Introductions to ArithmeticGeometricMean

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1 Intoductions to AitheticGeoeticMen Intoduction to the Aithetic-Geoetic Men Genel The ithetic-geoetic en eed in the woks of J Lnden (77, 775) nd J-L Lgnge ( ) who defined it though the following quite-ntul liit ocedue: g, li n n li n n ; n n n g, ϑ 3, z n n n n g, ϑ 4, z n z q C F Guss (79 799, 8, 876) continued to esech this liit nd in 8 deived its eesenttion though the hyegeoetic function F, ; c; z Definition of ithetic-geoetic en The ithetic-geoetic en g, is defined though the eciocl vlue of the colete ellitic integl Kz y the foul: g, Π 4 K A quick look t the ithetic-geoetic en Hee is quick look t the ghic fo the ithetic-geoetic en ove the el -lne Re I GhicsAy Connections within the ithetic-geoetic en gou nd with othe function gous Reesenttions though oe genel functions

2 htt://functionswolfco The ithetic-geoetic en g, cn e eesented though the eciocl function of the ticul cses of hyegeoetic nd Meije G functions: g, Π g, F, ; ; G,,,, Reesenttions though elted equivlent functions The definition of the ithetic-geoetic en g, cn e inteeted s eesenttion of g, though elted equivlent functions the eciocl of the colete ellitic integl Kz with z : g, Π 4 K The est-known oeties nd fouls fo the ithetic-geoetic en Vlues in oints The ithetic-geoetic en g, cn e exctly evluted in soe oints, fo exle: g, g, Π g, K g, Π 3 4 g, ϑ 4, z ϑ 3, z ; z ϑ 3, z g, Rel vlues fo el guents Fo el vlues of guents, (with ), the vlues of the ithetic-geoetic en g, e el Anlyticity The ithetic-geoetic en g, is n nlyticl function of nd tht is defined ove Poles nd essentil singulities The ithetic-geoetic en g, does not hve oles nd essentil singulities

3 htt://functionswolfco 3 Bnch oints nd nch cuts The ithetic-geoetic en g, on the -lne hs two nch oints: nd It is single-vlued function on the -lne cut long the intevl,, whee it is continuous fo ove: li g Ε, g, ; Ε li g Ε, g, Ε ; Peiodicity The ithetic-geoetic en g, does not hve eiodicity Pity nd syeties The ithetic-geoetic en g, is n odd function nd hs io nd euttion syety: g, g, ; g, g, ;, g, g, The ithetic-geoetic en g, is the hoogenous function: gc, c c g, ; c Seies eesenttions The ithetic-geoetic en g, hs the following seies eesenttions t the oints,, nd : Π g, log4 log Π log 4 ; 8 log4 log g, ; 6 g, Π log 4 Π log 4 8 log 4 ; Poduct eesenttion The ithetic-geoetic en g, hs the following infinite oduct eesenttion:

4 htt://functionswolfco 4 g, q k ; q q k k Integl eesenttion q k q k The ithetic-geoetic en g, hs the following integl eesenttion: g, Π Π cos t sin t t ; Liit eesenttion The ithetic-geoetic en g, hs the following liit eesenttion, which is often used fo the definition of g, : g, li n n li n n ; n n n g, ϑ 3, z n n n n g, ϑ 4, z n z q Tnsfotions The hoogeneity oety of the ithetic-geoetic en g, leds to the following tnsfotions: g, g, ; gc, c c g, ; c g, g, ; g, z g, z ; Anothe gou of tnsfotions is sed on the fist of the following oeties: g, g, g, z gz, z g, g, Reesenttions of deivtives The fist deivtives of the ithetic-geoetic en g, hve the sile eesenttions:

5 htt://functionswolfco 5 g, g, Π Π g, E g, g, g, E Π Π The n th -ode syolic deivtives e uch oe colicted Hee is n exle: n g, g, n Π n 4 n n K n q n n q n q K q q n j k k j n j k j k n j k j k q q n j kj k q q Ak i,, A n k j,, i j n q n q n q K q q n j k j n j k j k k n j k j k q q n j kj k q Ak i,, A n k j,, ; A,, K i q j Π s s s s s F, ; s; q q q q u u u q q q,i ui u u u q ; u, u,, u q i ui u i u i u i u i ; n Diffeentil equtions The ithetic-geoetic en g, stisfies the following second-ode odiny nonline diffeentil eqution: w w 3 w w w ; w g, It cn lso e eesented s til solutions of the following til diffeentil eqution: g, g, g, Inequlities The ithetic-geoetic en g, lies etween the iddle geoetic en nd iddle ithetic en, which is shown in the following fous inequlity: g,

6 htt://functionswolfco 6 Alictions of the ithetic-geoetic en Alictions of the ithetic-geoetic en include fst high-ecision couttion of Π, logz, z, sinz, cosz, nd so on

7 htt://functionswolfco 7 Coyight This docuent ws downloded fo functionswolfco, coehensive online coendiu of fouls involving the secil functions of thetics Fo key to the nottions used hee, see htt://functionswolfco/nottions/ Plese cite this docuent y efeing to the functionswolfco ge fo which it ws downloded, fo exle: htt://functionswolfco/constnts/e/ To efe to ticul foul, cite functionswolfco followed y the cittion nue eg: htt://functionswolfco/33 This docuent is cuently in eliiny fo If you hve coents o suggestions, lese eil coents@functionswolfco -8, Wolf Resech, Inc

On Some Hadamard-Type Inequalıtıes for Convex Functıons

On Some Hadamard-Type Inequalıtıes for Convex Functıons Aville t htt://vuedu/ Al Al Mth ISSN: 93-9466 Vol 9, Issue June 4, 388-4 Alictions nd Alied Mthetics: An Intentionl Jounl AAM On Soe Hdd-Tye Inequlıtıes o, Convex Functıons M Ein Özdei Detent o Mthetics