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variants of this functions
Fibonacci




Mathematica Notation

Traditional Notation









Hypergeometric Functions > Fibonacci[nu,z] > Integral transforms > Laplace transforms





http://functions.wolfram.com/07.06.22.0001.01









  


  








Input Form





LaplaceTransform[Fibonacci[t, z], t, w] == (1/(2 Sqrt[z^2 + 4])) (2/(w - Log[(z + Sqrt[z^2 + 4])/2]) - 1/(w - Pi I + Log[(z + Sqrt[z^2 + 4])/2]) - 1/(w + Pi I + Log[(z + Sqrt[z^2 + 4])/2])) /; Re[w] > Abs[Log[(z + Sqrt[z^2 + 4])/2]]








Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msub> <mi> &#8466; </mi> <mi> t </mi> </msub> <mo> [ </mo> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> t </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 4 </mn> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 2 </mn> <mrow> <mi> w </mi> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 4 </mn> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> w </mi> <mo> + </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 4 </mn> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> w </mi> <mo> - </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 4 </mn> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LaplaceTransform", "[", RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["t_", ",", "z_"]], "]"]], ",", "t_", ",", "w_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[FractionBox["2", RowBox[List["w", "-", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "4"]]]]], ")"]]]], "]"]]]]], "-", FractionBox["1", RowBox[List["w", "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "+", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "4"]]]]], ")"]]]], "]"]]]]], "-", FractionBox["1", RowBox[List["w", "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "+", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "4"]]]]], ")"]]]], "]"]]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "4"]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "w", "]"]], ">", RowBox[List["Abs", "[", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "4"]]]]], ")"]]]], "]"]], "]"]]]]]]]]]]


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





2001-10-29