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| {{Unreferenced|date=January 2009}}
| | Jayson Berryhill is how I'm called and my spouse doesn't like it at all. Mississippi is where his house is. Credit authorising is exactly where my main earnings arrives from. What me and my family members love is to climb but I'm thinking on beginning some thing new.<br><br>Feel free to surf to my homepage ... clairvoyants ([http://www.streamfurther.com/bellin_stream/uprofile.php?UID=61573 http://www.streamfurther.com/bellin_stream/uprofile.php?UID=61573]) |
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| The '''analytization trick''' is a [[heuristic]] often applied by [[physicist]]s.
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| Suppose we have a [[function (mathematics)|function]] ''f'' of a [[complex variable]] ''z'' which is '''not''' [[analytic function|analytic]], but happens to be [[differentiable]] with respect to its [[real number|real]] and [[imaginary number|imaginary]] components separately. Differentiating ''f'' with respect to ''z'' is out of the question, but it turns out if
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| :<math>f(z)=g(\bar{z},z)</math>
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| for some [[analytic function]] ''g'' of [[two complex variables]], we can pretend ''f'' is ''g'' (physicists do this sort of thing all the time) and work with
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| :<math>\left.\frac{\partial}{\partial z_1}g\right|_{z_1=\bar{z};z_2=z}</math>
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| and
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| :<math>\left.\frac{\partial}{\partial z_2}g\right|_{z_1=\bar{z};z_2=z}</math>
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| instead. Physicists write these as
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| :<math>\frac{\partial}{\partial \bar{z}}f(\bar{z},z)</math>
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| and
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| :<math>\frac{\partial}{\partial z}f(\bar{z},z)</math> | |
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| and give some [[handwaving]] explanation as to why <math>\bar{z}</math> and ''z'' may be treated as if they are "independent" when they really are not.
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| Note that if ''g'' exists, it is unique (due to the theorem about the uniqueness of [[analytic continuation]]s), at least if we ignore complications like [[branch cut]]s and so on.
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| Conceptually, whenever this trick is used, it probably means on a physical level that the variable z that they are working with "really" has a real structure and physicists are merely pigeonholing it into a complex variable.
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| Actually, it's not even necessary for there to be an analytic ''g''. It's enough for ''f'' to be [[differentiable]] with respect to its [[real number|real]] and [[imaginary number|imaginary]] components (or n times differentiable, as the case may be). In that case,
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| :<math>f(\bar{z},z)</math>
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| has to be treated purely formally.
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| [[Category:Several complex variables]]
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| {{Mathanalysis-stub}}
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Jayson Berryhill is how I'm called and my spouse doesn't like it at all. Mississippi is where his house is. Credit authorising is exactly where my main earnings arrives from. What me and my family members love is to climb but I'm thinking on beginning some thing new.
Feel free to surf to my homepage ... clairvoyants (http://www.streamfurther.com/bellin_stream/uprofile.php?UID=61573)