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infi2HW12 - f x,y B r = x,y | x 2 y 2 ≤ r 2 “:egiked...

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12 'qn milibxz oeilb - 104281 - 2 itpi` .mixdva 12 : 00 dry cr -- 2008 lixt`l 10 :dybd jix`z A 4 lcebn xiip lr yibdl yi .qxewd ly mi`zd cg`l ec`n` oiipaa qt` dnewa dybdd :zxekfz sca jxev oi` .oeilib xtqne ,f"z ,zeny xexiaa oiivl `p .wcedn yibdl `p .mixqlw e` zeiwy `ll .xry 1 libxz :zedfd z` egiked Z 1 0 x p (log x ) m dx = ( - 1) m m ! ( p + 1) m +1 m N ,p 1 :mi`ad mialyd t"r n f ∂p n zeiwlgd zexfbpd mby e`xd . [0 , 1] × [1 , ) drevxa dtivx f ( x,p ) = x p - y egiked .1 .qt`zn x oda zecewpa wiicl ecitwd .drevx dze`a zetivx .uipail llka yeniy i"r m lr divwecpi`a zedfd z` egiked .2 2 libxz :`ad lxbhpi`d z` eayg I = Z 1 0 log(1 + x ) 1 + x 2 dx :dkxcd :xear I = I (1) miiwzny ewca .1 I ( α ) = Z α 0 log(1 + αx ) 1 + x 2 dx .mi`zn oalna I ( α ) xear miniiwzn uipail llk ly mi`pzdy ewca .2 .( α 1+ α 2 arctan α + log(1+ α 2 ) 2(1+ α 2 ) :daeyz) . ∂α I ( α ) z` eayg .3 aygl n"r α = 1 eaivde miieqn `l lxbhpi` i"r ∂α I ( α ) ly dnecw divwpetk I ( α ) z` e`vn .4 . I z` 3 libxz :onqp .ziy`xa dtivxe [ - M,M ] × [ - M,M ] reaixa ziliaxbhpi`
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Unformatted text preview: f ( x,y ) B r = ' ( x,y ) | x 2 + y 2 ≤ r 2 “ :egiked f (0 , 0) = lim r → + 1 πr 2 ZZ B r f ( x,y ) dxdy 1 4 libxz :lxbhpi`d oezp I = Z 1 Z 1-√ 1-y f ( x,y ) dxdy + Z 1 Z √ 4-y 2 1+ √ 1-y f ( x,y ) dxdy + Z 2 1 Z √ 4-y 2 f ( x,y ) dxdy . D divxbhpi`d megz z` exiv .1 :d`ad dxeva lxbhpi`d z` enyx ,xnelk .divxbhpi` xcq etilgd .2 Z b a ˆ Z β ( x ) α ( x ) f ( x,y ) dy ! dx eynzyd) lxbhpi`d ly xiyi aeyig i"re xeiva zeppeazd i"r :mikxc izya D ghy z` eayg .3 .( R √ a 2-x 2 dx = x 2 √ a 2-x 2 + a 2 2 arcsin ( x 2 ) :`gqepa 5 libxz :lxbhpi`d oezp I = ZZ [0 , 1] × [0 , 1] f ( x,y ) dxdy ghynd z` ze`xl eqp .eze` eayge miiw lxbhpi`d dnl exiaqd . f ( x,y ) = min { 1 , 2 y, 1 2 x } xy`k .xvepy 2...
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