08000MA201122001 Pages: 2

Reg No.: Name:

Max. Marks: 100

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APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY
Third Semester B.Tech Degree (S,FE) Examination December 2020 (2015 scheme)

Course Code: MA201
Course Name: LINEAR ALGEBRA AND COMPLEX ANALYSIS

PARTA
Answer any two full questions, each carries 15 marks
Show that the function f(z) defined by.

Re(z) 5
f(z) = | 2 ° is not continuous at 2 = 0.
0, z=0

Show that u = x? — 3xy? is harmonic. Hence find its harmonic conjugate.
Determine the Linear fractional transformation that maps 21 = 0, 22 = 21, ‏وع‎ -
—2i onto w, = —1,w2 = 0, ४३ = © respectively.

Fi ⋅ ∙↕ ⋅
ind the image of the strip 2323 1 under the transformation - 22.

Show that f(z) = z? is analytic everywhere and find its derivative.
Under the transformation w = ~ find the image of x 2 1.
PART छ
Answer any two full questions, each carries 15 marks

Evaluate In Re(z)dz where C is the parabola y = 1 + न (x - 1). from

1+1 03 + 3
2
630052
Use Cauchy’s integral formula to evaluate f 7 ‏عكر‎ dz where C is the unit
(2-3)
circle counterclockwise.
Fi ⋅ ള്‌
ind the poles and residues of f(z) = त्र
Find the Taylor series and Laurent series expansions of f(z) = ಗ್‌ about 2 = —i

Evaluate using Cauchy’s residue theorem ழ்‌. tan2nz dz where C is the circle

|2 0.2 = 0.2

Duration: 3 Hours

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