Find the points where the function is not analytic.
To find the points where the function \[ f(z) = \frac{\log(z+4)}{z^2 + i} \] is not analytic, we need to determine where the function fails to be holomorphic. This happens where: 1. The function is not defined (i.e., points where the denominator is zero). 2. The function is not differentiable (i.e., points where the logarithm function is not analytic). --- ### ________ ____ ______ _____ _____.
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22-1 Prove that the linear fractional transformation maps the circle
into itself. Also prove that
is conformal in
Find the radius of convergence of the following series.
i) ii)
Evaluate de where e is the circle
Determine whether each of the following statement is true or false. Justify your answer with a short proof or a counter example.
i) where a and b are integers, the
ii) 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 w(x,y) is harmonic in D.
iv)
v) The inequality holds for
vii) If a power series converges for
and if
is such that
then converges for r | z |
1.
viii) If is entire and
for all, then there exists an entire function g such that
ix) A mobius transformation which maps the upper half plane onto itself and fixing 0,
and no other points, must be of the form
for some
and
x) If is entire and Re
is bounded as
then
is constant.
Find the maximum modulus on the closed circular region defined by
Find the image of the circle under the mapping
What happens when r = 1 ?
Find the points where the function is not analytic.