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Sinh






Mathematica Notation

Traditional Notation









Elementary Functions > Sinh[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving trigonometric and exponential functions > Involving sin and exp > Involving ep z sin(b zr)sinh(c zr)





http://functions.wolfram.com/01.19.21.1194.01









  


  










Input Form





Integrate[E^(p z) Sin[b z^2] Sinh[c z^2], z] == (1/8) (-1)^(1/4) Sqrt[Pi] ((I E^((I p^2)/(-4 b + 4 I c)) Erfi[((-1)^(3/4) (I p + 2 (b - I c) z))/ (2 Sqrt[b - I c])])/Sqrt[b - I c] + (E^((I p^2)/(4 (b + I c))) Erfi[((-1)^(1/4) ((-I) p + 2 b z + 2 I c z))/ (2 Sqrt[b + I c])])/Sqrt[b + I c] - (I Erfi[((-1)^(3/4) (I p + 2 b z + 2 I c z))/(2 Sqrt[b + I c])])/ (E^((I p^2)/(4 (b + I c))) Sqrt[b + I c]) + (E^((I p^2)/(4 b - 4 I c)) Erfi[((-1)^(3/4) (p + 2 I b z + 2 c z))/ (2 Sqrt[b - I c])])/Sqrt[b - I c])










Standard Form





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MathML Form







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</apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", RowBox[List["Sin", "[", RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "8"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "+", RowBox[List["4", " ", "\[ImaginaryI]", " ", "c"]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]]]], "]"]]]], SqrtBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "p"]], "+", RowBox[List["2", " ", "b", " ", "z"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]]]], "]"]]]], SqrtBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", "b", " ", "z"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]]]], "]"]]]], SqrtBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List[RowBox[List["4", " ", "b"]], "-", RowBox[List["4", " ", "\[ImaginaryI]", " ", "c"]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]], "+", RowBox[List["2", " ", "c", " ", "z"]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]]]], "]"]]]], SqrtBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]]]], ")"]]]]]]]]










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





2002-12-18