s + 1))) (E^((p - c (-2 s + u)) z) HypergeometricPFQ[{(c (2 s + 1) + p)/(2 c), u + 1}, {1 + (c (2 s + 1) + p)/(2 c)}, E^(2 c z)]) + (E^((p + c (-2 s + u)) z) HypergeometricPFQ[ {(c (2 u - 2 s + 1) + p)/(2 c), u + 1}, {1 + (c (2 u - 2 s + 1) + p)/(2 c)}, E^(2 c z)])/ (p + c (-2 s + 2 u + 1))), {s, 0, Floor[(1/2) (-1 + u)]}]) + 2^(-m - u) (1 - E^(2 c z))^(u + 1) Csch[c z]^(u + 1) Sum[(-1)^k Binomial[m, k] (E^((I m Pi)/2) ((E^(((-I) a (-2 k + m) + p) z) Binomial[u, u/2] HypergeometricPFQ[ {(c (u + 1) + ((-I) a (-2 k + m) + p))/(2 c), u + 1}, {1 + (c (u + 1) + ((-I) a (-2 k + m) + p))/(2 c)}, E^(2 c z)] (1 - Mod[u, 2]))/((-I) a (-2 k + m) + p + c (u + 1)) + Sum[Binomial[u, s] ((E^(((-I) a (-2 k + m) + p - c (-2 s + u)) z) HypergeometricPFQ[{(c (2 s + 1) + ((-I) a (-2 k + m) + p))/ (2 c), u + 1}, {1 + (c (2 s + 1) + ((-I) a (-2 k + m) + p))/ (2 c)}, E^(2 c z)])/((-I) a (-2 k + m) + p + c (2 s + 1)) + (E^(((-I) a (-2 k + m) + p + c (-2 s + u)) z) HypergeometricPFQ[{ (c (2 u - 2 s + 1) + ((-I) a (-2 k + m) + p))/(2 c), u + 1}, { 1 + (c (2 u - 2 s + 1) + ((-I) a (-2 k + m) + p))/(2 c)}, E^ (2 c z)])/((-I) a (-2 k + m) + p + c (-2 s + 2 u + 1))), {s, 0, Floor[(1/2) (-1 + u)]}]) + ((E^((I a (-2 k + m) + p) z) Binomial[u, u/2] HypergeometricPFQ[ {(c (u + 1) + (I a (-2 k + m) + p))/(2 c), u + 1}, {1 + (c (u + 1) + (I a (-2 k + m) + p))/(2 c)}, E^(2 c z)] (1 - Mod[u, 2]))/(I a (-2 k + m) + p + c (u + 1)) + Sum[Binomial[u, s] ((E^((I a (-2 k + m) + p - c (-2 s + u)) z) HypergeometricPFQ[{(c (2 s + 1) + (I a (-2 k + m) + p))/(2 c), u + 1}, {1 + (c (2 s + 1) + (I a (-2 k + m) + p))/(2 c)}, E^ (2 c z)])/(I a (-2 k + m) + p + c (2 s + 1)) + (E^((I a (-2 k + m) + p + c (-2 s + u)) z) HypergeometricPFQ[{ (c (2 u - 2 s + 1) + (I a (-2 k + m) + p))/(2 c), u + 1}, { 1 + (c (2 u - 2 s + 1) + (I a (-2 k + m) + p))/(2 c)}, E^ (2 c z)])/(I a (-2 k + m) + p + c (-2 s + 2 u + 1))), {s, 0, Floor[(1/2) (-1 + u)]}])/E^((1/2) I m Pi)), {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0 && Element[u, Integers] && u > 0

