Cosine: Integration (formula 01.07.21.0953)

Cos

$Mathematica Notation$

Elementary Functions

Cos[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving trigonometric and exponential functions

Involving powers of sin and exp

Involving e^{b z^r+e} sin^m(a z^r+q) cos(c z^r+g)

http://functions.wolfram.com/01.07.21.0953.01

Input Form

Integrate[E^(b z^2 + e) Sin[a z^2 + q]^m Cos[c z^2 + g], z] == 2^(-2 - m) E^(e - I g) Sqrt[Pi] Binomial[m, m/2] (Erfi[Sqrt[b - I c] z]/Sqrt[b - I c] + (E^(2 I g) Erfi[Sqrt[b + I c] z])/ Sqrt[b + I c]) (1 - Mod[m, 2]) + 2^(-2 - m) Sqrt[Pi] Sum[(-1)^k Binomial[m, k] ((E^(e - I g + (I m Pi)/2 - I (-2 k + m) q) Erfi[Sqrt[b - I c - I a (-2 k + m)] z])/ Sqrt[b - I c - I a (-2 k + m)] + (E^(e + I g + (I m Pi)/2 - I (-2 k + m) q) Erfi[Sqrt[b + I c - I a (-2 k + m)] z])/ Sqrt[b + I c - I a (-2 k + m)] + (E^(e - I g - (I m Pi)/2 + I (-2 k + m) q) Erfi[Sqrt[b - I c + I a (-2 k + m)] z])/ Sqrt[b - I c + I a (-2 k + m)] + (E^(e + I g - (I m Pi)/2 + I (-2 k + m) q) Erfi[Sqrt[b + I c + I a (-2 k + m)] z])/ Sqrt[b + I c + I a (-2 k + m)]), {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0

Standard Form

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MathML Form

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2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> e </ci> </apply> </apply> <apply> <power /> <apply> <sin /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> q </ci> </apply> </apply> <ci> m </ci> </apply> <apply> <cos /> <apply> <plus /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> g </ci> </apply> 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Rule Form

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

2002-12-18