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Laurent Series Expansion and Its Applications

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Portland State University PDXScholar University Honors Theses University Honors College 5-15-2020 Laurent Series Expansion and its Applications Anna Sobczyk Portland State University Follow this and additional works at: https://pdxscholar.library.pdx.edu/honorstheses Part of the Mathematics Commons Let us know how access to this document benefits ou.y Recommended Citation Sobczyk, Anna, "Laurent Series Expansion and its Applications" (2020). University Honors Theses. Paper 839. https://doi.org/10.15760/honors.858 This Thesis is brought to you for free and open access. It has been accepted for inclusion in University Honors Theses by an authorized administrator of PDXScholar. Please contact us if we can make this document more accessible: [email protected]. Laurent Series Expansion and its Applications By Anna Sobczyk An undergraduate honors thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Science In University Honors And Mathematics Thesis Advisor Dr. Bin Jiang Portland State University 2020 Abstract The Laurent expansion is a well-known topic in complex analysis for its application in obtaining residues of complex functions around their singularities. Computing the Laurent series of a function around its singularities turns out to be an efficient way to determine the residue of the function as well as to compute the integral of the function along any closed curves around its singularities. Based on the theory of the Laurent series, this paper provides several working examples where the Laurent series of a function is determined and then used to calculate the integral of the function along any closed curve around the singularities of the function. A brief description of the Frobenius method in solving ordinary differential equations is also provided. Section I. Introduction The method of Laurent series expansions is an important tool in complex analysis. Where a Taylor series can only be used to describe the analytic part of a function, Laurent series allows us to work around the singularities of a complex function. To do this, we need to determine the singularities of the function and can then construct several concentric rings with the same center 푧0 based on those singularities and obtain a unique Laurent series of 푧 − 푧0inside each ring where the function is analytic. 1 The construction of Laurent series is important because the coefficient corresponding to the 푧−푧0 term gives the residue of the function. The calculation of the integral of the function along any closed curve can be done efficiently by using such residue based on the Residue Theorem. Not only does the Laurent series create an efficient method for the integration, it also has many other applications in physics and engineering. While the residue of the function has been used extensively in calculating both complex and real 1 integration, we seldom investigate the coefficient of the term that occur in the outer rings of 푧−푧0 a Laurent series expansion. This paper serves to speak on the significance of this coefficient in the outer rings by providing several working examples of the Laurent series outside of the center annulus and using them to compute the integral of the function along any closed curve outside of the center annulus. This paper also describes the Frobenius method, a method very similar to Laurent series, in solving second-order ordinary differential equations around their singularities. 2 Section II. Background Theory First, let us define a Laurent series.1 Theorem 1.1 Let 0 ≤ 푟1 < 푟2 and 푧0 ∈ ℂ. Suppose 푓(푧) is analytic on the region 퐴 = {푧 ∈ ℂ| 푟1 < |푧 − 푧0| < 푟2} where 푟1 ≥ 0 and 푟2 ≤ ∞. Then we have: ∞ ∞ 푛 푏푛 푓(푧) = ∑ 푎푛(푧 − 푧0) + ∑ 푛 (푧 − 푧0) 푛=0 푛=1 which converge absolutely on 퐴. This series for 푓(푧) is called the Laurent series or Laurent expansion around 푧0 in the annulus 퐴. The coefficients can be determined by: 1 푓(휁) 푎푛 = ∫ 푛+1 푑휁 , 푛 = 0, 1, 2, … 2휋 훾 (휁 − 푧0) 1 푛−1 푏푛 = ∫푓(휁)(휁 − 푧0) 푑휁 , 푛 = 1, 2, … 2휋 훾 Where 훾 can be any circle with center 푧0 and radius 푟, 푟1 < 푟 < 푟2. Furthermore, 푎푛and 푏푛are unique. However, the equations for finding the coefficients 푎푛and 푏푛in the Laurent Series are impractical for a given function. Although the formulas exist, they are seldom used as there are always easier tricks to obtain the Laurent expansion. For instance, we can use well-known Taylor expansions of some fundamental function to obtain the Laurent series. Another common trick is to creatively manipulate a function 푓 to fit into the form of a common geometric series. See the working examples for demonstrations of these tricks. Next, let