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 ArithmeticGeometricMean

 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

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

 Rule Form

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

 2001-10-29