DIFFERENTIABILITY OF COMPLEX FUNCTIONS Contents

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lim h→0 f(z + h)− f(z) h if this limit exists. If the derivative of f exists at z, we denote its value by f ′(z), and we say f isholomorphic at z. If f is holomorphic at every point of an open set U we say that f is holomorphic on U . This definition naturally leads to several basic remarks. First, the definition formally looks identical to the limit definition of a derivative of a function of a real variable, which is inspired by trying to approximate a tangent line using secant lines. However, in the limit as h→ 0, we are allowing h to vary over all complex numbers that approach 0, not just real numbers. One of the main principles of this class is that this seemingly minor difference actually makes a gigantic difference in the behavior of holomorphic functions. We insist that z be an interior point of Ω to ensure that as we let h → 0, we can approach h in any direction. This is similar to the fact that derivatives (at least, derivatives which are not one-sided) of real functions are only defined at interior points of intervals, not at the endpoints of closed intervals. Why is the term holomorphic used instead of differentiable? We could use differentiable, or perhaps the more specific term complex differentiable, but the convention in mathematics is that the term holomorphic refers to differentiability of complex functions, not real functions. In particular, we will see that a complex function being holomorphic is substantially more restrictive than the corresponding real function it induces being differentiable. Perhaps you might be wondering what exactly a limit of a complex function is. After all, when we say that h → 0 as h ranges over complex numbers, what exactly do we mean? The rigorous definition is just the natural extension of the ε−δ definition used for real functions. More precisely, we say that lim z→z0 f(z) = w if for all ε > 0

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تاریخ انتشار 2012