
 Analytic Functions, The Magnus Effect, and Wings 
  When dealing with a realvalued function of a single real variable, the derivative of the function with respect to that variable is unambiguous, because there is only one way to vary a real number. 
  Analytic functions are a subset of all possible functions of a complex variable, because we they satisfy the requirement that the derivative of f(z) is unambiguous. 
  Now, a remarkable fact, first noticed by Joukowsky, is that the very same analytic function that transforms a unit circle (centered at the origin) into a flat plate also transforms circles of a different magnitude and centered away from the origin into shapes that are not perfectly flat, but that closely resemble practical airfoils. 
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