Expand in a Laurent series valid for
i) ii)
To find the Laurent series expansion of \( f'(z) = \frac{1}{(z-1)^2 (z-3)} \), we need to expand the function in two different regions: one around \( z = 1 \) and the other around \( z = 3 \). ### i) For \( 0 < |z - 1| < 2 \) In this region, we can treat \( z - 1 \) as small and use the fact that \( \frac{1}{z - 3} \) can be expanded around \( z = 1 \). First, express \( \frac{1}{z - 3} \) as a geometric series: ___ ________ ________ ______ __________ _________.
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then show that there exists a real R>0 such that
for
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 zeros and singularities of the function Also find the residue at the poles.
Find the points where the function is not analytic.
Expand in a Laurent series valid for
i) ii)
Evaluate the following integrals:
i)
ii)
Consider and the closed circular region
Find points in R where |
| has its maximum and minimum values.