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Singularities of Complex Functions Thursday, November 07, 2013 1:56 PM Singularities of Complex Functions Thursday, November 07, 2013 1:56 PM Homework 3 due Friday, November 15 at 5 PM. From last time, we saw that if we have in hand a function that is known to be analytic over some annular domain, then one can represent that function as a Laurent series which is convergent over that annulus and uniformly convergent over any compact subset of that annulus. In particular, if a function is known to be analytic over an open disc, then one can write a Taylor series representation for that function which converges over that disc, and is uniformly convergent over any compact subset. One interesting consequence of this theorem is it gives an alternative, and often easier way of determining the radius of convergence of a Taylor series, other than using the root test. In particular, a Taylor series representation of an analytic function, centered at a point will have a radius of convergence equal to the distance to the nearest singularity (point where analyticity fails). This property is not true for real analytic functions. Consider the complex analytic function Imagine we compute the Taylor series for this function about Because Taylor series representations are unique, any way we get the Taylor series through power series manipulations is OK, within the radius of convergence. So for example, expand it as a geometric series: Radius of convergence by root test: This simple geometric observation doesn't work on real analytic functions. Consider the real- valued function New Section 2 Page 1 valued function The real-valued Taylor series will be the same (with x instead of z), so its radius of convergence will be the same. But the function itself has no singularities on the real axis. In addition to analytic functions being shown to have Taylor series representations expanded about arbitrary points of analyticity, one can show conversely that a convergent power series (Taylor series) defines an analytic function. That is, if we're given a power series: and we show (by say the root test) that this converges over a domain at least as big as then this power series defines an analytic function inside that domain, whose derivatives can be obtained by differentiating the power series term by term. Proof: By the Weierstrass M-test we know that the power series will actually converge uniformly over any closed bounded region if Notice that each term in the power series is analytic. A uniformly convergent series of analytic functions must itself be analytic because of Morera's theorem. Therefore the power series defines an analytic function on but is any number so we have analyticity on The fact that the derivative of the power series is simply the sum of the derivatives: because one can use the Cauchy integral formula for the derivative and interchange integrals and sums by uniform convergence. Between these theorems, and Cauchy's theorem, and Morera's theorem, we see that there are three equivalent ways to characterize analyticity (holomorphicity) over a open domain: • Existence of the complex derivative over the open domain • The integral of the function over any closed contour in the domain is zero • The ability to express the function as a power series anywhere within the domain. The interaction of power series with analyticity enables an interesting procedure called analytic continuation which can take a power series representation of a function, with only a limited domain of convergence, and extend it to define an analytic function over a broader domain of the complex plane. More generally analytic continuation is a way to extend a limited domain of definition of a complex function to a broader domain. Analytic continuation can be shown to prescribe a unique analytic function over the broader domain so long as it remains simply connected. But sometimes (very important in practice) the analytic continuation New Section 2 Page 2 connected. But sometimes (very important in practice) the analytic continuation leads to a multivalued function. Simple example: imagine power series expansions of Log z. This is the notion of monodromy. One other consequence of note from the above developments is that Laurent series can be added, multiplied, composed, integrated, differentiated by formal series manipulations within their domain of convergence, justified by uniform convergence and the special consideration for differentiation. Singularities of Complex Functions If singularities of complex functions are not isolated, then no particular good theory. Isolated: the function is analytic in a punctured neighborhood of the singularity : Examples of non-isolated singularities: • has a non-isolated singularity at z=0: Cluster singularity You can find examples of natural barriers in the texts corresponding to certain power series. New Section 2 Page 3 But isolated singularities have a good comprehensive theory because one can develop a Laurent series representation of the complex function at least over the punctured neighborhood of analyticity of that singularity. And therefore one can classify isolated singularities by the behavior of the Laurent series in this punctured neighborhood. converges absolutely over a punctured neighborhood and uniformly over any compact subset of this neighborhood. Principal part of the Laurent series is Three cases: 1. There is no principal part ( if ): removable singularity 2. The principal part has a finite number of nonzero terms: pole 3. The principal part has infinitely many nonzero terms: essential singularity Removable singularity: Example: We'll show later that: converge for Has no principal part. But we might have been hesitant to say that was analytic at z=0 because it looks like we're dividing by zero there. The apparent singularity at z=0 can easily be removed by simply defining the function to take the value at the singularity by using the Laurent series (which just looks like a Taylor series) and plugging in the value of z at the removable singularity. In other words, there is no problem in simply extending the definition of the Laurent series to be a Taylor series by continuous extension to the removable singularity (here at z=0). New Section 2 Page 4 (here at z=0). By defining We get an analytic function defined on the non-punctured neighborhood and represented by the Taylor series (which is the same as the Laurent series). Note in particular that since the Laurent series will have a well-defined limit at the removable singularity, a Laurent series is bounded near a removable singularity. Poles: Principal part of the Laurent series about a pole has the form: N is the order of the pole is the strength of the pole. Examples of functions with poles: Notice that if f(z) has a pole of order N, then will have a removable singularity at , and has a limit there. So, by unwinding the defintion of continuity, etc.: Given any when R is the outer radius of convergence of the Laurent series, there exist constants such that: New Section 2 Page 5 That is, at a pole, a function diverges as a negative power law with exponent equal to the order of the pole. Actually this is fairly nice behavior on the Riemann sphere; it implies that: at a pole . Essential singularity Two theorems that characterize how nasty functions behave near essential singularities: Casarati-Weierstrass theorem: At an essential singularity, one can specify any complex value on the extended complex plane and a tolerance and one can find arbitrarily small neighborhoods of the essential singularity where . Picard theorem: In any neighborhood of an essential singularity, the complex function will assume every value on the complex plane, except possibly for one value. Removable singularity Pole Essential singularity Prototype example of an essential singularity is at z=0. New Section 2 Page 6 Principal part has infinitely many negative terms. Let's see how Picard theorem applies to this function. Realize that, by elementary properties of exponential function: In a neighborhood the function will cover the exterior of a disc of radius One other note: What about branch points? They don't even have Laurent series representations about them because...they're multivalued. To analyze them in the same way requires the use of Puiseux series (fractional powers); can be done but we won't. New Section 2 Page 7 .
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