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Erfi






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Erfi[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving hyperbolic functions and a power function > Involving sinh and power





http://functions.wolfram.com/06.28.21.0079.01









  


  










Input Form





Integrate[z^3 Sinh[b z] Erfi[a z], z] == (1/(16 a^6 b^4 Sqrt[Pi])) (E^(-(b^2/(4 a^2)) - b z) (-48 a^5 b E^(b^2/(4 a^2) + a^2 z^2) - 4 a^3 b^3 E^(b^2/(4 a^2) + a^2 z^2) - 2 a b^5 E^(b^2/(4 a^2) + a^2 z^2) - 48 a^5 b E^(b^2/(4 a^2) + 2 b z + a^2 z^2) - 4 a^3 b^3 E^(b^2/(4 a^2) + 2 b z + a^2 z^2) - 2 a b^5 E^(b^2/(4 a^2) + 2 b z + a^2 z^2) - 24 a^5 b^2 E^(b^2/(4 a^2) + a^2 z^2) z - 4 a^3 b^4 E^(b^2/(4 a^2) + a^2 z^2) z + 24 a^5 b^2 E^(b^2/(4 a^2) + 2 b z + a^2 z^2) z + 4 a^3 b^4 E^(b^2/(4 a^2) + 2 b z + a^2 z^2) z - 8 a^5 b^3 E^(b^2/(4 a^2) + a^2 z^2) z^2 - 8 a^5 b^3 E^(b^2/(4 a^2) + 2 b z + a^2 z^2) z^2 + 8 a^6 E^(b^2/(4 a^2)) Sqrt[Pi] (6 + 6 b z + 3 b^2 z^2 + b^3 z^3 + E^(2 b z) (-6 + 6 b z - 3 b^2 z^2 + b^3 z^3)) Erfi[a z] + (48 a^6 + 12 a^4 b^2 + b^6) E^(b z) Sqrt[Pi] Erfi[b/(2 a) - a z] + 48 a^6 E^(b z) Sqrt[Pi] Erfi[b/(2 a) + a z] + 12 a^4 b^2 E^(b z) Sqrt[Pi] Erfi[b/(2 a) + a z] + b^6 E^(b z) Sqrt[Pi] Erfi[b/(2 a) + a z]))










Standard Form





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MathML Form







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z </mi> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mi> b </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 48 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 6 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mi> b </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mfrac> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <sinh /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 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<apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <ci> b </ci> </apply> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <ci> a </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <ci> b </ci> </apply> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <ci> a </ci> </apply> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erfi </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 48 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erfi </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





© 1998-2014 Wolfram Research, Inc.