QUESTION 5Consider the function f(2)=12³(2-3)*(5.1) Use a known Maclaurin series to find the Laurent series expansion of f(z) in the followingregions:(5.1.1) 0<2<3(5.1.2) 3<|2|<∞0(5.2) Classify the singularity z = 0. Find the residue of f at z = 0.(5.3) Find the residues of the f at all singular points without using the Laurent series.(5.4) Hence, evaluate the contour integral fc f(z)dz, where C is the circle |2 - 2| = 3.

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Answered: QUESTION 5 Consider the function f(2)=…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.2: Arithmetic Sequences

Problem 68E

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First-Order Equations. The series methods discussed in this section are directly applicable to the first-order linear differential equation P(x)y′ + Q(x)y = 0 at a point x0, if the function p = Q/P has a Taylor series expansion about that point. Such a point is called an ordinary point, and further, the radius of convergence of the series y=∞∑n=0an(x−x0)ny=∑n=0∞anx−x0n is at least as large as the radius of convergence of the series for Q/P. In each of Problems 13 through 16, solve the given differential equation by a series in powers of x and verify that a0 is arbitrary in each case. Problem 17 involves a nonhomogeneous differential equation to which series methods can be easily extended. Where possible, compare the series solution with the solution obtained by using the methods of Chapter 2. 14.y′ − xy = 0
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