Find the zeros and singularities of the function in
. Also find the residue at the poles
The given function is [ f(z) = \frac{z}{4\cos^2 z - 1} = \frac{z}{(2\cos z - 1)(2\cos z + 1)}. ] Step 1: Find zeros of (f(z)) The zeros occur when the numerator (z = 0). So, the only zero is: [ z = 0. ] Step 2: Find singularities (poles) Singularities occur when the denominator (4\cos^2 z - 1 ___ ____ _______ ________ __________ ___ ___ _____ ____ _________ ____ _____.
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Determine whether each of the following statement is true or false. Justify your answer with a short proof or a counter example.
i) If , where a and b are integers, then
if a > 0.
ii) If f(z) and are analytic functions in a domain, then f is necessarily a constant.
iii) A real-valued function u(x, y) is harmonic in D iff u(x, -y) is harmonic in D.
iv) .
v) The inequality holds for
.
vi) If has the property that
converges, then f is necessarily an entire function.
vii) If a power series converges for |z| < 1 and if
is such that |bn| < n2 |an| for all
, then
converges for |z| < 1.
viii) If f is entire and for all z, then there exists an entire function g such that
for all
.
ix) A mobius transformation which maps the upper half plane onto itself and fixing
and no other points, must be of the form
for some
and
.
x) If f is entire and is bounded as
, then f is constant.
Which of the following binary codes are linear?
i)
ii)
Justify your answer.
Prove that the linear fractional transformation maps the circle
into itself. Also prove
that f(z) is conformal in .
) Let C be the ternary [8,3] narrow-sense BCH code of designed distance , which has defining set
. Use the primitive root 8th root of unity you chose in 4a) to avoid recomputing the the table of powers. If
is the generator polynomial of C and
is the received word, find the transmitted codeword. Use the following table in 1.
Show that there is only one linear fractional transformation that maps three given distinct points z1, z2 and z3 in the extended z plane onto three specified distinct points w1, w2 and w3 in the extended w plane.
) For the binary, (6,3) linear code C with generator matrix
prepare a standard array for decoding. Use it to decode the vectors (1, 1, 1, 0, 1, 1), and (1, 1, 0, 1, 1, 1). \hfill (7)
Let and
be two binary codes with generator matrices
respectively.
i) Find the minimum distance of both the codes.
Table 1: Table for F16.
| 0000 | 0 | 1000 | α³ | 1011 | α⁷ | 1110 | α¹¹ |
|---|---|---|---|---|---|---|---|
| 0001 | 1 | 0011 | α⁴ | 0101 | α⁸ | 1111 | α¹² |
| 0010 | α | 0110 | α⁵ | 1010 | α⁹ | 1101 | α¹³ |
| 0100 | α² | 1100 | α⁶ | 0111 | α¹⁰ | 1001 | α¹⁴ |