Consider and the closed circular region
Find points in R where |
| has its maximum and minimum values.
We need to find the maximum and minimum values of \( |f(z)| \) for the function: \[ f(z) = z^2 - z \] on the closed circular region: \[ R = \{z : |z| \leq 1\} \] ### Step 1: Evaluating \( |f(z)| \) at Critical Points Inside \( R \) To find the critical __________ ________ ________ ________ ____ _____ ________ ______ ____ ______ _____ __________.
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Find the image of the semi-infinite strip when
. Sketch the strip and its image.
Consider and the closed circular region
Find points in R where |
| has its maximum and minimum values.
Find the zeros and singularities of the function Also find the residue at the poles.
Find the image of the circle under the mapping
What happens when r = 1 ?
Find the maximum modulus on the closed circular region defined by
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 all the singularities of the function