Consider and the closed circular region
. Find points in R where | f (z)| has its maximum and minimum values
We are given the function \( f(z) = z^2 - z \) and asked to find the points in the region \( R = \{ z: |z| \leq 1 \} \) where \( |f(z)| \) has its maximum and minimum values. ### Approach: 1. Function Analysis: We ______ ____ _____ _______ ________ ____ ________ _____ _____ _______ _______ _________.
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Evaluate the following integrals
(a)
(b)
If , then show that there exists a real
such that
for
.
Evaluatewhere c is the circle | z | = 4.
a) Find the image of the semi-infinite stripwhen w= i/z . 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 z = a +ib , where a and b are integers, then n 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
and if bn
is such that
for all
, then
converges for
.
viii) If f is entire and f (z) = f (−z) for all z , 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 Tz =
z for some
0 and
.
x) If f is entire and Re f (z) is bounded as | z| → , then f is constant.
Find the image of the circle | z | = under the mapping
. What happens when r = 1 ?