Find the points where the function is not analytic.
To determine where the function \( f(z) = \frac{\log(z+4)}{z^2 + i} \) is not analytic, we need to find the points where the function is not differentiable.
A function is not analytic at a point if it's not differentiable there. The most common reason for a function not being differentiable is if it has a singularity __________ __________ ______ ______ _____ ___ _______ ____ ________ _______ ________.
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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 and
are analytic functions in
domain, then
is necessarily a constant.
iii) A real-valued function u(x, y) is harmonic in iff u(x, − y)
is harmonic in .
iv)
v) The inequality holds for
.
vi) If has the property that
converges, then
is necessarily an entire function.
vii) If a power series converges for
and if
is such that
for all
, then
converges for
.
viii) If 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 is entire and Re f(z) is bounded as
, then
is constant.
Find the constant c such that can be extended to be analytic at
, when
is fixed.
Evaluate , where
is the eight like figure shown in Fig. 1
Expand in a Laurent series valid for i)
and ii)
Find all the singularities of the function
Show that there is only one linear fractional transformation that maps three given distinct points and
in the extended
plane onto three specified distinct points
and
in the extended
plane.
Evaluate the following integrals:
i)
ii)