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7: Derivados complejos

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    Hemos estudiado funciones que toman entradas reales y dan salidas complejas (por ejemplo, soluciones complejas al oscilador armónico amortiguado, que son funciones complejas del tiempo). Para tales funciones, la derivada con respecto a su entrada real es muy parecida a la derivada de una función real de entradas reales. Equivale a tomar las derivadas de las partes real e imaginaria, por separado:\[\frac{d\psi}{dx} = \frac{d\mathrm{Re}(\psi)}{dx} + i \frac{d\mathrm{Im}(\psi)}{dx}.\] Ahora consideremos el caso más complicado de una función de una variable compleja:\[f(z) \in \mathbb{C}, \;\;\mathrm{where}\;\; z \in \mathbb{C}.\] En un nivel, podríamos simplemente tratar esto como una función de dos entradas reales independientes:\(f(x,y)\), donde\(z = x + i y\). Sin embargo, al hacerlo estaríamos haciendo caso omiso de la estructura matemática de la entrada compleja, el hecho de que no\(z\) es simplemente una colección de dos números reales, sino un número complejo que puede participar en operaciones algebraicas. Esta estructura tiene importantes implicaciones para el cálculo diferencial de funciones complejas.


    This page titled 7: Derivados complejos is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to the style and standards of the LibreTexts platform.