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( F ´ ıs ica M a te m ´ atica I - FIS01207 - Unidade I ) ( F un ¸ c ˜ oes de V ar i ´ a v el Complexa: Lista 2 - FUN C ¸ O ˜ ES ANA L ´ ITICAS ) ( ) ( 1 ) ( 1 ) ( Pr o v e, diretame n te da de fi ni ¸ c ˜ a o de der i v ada, que f r ( z ) = − quando f ( z ) = e z / = 0. ) ( 1. ) ( 0 ) ( 0 ) ( z 2 ) ( z ) ( 0 ) ( Apli c ando d iretame n te a de fi ni ¸ c ˜ ao de d eri v ada, determine se existe f r ( z ) em algum p o n to para f ( z ) = Re ( z ). ) ( 2. ) ( 3. ) ( De t ermine se existe e qual ´ e a deri v ada das segui n tes fun ¸ c ˜ oes. ) ( a ) f ( z ) = 3 x + y + i (3 y − x ) f ( z ) = iz + 2 f ( z ) = x − iy d ) f ( z ) = 3 z 2 − 2 z + 4 e ) f ( z ) = e − x (cos y − i sin y ) f ) f ( z ) = z − z ¯ ) ( z − 1 ) ( g ) f ( z ) = ) ( 2 z + 1 ) ( f ( z ) = e x (cos y − i sin y ) f ( z ) = 2 x + ixy 2 j ) f ( z ) = x 3 − 3 xy 2 + i (3 x 2 y − y 3 ) k ) f ( z ) = z Im( z ) ) ( z 3 + i ) ( l ) f ( z ) = ) ( z 2 − 3 z + 2 ) ( f ( z ) = cos x cosh y − i sin x sinh y f ( z ) = x 2 + iy 2 ) ( 4. ) ( V erifique se u ´ e harm ˆ onica e a c he, e n t ˜ ao, uma conjugada harm ˆ onica v para: ) ( 3 2 a ) u = 2 x (1 − y ); b ) u = 2 x − x + 3 xy c ) u = sinh x sin y ; d ) u = x 2 y − y 3 Sej a f ( z ) = u ( r , θ ) + i v ( r , θ ) uma fun ¸ c ˜ ao ana l ´ ıtica n um certo do m ´ ınio D . Utilizando as condi ¸ c ˜ oes de Cau c h y-Riemann em c o ordenadas p olares, mostre que u e v satisfazem a equa ¸ c ˜ ao de Laplace ) ( 5. ) ( 2 ∂ 2 u ∂r 2 ) ( ∂u ∂ 2 u ) ( r ) ( + r + ∂r ) ( = 0 ) ( ∂θ 2 ) ( Respostas ) ( Questao 3: ) ( ( a ) f r ( z ) = 3 − i ) ( ( b ) f r ( z ) = i ( c ) N ˜ ao existe ) ( ( d ) f r ( z ) = 6 z − 2 ) ( ( e ) f r ( z ) = − e − x cos y + i ( e − x sen y ) ) ( ( f ) N ˜ ao existe 3 ) ( ( g ) f r ( z ) = ) ( (2 z +1) 2 ) ( N ˜ ao existe N ˜ ao existe ) ( ( j ) f r ( z ) = 3 x 2 − 3 y 2 + 6 ixy ) ( ( k ) N ˜ ao existe ) ( z 4 − 6 z 3 +6 z 2 − 2 iz +3 i ) ( ( l ) f r ( z ) = ) ( ( z 2 − 3 z +2) 2 ) ( ( m ) f r ( z ) = − sen x cosh y − i cos x senh y ) ( ( n ) N ˜ ao existe ) ( Questao 4: ) ( ( a ) v = x 2 − y 2 + 2 y + C ) ( ( b ) v = 2 y − 3 x 2 y + y 3 + C ) ( ( c ) v = − cosh x cos y + C ) ( ( d ) N ˜ ao ´ e harm ˆ onica )
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