Find the radius of convergence of the following series.
i)
ii)
i) Consider the series (\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k!} (z - 1 - i)^k). The general term is (a_k = \frac{(-1)^{k+1}}{k!} (z - 1 - i)^k). To find the radius of convergence (R), use the formula: [ \frac{1}{R} = \lim_{k \to _____ _________ __________ _____ ______.
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Find the image of the circle under the mapping
. What happens when
?
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.
Prove that the linear fractional transformation maps the circle
into itself. Also prove
that f(z) is conformal in .
For each of the linear codes, find the degree, a generator matrix and a parity check matrix.
Expand in a Laurent series valid for
i) 0 < |z - 1| < 2 and ii) 0 < |z - 3| < 2.
Evaluate , where c is the eight like figure shown in Fig. 1.
If is entire such that
in
then show that f has the form
where
are constants with
.