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variants of this functions
Erf






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Erf[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving power, exponential and hyperbolic functions > Involving power, exp and cosh





http://functions.wolfram.com/06.25.21.0108.01









  


  










Input Form





Integrate[z^n E^(b z) Cosh[c z] Erf[a z], z] == (-(1/2)) (-b + c)^(-1 - n) Erf[a z] Gamma[1 + n, (-b + c) z] - (1/2) (-b - c)^(-1 - n) Erf[a z] Gamma[1 + n, (-b - c) z] - (1/(2 Sqrt[Pi])) (a (-b - c)^(-1 - n) E^((b + c)^2/(4 a^2)) n! Sum[(1/m!) ((-(b + c))^m (-a^2)^((1/2) (-1 - m)) Sum[Binomial[m, k] (-((b + c)/(2 Sqrt[-a^2])))^(m - k) (Sqrt[-a^2] z + (b + c)/(2 Sqrt[-a^2]))^(1 + k) (-(Sqrt[-a^2] z + (b + c)/(2 Sqrt[-a^2]))^2)^((1/2) (-1 - k)) Gamma[(1 + k)/2, -(Sqrt[-a^2] z + (b + c)/(2 Sqrt[-a^2]))^2], {k, 0, m}]), {m, 0, n}]) - (1/(2 Sqrt[Pi])) (a (-b + c)^(-1 - n) E^((b - c)^2/(4 a^2)) n! Sum[(1/m!) ((-(b - c))^m (-a^2)^((1/2) (-1 - m)) Sum[Binomial[m, k] (-((b - c)/(2 Sqrt[-a^2])))^(m - k) (Sqrt[-a^2] z + (b - c)/(2 Sqrt[-a^2]))^(1 + k) (-(Sqrt[-a^2] z + (b - c)/(2 Sqrt[-a^2]))^2)^((1/2) (-1 - k)) Gamma[(1 + k)/2, -(Sqrt[-a^2] z + (b - c)/(2 Sqrt[-a^2]))^2], {k, 0, m}]), {m, 0, n}]) /; Element[n, Integers] && n >= 0










Standard Form





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MathML Form







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1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <ci> m </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> 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</ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> 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Date Added to functions.wolfram.com (modification date)





2001-10-29