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 ArithmeticGeometricMean

 http://functions.wolfram.com/09.54.06.0001.01

 Input Form

 ArithmeticGeometricMean[1, b] \[Proportional] Pi/(2 (Log[4] - Log[b])) + ((Pi (1 + Log[b/4]))/(8 (Log[4] - Log[b])^2)) b^2 + (b^4 Pi (8 - 40 Log[2]^2 + Log[1024] + 5 (-1 + Log[256] - 2 Log[b]) Log[b]))/(256 (Log[4] - Log[b])^3) + (b^6 Pi (24 + 4 Log[2] (27 + (23 - 132 Log[2]) Log[2]) + Log[b] (-54 - 92 Log[2] + 792 Log[2]^2 + Log[b] (23 - 396 Log[2] + 66 Log[b]))))/(3072 (Log[4] - Log[b])^4) + (b^8 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 (Log[4] - Log[b])^5) + (b^10 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 Log[2] (-17153 + 413700 Log[2]) + Log[b] (17153 - 827400 Log[2] + 82740 Log[b]))))))/ (7864320 (Log[4] - Log[b])^6) + O[b^12/Log[b]] /; (b -> 0)

 Standard Form

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 MathML Form

 agm ArithmeticGeometricMean ( 1 , b ) π 2 ( log ( 4 ) - log ( b ) ) + ( π ( log ( b 4 ) + 1 ) ) b 2 8 ( log ( 4 ) - log ( b ) ) 2 + ( b 4 π ( - 40 log 2 ( 2 ) + log ( 1024 ) + 5 log ( b ) ( log ( 256 ) - 2 log ( b ) - 1 ) + 8 ) ) / ( 256 ( log ( 4 ) - log ( b ) ) 3 ) + ( b 6 π ( 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 ) ) / ( 3072 ( log ( 4 ) - log ( b ) ) 4 ) + ( b 8 π ( 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 ) ) ) ) / ( 393216 ( log ( 4 ) - log ( b ) ) 5 ) + ( b 10 π ( 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 ) ( 413700 log ( 2 ) - 17153 ) + log ( b ) ( - 827400 log ( 2 ) + 82740 log ( b ) + 17153 ) - 63130 ) ) ) ) ) / ( 7864320 ( log ( 4 ) - log ( b ) ) 6 ) + O ( b 12 log ( b ) ) /; ( b "\[Rule]" 0 ) Condition Proportional ArithmeticGeometricMean 1 b 2 4 -1 b -1 b 4 -1 1 b 2 8 4 -1 b 2 -1 b 4 -40 2 2 1024 5 b 256 -1 2 b -1 8 256 4 -1 b 3 -1 b 6 4 2 2 23 -1 132 2 27 b 792 2 2 -1 92 2 b -396 2 66 b 23 -54 24 3072 4 -1 b 4 -1 b 8 8 2 2 2 1487 -1 11256 2 2230 744 96 b 4 2 45024 2 2 -1 4461 2 -4460 -744 b 6 2 1487 -1 22512 2 b 45024 2 -1 5628 b -1487 4460 393216 4 -1 b 5 -1 b 10 16 2 2 2 2 17153 -1 165480 2 31565 12765 2640 240 b 8 2 2 827400 2 2 -1 68612 2 -94695 -25530 -2640 b 12 2 -551600 2 2 34306 2 31565 4255 b 8 2 413700 2 -17153 b -827400 2 82740 b 17153 -63130 7864320 4 -1 b 6 -1 O b 12 b -1 Rule b 0 [/itex]

 Rule Form

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

 2001-10-29