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Consider the function f(2)= 1 2³(2-3)* (5.1) Use a known Maclaurin series to find the Laurent series expansion of f(z) in the following regions: (5.1.1) 0<2<3 (5.1.2) 3<|2|<∞ (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 |z − 21 = 3.

Consider the function f(2)= 1 2³(2-3)* (5.1) Use a known Maclaurin series to find the Laurent series expansion of f(z) in the following regions: (5.1.1) 0<2<3 (5.1.2) 3<|2|<∞ (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 |z − 21 = 3.

Advanced Engineering Mathematics
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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QUESTION 5 Consider the function f(2)= 1 2³ (z-3)* (5.1) Use a known Maclaurin series to find the Laurent series expansion of f(z) in the following regions: (5.1.1) 0<2<3 (5.1.2) 3<|2|<∞ (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 fo f(z)dz, where C is the circle |z2| = 3.
Transcribed Image Text:QUESTION 5 Consider the function f(2)= 1 2³ (z-3)* (5.1) Use a known Maclaurin series to find the Laurent series expansion of f(z) in the following regions: (5.1.1) 0<2<3 (5.1.2) 3<|2|<∞ (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 fo f(z)dz, where C is the circle |z2| = 3.
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