How to get the first variation of a complex lagrangian?











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How do I get the first variation for this:
$$ int_C Lleft(z,phi,frac{mathrm{d}phi}{mathrm{d}z}right)mathrm{d}z$$
where:
$$z=x+iy,$$
$$mathrm{d}z=mathrm{d}x+imathrm{d}y,$$
$$phi=f(x,y)+ig(x,y).$$
The integral is a complex line integral and $phi$ is an analytical function.
I'm not sure whether the process is identical to that of real analysis. Is it even possible to get such a variation? Does it even make any sense?










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    down vote

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    How do I get the first variation for this:
    $$ int_C Lleft(z,phi,frac{mathrm{d}phi}{mathrm{d}z}right)mathrm{d}z$$
    where:
    $$z=x+iy,$$
    $$mathrm{d}z=mathrm{d}x+imathrm{d}y,$$
    $$phi=f(x,y)+ig(x,y).$$
    The integral is a complex line integral and $phi$ is an analytical function.
    I'm not sure whether the process is identical to that of real analysis. Is it even possible to get such a variation? Does it even make any sense?










    share|cite|improve this question


























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      How do I get the first variation for this:
      $$ int_C Lleft(z,phi,frac{mathrm{d}phi}{mathrm{d}z}right)mathrm{d}z$$
      where:
      $$z=x+iy,$$
      $$mathrm{d}z=mathrm{d}x+imathrm{d}y,$$
      $$phi=f(x,y)+ig(x,y).$$
      The integral is a complex line integral and $phi$ is an analytical function.
      I'm not sure whether the process is identical to that of real analysis. Is it even possible to get such a variation? Does it even make any sense?










      share|cite|improve this question















      How do I get the first variation for this:
      $$ int_C Lleft(z,phi,frac{mathrm{d}phi}{mathrm{d}z}right)mathrm{d}z$$
      where:
      $$z=x+iy,$$
      $$mathrm{d}z=mathrm{d}x+imathrm{d}y,$$
      $$phi=f(x,y)+ig(x,y).$$
      The integral is a complex line integral and $phi$ is an analytical function.
      I'm not sure whether the process is identical to that of real analysis. Is it even possible to get such a variation? Does it even make any sense?







      complex-analysis calculus-of-variations






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      edited Nov 22 at 13:04









      amWhy

      191k27223439




      191k27223439










      asked Nov 18 at 13:27









      BinaryBurst

      356110




      356110



























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