To evaluate the given integral, we can use the integral representation of the average value of a continuous function over an interval. The function in question here is f(e^(iθ)) multiplied by sin^2(θ)/2. The interval is [0, 2π].
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Find the maximum modulus of f (z) = 2z + 5i on the closed circular region defined by
Find all solutions to the equation sin z = 5 .
Evaluate the following integrals:
Expand in a Laurent series valid for
(i) and
(ii) .
Evaluate , where c is the eight like figure shown in Fig. 1
a) Find the image of the semi-infinite stripwhen w= i/z . 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 z = a +ib , where a and b are integers, then n 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
and if bn
is such that
for all
, then
converges for
.
viii) If f is entire and f (z) = f (−z) for all z , 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 Tz =
z for some
0 and
.
x) If f is entire and Re f (z) is bounded as | z| → , then f is constant.