Harmonic holomorphic function in Ω












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$Ω$ is simply connected in $C$, $u$ is a harmonic function in $Ω$ , $v$ in $Ω$



$$v(x,y) = int_0^1 (yu{Tiny x} (tx,ty)-xu{Tiny y} (tx,ty)) dt$$



Prove that there exists a holomorphic function $u+iv$ in $Ω$



I have the solution - can someone explain me the equalities to where i put the orange question marks? I dont understand why the first part of the term disappears.



Photo/solution










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    0














    $Ω$ is simply connected in $C$, $u$ is a harmonic function in $Ω$ , $v$ in $Ω$



    $$v(x,y) = int_0^1 (yu{Tiny x} (tx,ty)-xu{Tiny y} (tx,ty)) dt$$



    Prove that there exists a holomorphic function $u+iv$ in $Ω$



    I have the solution - can someone explain me the equalities to where i put the orange question marks? I dont understand why the first part of the term disappears.



    Photo/solution










    share|cite|improve this question

























      0












      0








      0







      $Ω$ is simply connected in $C$, $u$ is a harmonic function in $Ω$ , $v$ in $Ω$



      $$v(x,y) = int_0^1 (yu{Tiny x} (tx,ty)-xu{Tiny y} (tx,ty)) dt$$



      Prove that there exists a holomorphic function $u+iv$ in $Ω$



      I have the solution - can someone explain me the equalities to where i put the orange question marks? I dont understand why the first part of the term disappears.



      Photo/solution










      share|cite|improve this question













      $Ω$ is simply connected in $C$, $u$ is a harmonic function in $Ω$ , $v$ in $Ω$



      $$v(x,y) = int_0^1 (yu{Tiny x} (tx,ty)-xu{Tiny y} (tx,ty)) dt$$



      Prove that there exists a holomorphic function $u+iv$ in $Ω$



      I have the solution - can someone explain me the equalities to where i put the orange question marks? I dont understand why the first part of the term disappears.



      Photo/solution







      holomorphic-functions






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      asked Dec 4 '18 at 10:29









      malilini

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          Let $g(t):= tu_y(tx,ty)$, then



          $ int_0^1 frac{d}{dt}(tu_y(tx,ty)) dt = int_0^1 g'(t) dt = g(1)-g(0) = u_y(x,y).$






          share|cite|improve this answer





















          • thanks, this i can understand. but what happened with $$-u{Tiny y} (tx,ty) $$
            – malilini
            Dec 4 '18 at 10:53












          • I didnt think about tagging you @Fred
            – malilini
            Dec 4 '18 at 19:45













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          0














          Let $g(t):= tu_y(tx,ty)$, then



          $ int_0^1 frac{d}{dt}(tu_y(tx,ty)) dt = int_0^1 g'(t) dt = g(1)-g(0) = u_y(x,y).$






          share|cite|improve this answer





















          • thanks, this i can understand. but what happened with $$-u{Tiny y} (tx,ty) $$
            – malilini
            Dec 4 '18 at 10:53












          • I didnt think about tagging you @Fred
            – malilini
            Dec 4 '18 at 19:45


















          0














          Let $g(t):= tu_y(tx,ty)$, then



          $ int_0^1 frac{d}{dt}(tu_y(tx,ty)) dt = int_0^1 g'(t) dt = g(1)-g(0) = u_y(x,y).$






          share|cite|improve this answer





















          • thanks, this i can understand. but what happened with $$-u{Tiny y} (tx,ty) $$
            – malilini
            Dec 4 '18 at 10:53












          • I didnt think about tagging you @Fred
            – malilini
            Dec 4 '18 at 19:45
















          0












          0








          0






          Let $g(t):= tu_y(tx,ty)$, then



          $ int_0^1 frac{d}{dt}(tu_y(tx,ty)) dt = int_0^1 g'(t) dt = g(1)-g(0) = u_y(x,y).$






          share|cite|improve this answer












          Let $g(t):= tu_y(tx,ty)$, then



          $ int_0^1 frac{d}{dt}(tu_y(tx,ty)) dt = int_0^1 g'(t) dt = g(1)-g(0) = u_y(x,y).$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 4 '18 at 10:40









          Fred

          44.2k1845




          44.2k1845












          • thanks, this i can understand. but what happened with $$-u{Tiny y} (tx,ty) $$
            – malilini
            Dec 4 '18 at 10:53












          • I didnt think about tagging you @Fred
            – malilini
            Dec 4 '18 at 19:45




















          • thanks, this i can understand. but what happened with $$-u{Tiny y} (tx,ty) $$
            – malilini
            Dec 4 '18 at 10:53












          • I didnt think about tagging you @Fred
            – malilini
            Dec 4 '18 at 19:45


















          thanks, this i can understand. but what happened with $$-u{Tiny y} (tx,ty) $$
          – malilini
          Dec 4 '18 at 10:53






          thanks, this i can understand. but what happened with $$-u{Tiny y} (tx,ty) $$
          – malilini
          Dec 4 '18 at 10:53














          I didnt think about tagging you @Fred
          – malilini
          Dec 4 '18 at 19:45






          I didnt think about tagging you @Fred
          – malilini
          Dec 4 '18 at 19:45




















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