us introduce the classification of singularities, which are central to constructing a Laurent series. Simply put, a singularity is a point 푧0 in which a function is differentiable at points arbitrarily close to but not including 푧0. Singularities are not always easily for a complicated function. There are different types and classifications of singularities that we will define now. Definition 1.1 If a function 푓 is analytic on a region 퐴 = {푧| 0 < |푧 − 푧0| < 푟2} 푤푡ℎ 푟1 = 0, which is a deleted neighborhood of 푧0, then 푧0 is called an isolated singularity2 and is expressed as: 푏푛 푏1 푓(푧) = ⋯ + 푛 + ⋯ + + 푎푛 + 푎1(푧 − 푧0) + ⋯ 푛 {푧| 0 < |푧 − 푧0| < 푟2} (푧−푧0) 푧−푧0 1 Theorem adapted from Marsden & Hoffman textbook. 2 Definition taken from Marsden & Hoffman and Apelian & Surace text books. 3 The convenience of Laurent series is that we can always find a Laurent expansion centered at an isolated singularity in an annulus that omits that point.3 The Laurent expansion allows for a series representation in both negative and positive powers of (푧 − 푧0) in a region excluding points where 푓 is not differentiable. If 푓 is differentiable in the entire region, then it is analytic and the Laurent series centered at 푧0 will reduce to the Taylor series of the function below: ∞ 푓푘(푐) 푓(푧) = ∑ (푧 − 푐)푘 푓표푟 |푧 − 푐| < 푟 푘! 푛=0 There are different classifications of isolated singularities as below: Definition 1.2 Let 푧0 be an isolated singularity of 푓. If all but a finite number of the 푏푛 terms are zero, then 푧0 is called a pole of 푓. The order of the pole is determined by the highest integer k such that 푏푘 ≠ 0 and is called a pole of order k. A pole of order one is commonly referred to as a simple pole or single pole. Definition 1.3 If an infinite number of 푏푘 are nonzero, 푧0 is called an essential singularity. This 푧0 is also sometimes called a pole of infinite order. Definition 1.4 We call 푧0 a removable singularity if all 푏푘’s are zero. A Taylor series expansion always exists for removable singularities. We focus on the main application of Laurent series: finding the residue of a function. While some complex functions have handy formulas for calculating the residue, it mainly depends on the type of singularity you are dealing with. For instance, there is no efficient way to find the residue of a function with an essential singularity. To find the residue in this case, you must find 풃 the Laurent expansion of the function and locate the ퟏ term: 푧−푧0 푏푛 풃ퟏ 푓(푧) = ⋯ + 푛 + ⋯ + + 푎푛 + 푎1(푧 − 푧0) + ⋯ 푛 {푧| 0 < |푧 − 푧0| < 푟2} (푧−푧0) 푧−푧0 1 In the Laurent expansion, 풃ퟏ is the coefficient of the term in the series. Even if the residue 푧−푧0 can be found easily through a formula, one might want to look at the Laurent series obtained in the outer annuli associated with the singularities. Once the Laurent expansion is known, we can 풃 find the corresponding ퟏ term. It can help us with calculating the integration of 푓(푧) along 푧−푧0 any closed curve located within those annuli. 3 See Apelian & Surace. 4 4 Theorem 1.2 Residue Theorem : Let 퐴 be a region and let 푧1,푧2, … , 푧푛 be 푛 distinct points in A. Let 푓 be analytic in 퐴 except for at the isolated singularities at 푧1,푧2, … , 푧푛. Let be a closed curve in 퐴 homotopic to a point in 퐴. Assume no 푧푖 lies on 훾. Then: 푛 ∫ 푓(푧)푑푧 = 2휋 ∑[푅푒푠(푓; 푧푖)]퐼(훾; 푧푖) 훾 푖=1 Here 푅푒푠(푓; 푧푖) is the residue of 푓 at 푧푖 and 퐼(훾; 푧푖) is the index of 훾 with respect to 푧푖. For simplicity, we assume 퐼(훾; 푧푖) = 1 for this paper. Simply put, for a clockwise 훾, the integral equals 2휋 times the sum of the residues of 푓 inside 훾. In complex analysis, residues play a critical role in allowing us to easily compute integrals. Using the Residue Theorem to solve a line integral is significantly more efficient than other methods because it greatly reduces computational time. The Residue Theorem in complex analysis also makes the integration of some real functions feasible without need of numerical approximation. The examples in this paper focus on obtaining the residue from a Laurent series. The residues obtained from the Laurent series would speed up the complex integration on closed curves. 4 Theorem adapted from Marsden & Hoffman textbook. 5 Section III. Working Examples In the following examples we will refer to a helpful geometric series and some common known Taylor expansions. Geometric Series ∞ ∑ 푧푛 , |푧| < 1 1 푛=0 = ∞ 1 − 푧 1 − ∑ , |푧| > 1 푧푛 { 푛=1 Table 1.1 Known Function Expansions Function Taylor Series around 0 Valid for ∞ 푧3 푧5 푧2푛−1 all 푧 sin (푧) 푧 − + − ⋯ = ∑(−1)푛+1 3! 5! (2푛 − 1)! 푛=1 ∞ cos (푧) 푧2 푧4 푧6 푧2푛 all 푧 1 − + − + ⋯ = ∑(−1)푛 2! 4! 6! (2푛)! 푛=1 푧 ∞ e 푧2 푧3 푧푛 all 푧 1 + 푧 + + + ⋯ = ∑ 2! 3! 푛! 푛=0 6 Example 1 1 Let 푓(푧) = .
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