LaplaceTransform[Fibonacci[t], t, z] == 1/(Sqrt[5] (z - ArcCsch[2])) - (z + ArcCsch[2])/(Sqrt[5] ((z + ArcCsch[2])^2 + Pi^2)) /; Re[z] > Log[GoldenRatio]

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<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> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> t </mi> </msub> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <msqrt> <mn> 5 </mn> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mrow> <msqrt> <mn> 5 </mn> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[List[], GoldenRatio]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> LaplaceTransform </ci> <apply> <ci> Fibonacci </ci> <ci> t </ci> </apply> <ci> t </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> z </ci> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <apply> <ln /> <ci> GoldenRatio </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29