Mathematical Methods for Physics and Engineering - Matematica.NET

COMPLEX VARIABLESWe then find that[ (z +∆z)f ′ 2 − z 2 ] [ (∆z) 2 ]+2z∆z(z) = lim= lim∆z→0 ∆z∆z→0 ∆z( )= lim ∆z +2z =2z,∆z→0from which we see immediately that the limit both exists and is independent ofthe way in which ∆z → 0. Thus we have verified that f(z) =z 2 is differentiablefor all (finite) z. We also note that the derivative is analogous to that found forreal variables.Although the definition of a differentiable function clearly includes a wideclass of functions, the concept of differentiability is restrictive and, indeed, somefunctions are not differentiable at any point in the complex plane.◮Show that the function f(z) =2y +ix is not differentiable anywhere in the complex plane.In this case f(z) cannot be written simply in terms of z, and so we must consider thelimit (24.1) in terms of x and y explicitly. Following the same procedure as in the previousexample we findf(z +∆z) − f(z)∆z2y +2∆y + ix + i∆x − 2y − ix=∆x + i∆y2∆y + i∆x=∆x + i∆y .In this case the limit will clearly depend on the direction from which ∆z → 0. Suppose∆z → 0 along a line through z of slope m, sothat∆y = m∆x, thenlim∆z→0[ f(z +∆z) − f(z)∆z] [ ]2∆y + i∆x= lim= 2m + i∆x, ∆y→0 ∆x + i∆y 1+im .This limit is dependent on m and hence on the direction from which ∆z → 0. Since thisconclusion is independent of the value of z, and hence true for all z, f(z) =2y + ix isnowhere differentiable. ◭A function that is single-valued and differentiable at all points of a domain Ris said to be analytic (or regular) inR. A function may be analytic in a domainexcept at a finite number of points (or an infinite number if the domain isinfinite); in this case it is said to be analytic except at these points, which arecalled the singularities of f(z). In our treatment we will not consider cases inwhich an infinite number of singularities occur in a finite domain.826