Complex derivatives - Complex derivatives From Physics...

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Complex derivatives From Physics Notes Previously, we have studied considered functions that take inputs that are real numbers, and give outputs that are complex numbers (e.g., solutions to the damped harmonic oscillator that are complex functions of time). As we saw, the derivative of such a function with respect to its real input is handled in much the same way as for a real function of real inputs: Let us now consider the more complicated case of a function of a complex variable: At one level, we could treat such a function simply as a function of two independent real inputs: , where . However, in doing so we would be disregarding the mathematical "structure" of the complex input —i.e., the fact that is not just a mere collection of two real numbers, but a complex number which can be subjected to algebraic operations. This structure has far-reaching consequences for the differential calculus of complex functions. Contents 1 Complex continuity and differentiability 2 Analytic functions 2.1 Common analytic functions 3 Cauchy-Riemann equations 3.1 Proof 3.2 Consequences 4 Exercises Complex continuity and differentiability We define the concept of a continuous complex function in a manner similar to the real case. A complex function is said to be continuous at if, for any , we can find a such that In this definition, indicates the magnitude of a complex number, whereas in the definition of real continuity it indicated an absolute value. The basic meaning is the same: as we vary the input in some smooth way, there should be no abrupt jumps in the value of . If a function is continuous at a point , then we can define its complex derivative as
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Fig. 1: Starting from a point in the complex plane, we can make displacements in arbitrary direction, as shown by the arrows and . The quantity , in the limit , could depend on the displacement direction. If it does not, then is said to be complex differentiable at the point . This is very similar to the definition of the derivative for a function of a real variable. However, there's a complication which doesn't appear in the real case: the infinitesimal is a complex number, not just a real number. The above definition does not specify the argument of the complex number (i.e., the direction in the complex plane in which it's pointing, as illustrated in Fig. 1). In principle, we might get different results from the above formula when we plug in different infinitesimals , even in the limit where and even though is continuous. Example Consider the function . According to the formula for the complex derivative, But if we plug in a real , we get a different result than if we plug in an imaginary : . . In order to cope with this complication, we regard the complex derivative as well-defined only if the above definition gives the same answer regardless of the argument of . If a function satisfies this property at a point , then we say that the function is complex-differentiable at . In other words, a complex-differentiable function possesses an unambiguous complex derivative at the given point.
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