Evaluate the following integrals:
i)
ii)
To solve these integrals numerically, let's define the given integrals: \[ I_1 = \int_{0}^{2\pi} f(e^{i\theta})\cos^2(\theta/2)\,d\theta \] \[ I_2 = ______ ___ ____ ________ _______ __________ ____.
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Find the zeros and singularities of the function Also find the residue at the poles.
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.
22-1 Prove that the linear fractional transformation maps the circle
into itself. Also prove that
is conformal in
Evaluate the following integrals:
i)
ii)
Find all the singularities of the function
Find the image of the circle under the mapping
What happens when r = 1 ?
Evaluate where e is the eight like figure shown in Fig. 1.
