Some Properties of Analytic Functions

We mention briefly, for the sake of convenient reference, some properties of analytic functions which are required in various contexts. For more complete treatment of these topics, we refer to the many standard references on the topic, for example Ahlfors (1966). Some of the discussion is also based on a section of Morse and Feshbach (1953). 

Of all the functions defined on the xy plane, there is a very special class, termed analytic functions, which have the property that they are functions only of the combination z = x+iy and have a uniquely defined derivative with respect to z at each point in the region. This latter requirement is very restrictive in that it means that the derivative is independent of which of the infinite number of directions the limit is approached. If we write such a function F z) in the form 

These conditions are necessary consequences of the analyticity assumption. If the derivatives are continuous at a given point, it may also be shown that they are sufficient to ensure analyticity. 

Note that the Cauchy-Riemann equations imply that, if the real part of a complex function is known, its imaginary part is determined to within a constant. This point is discussed later, from another viewpoint. 

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