 Standard Form

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 MathML Form

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( u + 1 ) c + ( p - a ( m - 2 k ) ) 2 c + 1 ; 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[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]], " ", "c"]], "+", RowBox[List["(", RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["u", "+", "1"]], ")"]], " ", "c"]], "+", RowBox[List["(", RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ( 1 - u mod 2 \$CellContext`u 2 ) + s = 0 u - 1 2 ( u s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( - a ( m - 2 k ) + p - c ( u - 2 s ) ) z ( 2 s + 1 ) c - a ( m - 2 k ) + p 2 F 1 ( ( 2 s + 1 ) c + ( p - a ( m - 2 k ) ) 2 c , u + 1 ; ( 2 s + 1 ) c + ( p - a ( m - 2 k ) ) 2 c + 1 ; 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[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "+", "1"]], ")"]], " ", "c"]], "+", RowBox[List["(", RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "+", "1"]], ")"]], " ", "c"]], "+", RowBox[List["(", RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + ( ( - a ( m - 2 k ) + p + c ( u - 2 s ) ) z 2 F 1 ( ( - 2 s + 2 u + 1 ) c + ( p - a ( m - 2 k ) ) 2 c , u + 1 ; 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n m + u + Condition z p z a z m c z u c z 2 -1 m -1 u 1 -1 2 c z u 1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 p z u 1 c p -1 Binomial u u 2 -1 HypergeometricPFQ u 1 c p 2 c -1 u 1 u 1 c p 2 c -1 1 2 c z 1 -1 \$CellContext`u 2 s 0 u -1 2 -1 Binomial u s p -1 c u -1 2 s z 2 s 1 c p -1 HypergeometricPFQ 2 s 1 c p 2 c -1 u 1 2 s 1 c p 2 c -1 1 2 c z p c u -1 2 s z -2 s 2 u 1 c p -1 HypergeometricPFQ -2 s 2 u 1 c p 2 c -1 u 1 -2 s 2 u 1 c p 2 c -1 1 2 c z c z u 1 2 -1 m -1 u 1 -1 2 c z u 1 k 0 m -1 2 -1 -1 k Binomial m k m 2 -1 p -1 a m -1 2 k z u 1 c -1 a m -1 2 k p -1 Binomial u u 2 -1 HypergeometricPFQ u 1 c p -1 a m -1 2 k 2 c -1 u 1 u 1 c p -1 a m -1 2 k 2 c -1 1 2 c z 1 -1 \$CellContext`u 2 s 0 u -1 2 -1 Binomial u s -1 a m -1 2 k p -1 c u -1 2 s z 2 s 1 c -1 a m -1 2 k p -1 HypergeometricPFQ 2 s 1 c p -1 a m -1 2 k 2 c -1 u 1 2 s 1 c p -1 a m -1 2 k 2 c -1 1 2 c z -1 a m -1 2 k p c u -1 2 s z HypergeometricPFQ -2 s 2 u 1 c p -1 a m -1 2 k 2 c -1 u 1 -2 s 2 u 1 c p -1 a m -1 2 k 2 c -1 1 2 c z -2 s 2 u 1 c -1 a m -1 2 k p -1 -1 1 2 m a m -1 2 k p z u 1 c a m -1 2 k p -1 Binomial u u 2 -1 HypergeometricPFQ u 1 c a m -1 2 k p 2 c -1 u 1 u 1 c a m -1 2 k p 2 c -1 1 2 c z 1 -1 \$CellContext`u 2 s 0 u -1 2 -1 Binomial u s a m -1 2 k p -1 c u -1 2 s z 2 s 1 c a m -1 2 k p -1 HypergeometricPFQ 2 s 1 c a m -1 2 k p 2 c -1 u 1 2 s 1 c a m -1 2 k p 2 c -1 1 2 c z a m -1 2 k p c u -1 2 s z HypergeometricPFQ -2 s 2 u 1 c a m -1 2 k p 2 c -1 u 1 -2 s 2 u 1 c a m -1 2 k p 2 c -1 1 2 c z -2 s 2 u 1 c a m -1 2 k p -1 c z u 1 n m SuperPlus u SuperPlus [/itex]

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

 2002-12-18

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