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ArithmeticGeometricMean




Mathematica Notation

Traditional Notation









Elliptic Functions > ArithmeticGeometricMean[a,b] > Series representations > Generalized power series > Expansions at b==infinity for a==1





http://functions.wolfram.com/09.54.06.0005.01









  


  








Input Form





ArithmeticGeometricMean[1, b] \[Proportional] (Pi b)/(2 Log[4 b]) + (Pi (1 - Log[4 b]))/(8 Log[4 b]^2 b) + (Pi/(256 b^3 Log[4 b]^3)) (8 - 40 Log[2]^2 + Log[1024] - 5 Log[b] (-1 + Log[256] + 2 Log[b])) + (Pi (24 + 4 Log[2] (27 + (23 - 132 Log[2]) Log[2]) + Log[b] (54 + 92 Log[2] - 792 Log[2]^2 + (23 - 396 Log[2] - 66 Log[b]) Log[b])))/(3072 b^5 Log[4 b]^4) + (Pi (8 (96 + Log[2] (744 + Log[2] (2230 + (1487 - 11256 Log[2]) Log[2]))) + Log[b] (4 (744 + Log[2] (4460 + 4461 Log[2] - 45024 Log[2]^2)) + Log[b] (4460 + 6 (1487 - 22512 Log[2]) Log[2] + (1487 - 45024 Log[2] - 5628 Log[b]) Log[b]))))/ (393216 b^7 Log[4 b]^5) + (Pi (16 (240 + Log[2] (2640 + Log[2] (12765 + Log[2] (31565 + (17153 - 165480 Log[2]) Log[2])))) + Log[b] (8 (2640 + Log[2] (25530 + Log[2] (94695 + 68612 Log[2] - 827400 Log[2]^2))) + Log[b] (12 (4255 + Log[2] (31565 + 34306 Log[2] - 551600 Log[2]^2)) + Log[b] (63130 + 8 (17153 - 413700 Log[2]) Log[2] + (17153 - 827400 Log[2] - 82740 Log[b]) Log[b])))))/ (7864320 b^9 Log[4 b]^6) + O[1/(b^11 Log[b])] /; (Abs[b] -> Infinity)








Standard Form





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MathML Form







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</mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 12765 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 2640 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 240 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 827400 </mn> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 68612 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> 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</mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 4255 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 17153 </mn> <mo> - </mo> <mrow> <mn> 413700 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 827400 </mn> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 82740 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 17153 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 63130 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <msup> <mi> b </mi> <mrow> <mo> - </mo> <mn> 11 </mn> </mrow> </msup> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> b </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> ArithmeticGeometricMean </ci> <cn type='integer'> 1 </cn> <ci> b </ci> </apply> <apply> <plus /> <apply> <times /> <pi /> <ci> b </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> b </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <pi /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> b </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> b </ci> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 256 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -40 </cn> <apply> <power /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ln /> <cn type='integer'> 1024 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <ln /> <ci> b </ci> </apply> <apply> <plus /> <apply> <ln /> <cn type='integer'> 256 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 3072 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <cn type='integer'> 23 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 132 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 27 </cn> </apply> </apply> <apply> <times /> <apply> <ln /> <ci> b </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -792 </cn> <apply> <power /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 92 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <ln /> <ci> b </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -396 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 66 </cn> <apply> <ln /> <ci> b </ci> </apply> </apply> </apply> <cn type='integer'> 23 </cn> </apply> </apply> <cn type='integer'> 54 </cn> </apply> </apply> <cn type='integer'> 24 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 393216 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 7 </cn> </apply> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <plus /> <apply> <times /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <cn type='integer'> 1487 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 11256 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 2230 </cn> </apply> </apply> <cn type='integer'> 744 </cn> </apply> </apply> <cn type='integer'> 96 </cn> </apply> </apply> <apply> <times /> <apply> <ln /> <ci> b </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 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-45024 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5628 </cn> <apply> <ln /> <ci> b </ci> </apply> </apply> </apply> <cn type='integer'> 1487 </cn> </apply> </apply> <cn type='integer'> 4460 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 7864320 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 9 </cn> </apply> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> 6 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <plus /> <apply> <times /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <cn type='integer'> 17153 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 165480 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 31565 </cn> </apply> </apply> <cn type='integer'> 12765 </cn> </apply> </apply> <cn type='integer'> 2640 </cn> </apply> </apply> <cn type='integer'> 240 </cn> </apply> </apply> <apply> <times /> <apply> <ln /> <ci> b </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <plus /> <apply> <times /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -827400 </cn> <apply> <power /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 68612 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 94695 </cn> </apply> </apply> <cn type='integer'> 25530 </cn> </apply> </apply> <cn type='integer'> 2640 </cn> </apply> </apply> <apply> <times /> <apply> <ln /> <ci> b </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <plus /> <apply> <times /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -551600 </cn> <apply> <power /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 34306 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 31565 </cn> </apply> </apply> <cn type='integer'> 4255 </cn> </apply> </apply> <apply> <times /> <apply> <ln /> <ci> b </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <cn type='integer'> 17153 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 413700 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ln /> <ci> b </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -827400 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 82740 </cn> <apply> <ln /> <ci> b </ci> </apply> </apply> </apply> <cn type='integer'> 17153 </cn> </apply> </apply> <cn type='integer'> 63130 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> O </ci> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> -11 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> b </ci> 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Rule Form





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





2001-10-29