Prove that the linear fractional transformation maps the circle
into itself. Also prove
that f(z) is conformal in .
Consider the linear fractional transformation [ \phi(z)=\frac{2z-1}{2-z}. ] First, we show that the circle (C:\ |z|=1) is mapped into itself. Let (|z|=1). Then (\bar z = 1/z). _______ ____ _______ _______ _________ ______ _____.
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Find the image of the semi-infinite strip x > 0, 0 < y < 1 when . Sketch the strip and its image.
Find the radius of convergence of the following series.
i)
ii)
Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.
i) If the weight of each element in the generating matrix of a linear code is at least r, the minimum distance of the code is at least r.
ii) There is no linear self orthogonal code of odd length.
iii) There is no 3-cyclotomic coset modulo 121 of size 25.
iv) There is no duadic code of length 15 over .
v) There is no LDPC code with parameters ,
and
.
Consider and the closed circular region
. Find points in R where |f(z)| has its maximum and minimum values.
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.
Factor x8 - 1 over . Give the generator polynomials of all cyclic codes of length eight over
.
Find the zeros and singularities of the function in
. Also find the residue at the poles