Under what condition can we analytically continue an equation?

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Suppose we have an equation
$$(partial_x^2+bpartial_x+h(x))f(x)=g(x)$$
which we known is satisfied for real $x$.



Under what conditions, can we continue it to
$$(partial_z^2+bpartial_z+h(z))f(z)=g(z)$$
for complex $z$?







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    up vote
    2
    down vote

    favorite












    Suppose we have an equation
    $$(partial_x^2+bpartial_x+h(x))f(x)=g(x)$$
    which we known is satisfied for real $x$.



    Under what conditions, can we continue it to
    $$(partial_z^2+bpartial_z+h(z))f(z)=g(z)$$
    for complex $z$?







    share|cite|improve this question






















      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      Suppose we have an equation
      $$(partial_x^2+bpartial_x+h(x))f(x)=g(x)$$
      which we known is satisfied for real $x$.



      Under what conditions, can we continue it to
      $$(partial_z^2+bpartial_z+h(z))f(z)=g(z)$$
      for complex $z$?







      share|cite|improve this question












      Suppose we have an equation
      $$(partial_x^2+bpartial_x+h(x))f(x)=g(x)$$
      which we known is satisfied for real $x$.



      Under what conditions, can we continue it to
      $$(partial_z^2+bpartial_z+h(z))f(z)=g(z)$$
      for complex $z$?









      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Aug 22 at 9:34









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