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; }

 Csc

 http://functions.wolfram.com/01.10.21.0060.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["b_", " ", "z_"]], "]"]], "m_"], " ", RowBox[List["Csc", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "p"]], "+", "c"]], RowBox[List["2", " ", "c"]]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "p"]], "+", "c"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List["p", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "s"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["p", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], ")"]], " ", "z"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]], "+", "c"]], RowBox[List["2", " ", "c"]]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]], "+", "c"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List["p", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "2"], "+", RowBox[List[RowBox[List["(", RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], ")"]], " ", "z"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "p"]], "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "p"]], "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

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

 2002-12-18