Sma2110:Real Analysis Question Paper
Sma2110:Real Analysis
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2011
QUESTION ONE (30 MARKS)
a) Determine the singularity of the following function and classify each singularity. (5 Marks) i. ?? = sin 3 ii. ?? = cos 2 2-2+1 2 2+4 b) Find the Laurent series expansion of the function ?? = ??2 -1 3 ?? = 1 (5 Marks) c) Evaluate the integral i. 2-2 2-1 (+3)?? ?? where : = 2 using Corem. (5 Marks) ii. ???? 1+??4 0 (4 Marks) d) Show that = ??3 - 3??2 is harmonic and hence find the complex conjugate function of U such that ?? = + ?? is analytic. (4 Marks) e) Show that the function ?? = 2 transform straight lines x=1 and y=1 the z plane into parabolas in w-place. (6 Marks) f) Show that the function f(z) =2z+3 satisfies Schwarz reflection principle. (2 Marks)
QUESTION TWO (20 MARKS)
a) Prove that 1 -
1 22
1 -
1 32
1 -
1 42
… =
1 2
(7 Marks)
2
b) Show that the function ??1 = ?? 2??+1 ?? ??=0 and ??2 = (-??)?? (2-??)??+1 ??=0 are analytic continuation of each other and hence sketch the common region. (7 Marks) c) Obtain a Laurent series expansion of the function ?? = +1 (+2) at z = -2. (6 Marks)
QUESTION THREE (20 MARKS)
a) i) Define Residue of a function f(z) at a simple pole. (2 Marks) ii) Determine the residue of the function ?? = 1+4 at each of its pole. (6 Marks) b) Evaluate the integral i. sin ?? ??(1+??2) ???? - (6 Marks) ii. 4???? 5-4sin ?? 2 0 (6 Marks)
QUESTION FOUR (20 MARKS)
a) Show that the function = sin(-??)cos( - ) + sin(-??)sin(2 + ) is harmonic. (5 Marks) b) Show that the function ?? = 1 maps the line x = c into a circle centre 1 2 ,0 ???????? 1 2 . (6 Marks) c) Given that 1 = 2,2 = ?? ???? 3 = -1 is mapped into ??1 = ??,??2 = -1 ???? ??3 = 0. Determine the mobius transformation. (6 Marks)
d) Show that for closed polygon, the sum of the exterior angles ?1 - 1, ?2 - 1… ??? - 1 ???? (-2) (3 Marks)
QUESTION FIVE (20 MARKS)
a) Define Schwarz Christofel transformation. (2 Marks) b) Find the function which maps the interior of a triangle in w-plane onto upper half of the z-plane given that the vertices P (w=0) and Q(w=1) of a triangle maps 1 = 0 ???? 1( = 1) on the z-plane while the vertex R maps onto 1( = ). (10 Marks) c) Test whether the function ?? = 22 + 3 + 5?? satisfies Schwarz reflection principle. (4 Marks) d) Determine the real and the imaginary part of the function ?? = 2+1 2 . (4 Marks)
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