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Practice Sheet 5

The document contains homework problems for a Mathematics III course, focusing on Taylor and Laurent series expansions as well as residue evaluations. It includes specific tasks such as expanding functions at given points and evaluating integrals using residues around specified circles. The problems are structured to enhance understanding of complex analysis concepts.

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Tanjim Mostafa
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7 views1 page

Practice Sheet 5

The document contains homework problems for a Mathematics III course, focusing on Taylor and Laurent series expansions as well as residue evaluations. It includes specific tasks such as expanding functions at given points and evaluating integrals using residues around specified circles. The problems are structured to enhance understanding of complex analysis concepts.

Uploaded by

Tanjim Mostafa
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Home work: 05 Mohammad Hassan Murad, MNS, BRACU

MAT 215, Spring’15 Mathematics III

Problems based on Taylor, Laurent’s series and residue1


1. Expand each of the following functions in a Taylor series about the indicated points
(a) e−z at z = 0

π
(b) cos z at z =
2

(c) z 3 − z 2 + 4z − 2 at z = 2

(d) zez at z = −1.


z
2. Expand f (z) = in a Laurent series valid for
(z − 1)(2 − z)
(a) |z| < 1
(b) 1 < |z| < 2
(c) |z| > 2
(d) |z − 1| > 1
(e) 0 < |z − 2| < 1.
z2
I
3. Evaluate 2
dz using the residue at the poles, where C is the unit circle |z| = 1.
2z + 5z + 2
C

z2 + 4
I
4. Evaluate dz using the residue at the poles around the circle |z| = 3.
z3 + 2z 2 + 2z
C

z2 − z + 2
I
1
5. Evaluate dz using the residue at the poles around the circle |z| = 4.
2πi z4 + 10z 2 + 9
C

zeiπz
I
6. Evaluate dz using the residue at the poles where C is the upper half of the circle |z| = 2.
(z 2 + 2z + 5)(z 2 + 1)2
C

1 Follow the class lectures

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