Consider and the closed circular region
. Find points in
where
has its maximum and minimum values.
To find the maximum and minimum values of \(|f(z)|\) within the closed circular region \(R\), we need to examine the behavior of \(|f(z)|\) on the boundary of the disk \(|z| = 1\) and its interior.
First, let's compute \(|f(z)|\):
\[f(z) = z^2 - z\]
\[|f(z)| = |z^2 - z|\]
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Find all the singularities of the function
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i)
ii)
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in the extended
plane onto three specified distinct points
and
in the extended
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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 all solutions to the equation sin z = 5
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. Sketch the strip and its image