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 \infty} \left| \frac{a_{k+1}}{a_k} \right|. ] Compute the ratio: [ \left| _____ ________ ________ ______ _____ ______ ____ ___ ________ ___ ________ ______.
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Find the image of the semi-infinite strip x > 0, 0 < y < 1 when . Sketch the strip and its image.
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.
If , then show that there exists a real R > 0 such that
for
.
b) Find all solutions to the equation .
Find the maximum modulus of on the closed circular region defined by
.
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 |
Find the minimum distance for each of the codes.
If is entire such that
in
then show that f has the form
where
are constants with
.