Exponential function: Integration (formula 01.03.21.0762)

Exp

$Mathematica Notation$

Elementary Functions

Exp[z]

Integration

Indefinite integration

Involving functions of the direct function and a power function

Involving products of powers of the direct function and a power function

Involving product of powers of two direct functions and a power function

Involving zⁿ(e^{d z+e})^mu (e^{c z^r+f z+g})^nu

http://functions.wolfram.com/01.03.21.0762.01

Input Form

Integrate[z^n (E^(d z + e))^\[Mu] (E^(c Sqrt[z] + f z + g))^\[Nu], z] == (2^(-2 n - 1) E^((-d) z \[Mu] - c Sqrt[z] \[Nu] - f z \[Nu] - (c^2 \[Nu]^2)/(4 (d \[Mu] + f \[Nu]))) (E^(e + d z))^\[Mu] (E^(g + c Sqrt[z] + f z))^\[Nu] Sum[(-1)^(-h + k) 4^k (c \[Nu])^(-h - k + 2 n) (c \[Nu] + 2 Sqrt[z] (d \[Mu] + f \[Nu]))^(h + k) (-((c \[Nu] + 2 Sqrt[z] (d \[Mu] + f \[Nu]))^2/(d \[Mu] + f \[Nu])))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (c \[Nu] (c \[Nu] + 2 Sqrt[z] (d \[Mu] + f \[Nu])) Gamma[(1/2) (1 + h + k), -((c \[Nu] + 2 Sqrt[z] (d \[Mu] + f \[Nu]))^ 2/(4 (d \[Mu] + f \[Nu])))] + 2 (d \[Mu] + f \[Nu]) Sqrt[-((c \[Nu] + 2 Sqrt[z] (d \[Mu] + f \[Nu]))^2/ (d \[Mu] + f \[Nu]))] Gamma[(1/2) (2 + h + k), -((c \[Nu] + 2 Sqrt[z] (d \[Mu] + f \[Nu]))^2/ (4 (d \[Mu] + f \[Nu])))]), {k, 0, n}, {h, 0, k}])/ (d \[Mu] + f \[Nu])^(2 (1 + n)) /; Element[n, Integers] && n >= 0

Standard Form

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MathML Form

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</mo> <mrow> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> + </mo> <mrow> <mi> f </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> + </mo> <mrow> <mi> f </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> + </mo> <mrow> <mi> f </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> h </mi> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> 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<apply> <times /> <ci> c </ci> <ci> ν </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> h </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> c </ci> <ci> ν </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> d </ci> <ci> μ </ci> </apply> <apply> <times /> <ci> f </ci> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> h </ci> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> c </ci> <ci> ν </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 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Rule Form

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

2002-12-18