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Cos






Mathematica Notation

Traditional Notation









Elementary Functions > Cos[z] > Integration > Indefinite integration > Involving functions of the direct function > Involving products of the direct > Involving products of two direct functions > Involving cos(b zr+d z) cos(c zr+f z)





http://functions.wolfram.com/01.07.21.1086.01









  


  










Input Form





Integrate[Cos[b Sqrt[z] + d z] Cos[c Sqrt[z] + f z], z] == (1/2) ((-(1/(d - f)^(3/2))) ((b - c) Sqrt[Pi/2] (Cos[(b - c)^2/(4 (d - f))] FresnelC[(b - c + 2 (d - f) Sqrt[z])/ (Sqrt[d - f] Sqrt[2 Pi])] + FresnelS[(b - c + 2 (d - f) Sqrt[z])/(Sqrt[d - f] Sqrt[2 Pi])] Sin[(b - c)^2/(4 (d - f))])) - (1/(d + f)^(3/2)) ((b + c) Sqrt[Pi/2] (Cos[(b + c)^2/(4 (d + f))] FresnelC[(b + c + 2 (d + f) Sqrt[z])/(Sqrt[d + f] Sqrt[2 Pi])] + FresnelS[(b + c + 2 (d + f) Sqrt[z])/(Sqrt[d + f] Sqrt[2 Pi])] Sin[(b + c)^2/(4 (d + f))])) + Sin[(b - c) Sqrt[z] + (d - f) z]/ (d - f) + Sin[(b + c) Sqrt[z] + (d + f) z]/(d + f))










Standard Form





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MathML Form







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</apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <cos /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> d </ci> <ci> f </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> FresnelC </ci> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> d </ci> <ci> f </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> d </ci> <ci> f </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> FresnelS </ci> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> d </ci> <ci> f </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> d </ci> <ci> f </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> d </ci> <ci> f </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> d </ci> <ci> f </ci> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <sin /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> f </ci> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> f </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <sin /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> d </ci> <ci> f </ci> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> d </ci> <ci> f </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2002-12-18