Evaluate where c is the circle
.
The function ( \frac{1}{z^{2}+1} ) has singularities where ( z^{2}+1=0 ), that is at ( z=i ___ _____ ___ ________ _________ ________ ____ ___.
______ _______ _______ ________ _______ _____ _______ _______ _______ _______ ___.
_____ ________ _____ ___ _____.
_________ _____ __________ ____ ____ _____ _________ _____ ________.
_______ _______ __________ _____ __________ ________ _________ __________.
_________ _____ _____ _______ _______ ________ _______ _____ __________ _______.
_____ __________ _______ _________ ______.
__________ ___ ________ _________ ___ ________.
______ ____ _______ __________ ____.
_________ _____ __________ __________ ______ ______.
________ ______ ____ ____ ____ ______ ___ ____ _____ ____ _________.
__________.
Get Full Answer on WhatsApp
Find the image of the semi-infinite strip x > 0, 0 < y < 1 when . 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 , where a and b are integers, then
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 |z| < 1 and if
is such that |bn| < n2 |an| for all
, then
converges for |z| < 1.
viii) If f is entire and for all z, then there exists an entire function g such that
for all
.
ix) A mobius transformation which maps the upper half plane onto itself and fixing
and no other points, must be of the form
for some
and
.
x) If f is entire and is bounded as
, then f is constant.
Find the zeros and singularities of the function in
. Also find the residue at the poles.
Find the constant c such that can be extended to be analytic at
, when
is fixed.
b) Consider and the closed circular region
. Find points in R where |f(z)| has its maximum and minimum values.
Evaluate , where c is the eight like figure shown in Fig. 1.
Find the radius of convergence of the following series.
i)
ii)