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.
i) False. A counterexample: Let the equation be (x^2 + 1 = 0) with integers (a = 1, b = 1). There is no integer solution for (x), so the implication fails. ii) False. Analyticity of (f(z)) and another analytic function does not imply (f) is constant. For example, _______ __________ _________ _____ _______ _________ ____ ____ ________.
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Find all the singularities of the function .
Factor x5 - 1 over . Give the generator polynomials of all cyclic codes of length five over
.
) Evaluate the following integrals:
i) .
ii) .
Find the constant c such that can be extended to be analytic at
, when
is fixed.
Let C be the [5, 2] binary code generated by
Find the weight enumerator C of C . Use McWilliams identity to find the weight enumerator of C. Verify your answer by finding the generator matrix of C and finding the weight distribution of C.
Find the radius of convergence of the following series.
i)
ii)
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.