Complex Analysis III
Assessable learning outcomes: By the end of the module, students are expected to be able to: Identify different type of singularities of complex functions -Discuss analytic continuation and natural boundaries -Use residue calculus to study zeros and poles of given functions -Define and characterise some special functions, such as the Gamma and Zeta functions. Additional outcomes: The use of complex analytic arguments to derive basic results in algebra and real analysis.
Brief description of teaching and learning methods: Lectures supported by problem sheets. Penalties for late submission: Penalties for late submission on this module are in accordance with the University policy. You are strongly advised to ensure that coursework is submitted by the relevant deadline.
You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work. We now obtain Theorem 1. By the preceding Theorem 1. Now, for a circle 1. To make use of this fact, we need to give the following Definition 1.
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Describe the geometric meaning of the preceding representation of g z. We now have two possible situations.
Complex Analysis III
Then, obviously, 1. To see what this means, we shall apply the invariance of the double ratio. In this case, the fix-point equation 1. Next determine the fix-points of this composed transformation. More precisely, it must be that the end-points of a diameter are mapped onto end-points of another diameter as the mapping is bijective. Looking at this configuration draw a picture! Although the correspondence described above is not actually used here, it is important to know this fact, which is frequently applied in more advanced developments.
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The notion of trace is one of the two key notions in the classification, the second one being the set Fg of fix-points of g. A key result for classification is the following Theorem 1. This is again a routine computation, best being checked by mathematical software. In the situation of this theorem, we say that g and h are conjugate to each other.
Clearly, the relation of conjugation is an equivalence relation. By Theorem 1. Theorem 1. It remains to show that mk is independent of the choice of h.
We first look at transformations in the normal form: Theorem 1. Then, by Theorem 1. We still have to separate two possibilities in the loxodromic case: Definition 1. Otherwise, g is said to be properly loxodromic. We are now ready to prove our final result in this section: Theorem 1. This means that D must be a half-plane.
Hence, D is a half-plane the boundary of which is passing through the origin. We start now by proving the classical Poisson formula: Theorem 2. Proof of Theorem 2. Therefore, we may fix a branch of log g z to ensure that it will be analytic, and to be able to apply the Poisson formula. Also making use of Lemma 2. This follows from Theorem 2. Under the same assumptions as in Theorem 2. Nevanlinna theory: First Main Theorem Nevanlinna theory is a theory about the growth and value distribution of meromorphic functions.
First Main Theorem is actually nothing else than a reformulation of the Jensen formula from the preceding section. Despite of its simplicity, Nevanlinna theory, which is one of the greatest mathematical achievements in the last century, would not exist without First Main Theorem. Logarithmic Derivative Lemma on the other hand is a really deep result which has no direct predecessor. Second Main Theorem then follows from Logarithmic Derivative Lemma in a way which is technically somewhat complicated, though basically elementary.
The starting point for the necessary reformulation of the Jensen formula the following decomposition of logarithm in its positive and negative part as follows: Definition 3. Lemma 3. Definition 3. Non-integrated counting function. Counting function. Proximity function. Characteristic function.
Theorem 3. First Main Theorem. Let f be a meromorphic function not being identically equal to a constant. In the preceding theorem, the exact expression of O 1 depends on a, as shown by the following exact form of the First Main Theorem: Theorem 3. By Proposition 3. In what follows, Landau notations may also be applied for functions defined outside of a typically small exceptional set.
This also applies in the following exercises, as the characteristic function normally cannot be computed exactly, but only modulo a small error term, say of type O 1 , or something else. References  Jank, G. Nevanlinna theory: Basic results In this section, we include a collection of basic properties of the Nevanlinna functions which directly follow from their definitions and from previous complex analysis.
Theorem 4. For finitely many meromorphic functions f1 ,. First observe that the assertions concerning the proximity functions immediately follow from Lemma 3.
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Looking next at the non-integrated counting functions, a pole of a product at a point, say z0 , obviously is of multiplicity at most the sum of pole multiplicities of the components. By logarithmic integration, the assertions concerning the counting functions follow at once. Finally, the assertions concerning the characteristic functions are an immediate consequence of its definition as the sum of the proximity function and the counting function.
First observe that the function bw has exactly the same poles as w, counting multiplicity. Let f be a non-constant entire function.
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By Liouville theorem, f is not bounded. Therefore, the maximum modulus M r, f is not bounded as well. We may also assume that f is not a constant function. We next show that f has at most finitely many poles. Clearly, g is an entire function, as all poles of f are cancelled by the zeros of P. By the Jensen formula Corollary 2.
By the theorem of bounded convergence, see , Corollary 4. More precisely, we show the T r, f is increasing with respect to r which is easy to prove and convex with respect to log r. Same conclusions hold for the counting function N r, f as well, but not for the proximity function m r, f in general. To prove the first claim increasing , observe that the non-integrated counting function n r, f is trivially increasing with respect to r, and this property carries over to N r, f in integration.
That T r, f is increasing, is an immediate consequence of the Cartan theorem. A slight modification may be used to prove that T r, f is increasing as well, if f has a pole at the origin. The counting function and the characteristic function of a meromorphic function f are continuous functions of r. Therefore, the proximity function is continuous as well. Convex functions, Academic Press, New York, This first part towards Second Main Theorem is elementary needing, however, technical computations.
This phase is the deep aspect of the Nevanlinna theory of meromorphic functions. Before proceeding to the main task of this section, we prove Lemma 5. Clearly, P f has a pole at z if and only if f has a pole at this point. If the pole of f there is of multiplicity p, then the pole of P f at this point is of multiplicity np. To calculate the proximity function, recall Lemma 3. Combining this with the previous asymptotic equality for the coounting function, we get the assertion. Let f be a non-constant meromorphic function, and let c1 ,. Before starting the actual proof, it is important to understand the meaning of N1 r, f.
Therefore, N1 r, f counts multiple points of f , with multiplicity reduced by one, i. To this end, we apply First Main Theorem, preceding preparations and Lemma 5. To start this analysis, we first prove 33 Theorem 5. The proof of this theorem is relatively long, and so we divide it in several distinct parts. Hence, g is analytic in some neighborhood of Z0. Therefore, the real parts of two functions log f z and g z analytic in a neighborhood of z0 are identical. Theorem 5.
As we see immediately, this change is of no importance for non-rational functions f. The essential problem to understand the real contents of Theorem 5. To obtain control over this, we need to apply the following important lemma due to E. Borel in Lemma 5. Let T r be a continuous, non-decreasing function defined in the positive real axis. Since T r is continuous, the set E is closed. If the inductive process to determine the sequence rn terminates after finitely many steps, then clearly E is of finite linear measure E.
If the process is not finite, then the sequence rn has a limit point as an increasing sequence. Show that these two definitions of order are equal for an entire function. Looking at the expression of S r, f in Theorem 5. Therefore, the key for the final formulation of the Second Main Theorem is 38 Lemma 5. Let E be the exceptional set determined by the Borel lemma, lemma 5. In what follows, let S r, f be the same quantity as in Theorem 5. In particular, this means that the estimate included in the notation S r, f holds outside of a possible exceptional set of finite linear measure.
We then easily get the following simple modification of Second Main Theorem: Theorem 5. The second assertion now follows from the first one by adding on both sides of the inequality, and then applying the First Main Theorem. Suppose, contrary to the assertion, that f has three Picard values, say a, b, c. By Theorem 5. We next proceed to our final version of the Second Main Theorem by considering distinct a-points of f. In other words, each a-point will be counted only once, independently of its multiplicity.