Please assist with a solution for 3.3 (a) 3.2 Find and classify zeros of z (e1).3.3 Let f(z) =2.2ー2-2"(a) Find Laurent series valid in the region A = {z € C:1<|z| < 2}.(b) Give the principal part and the analytic part of the series in (a).(c) Classify the singularity z – 0 in A above.3.4 Find the residue of f(z) = (1 – 2*) sin(1/2).

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Answered: 3.3 Let f(2) = 2 -2. (a) Find Laurent…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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4.2 Let f(2) = e2z/(z+2). (a) Find Laurent series of f (z) around z = -2.
4.2 Let f(2) = and let A = {z € C: 2< |z+1| < 3}. 22-32+2 (a) Find Laurent series of f(z) valid in the annulus region A. (b) Classify the singularity for series in (a) in the annulus region A.
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1 Expand function f(z) =. in a Laurent series at z, = 0 and determine its az +b convergent area, where a and b are complex number, a ± 0 and b 0.
Let f(r) = (a) Find first four terms of the Maclaurin series for f(r). (b) Use your result to find the Maclaurin series for f(x) k (c) For the power series representation of f(x), f(x) Cna", Let Pe(r) = Cn", be the polynomial approxima- n=0 tion of f(r) with k+1 terms [that is, Po has a single term, P has two etc.]. Find, P1(1), P2(x), P3(x), P4(x) for the function given. 1 (d) Create a graph that shows f(r) P(r), P2(x), P3(x), Pa(r). Hint: Recommended graph window: [-2, 2] x [-2, 5].
(14) Use a Laurent series to find the residue of f(2) = (z +3) sin z + 3 at the singularity z = -3. Classify the singularity as a removable, an essential, or a pole.
Find the Taylor series for f(x) centered at 5 if fln) (5) (-1)"(n+1)! %3D In(6)4" n = 0 8.
1. Let f(z) = z3 cz - 1 ) Find Laurent series f(z) valid in the region A = {zec : 12-il > 13 2. Let f(z) = Find Laurent cinnulus region 6 (Z-4) (2+2) series of f(z) valid in the A = {₂2=C: KIZ+11 <5}
3.3 Let f(z) = 7* (a) Find Laurent series valid in the region A = {z € C :1 < [z[ < 2}. (b) Give the principal part and the analytic part of the series in (a). (c) Classify the singularity z = 0 in A above.
3. elemfe Consider the function f(x) = 2. (a) Find the Taylor series of f(x) at a = 5 using the definition of a Taylor series. (b) Find the radius of convergence of this power series. 1
z2-1 3. Expand f(z) = in a Laurent's series valid for 1 < ]z| < 3. (z-1)(z+3)
Find the Taylor series for f(x) centered at 8 if fln)(8) = (-1)"+1 (n²+3) 8" %3D n = 0

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3.3 Let f(2) = 2 -2. (a) Find Laurent series valid in the region A = {z € C :1< |z[ < 2}.