Exponential function: Integration (formula 01.03.21.0734)

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})^mu (e^{c z^r+g})^nu

http://functions.wolfram.com/01.03.21.0734.01

Input Form

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

Standard Form

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MathML Form

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</ci> <ci> h </ci> </apply> <apply> <ci> Binomial </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <plus /> <apply> <times /> <ci> c </ci> <ci> ν </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> d </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> μ </ci> </apply> <apply> <times /> <ci> c </ci> <ci> ν </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> h </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> d </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> μ </ci> </apply> <apply> <times /> <ci> c </ci> <ci> ν </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> d </ci> <ci> μ </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> d </ci> <ci> μ </ci> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> d </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> μ </ci> </apply> <apply> <times /> <ci> c </ci> <ci> ν </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> d </ci> <ci> μ </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> h </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> d </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> μ </ci> </apply> <apply> <times /> <ci> c </ci> <ci> ν </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> d </ci> <ci> μ </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2002-12-18