Find the radius of convergence of the following series.
i) ii)
To determine the radius of convergence of the given power series, we can use the radius of convergence formula based on the root test (Cauchy-Hadamard theorem). The radius of convergence \( R \) is given by: \[ \frac{1}{R} = _____ _______ _____ ______ ____ _____ __________ _________.
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Find all solutions to the equation sin z=5.
5.a) Find the constant c such that can be extended to be analytic at z =1 , when n∈ N is fixed.
Find the image of the semi-infinite strip when
. Sketch the strip and its image.
Evaluate where e is the eight like figure shown in Fig. 1.
Evaluate the following integrals:
i)
ii)
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.
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in the extended
plane onto three specified distinct points
, and w,3 in the extended w plane.
Evaluate de where e is the circle