Evaluate the following integrals
a) .
b) .
Consider first [ I_1=\int_{0}^{\infty}\frac{x^{2}+2}{(x^{2}+1)(x^{2}+4)},dx . ] Decompose the integrand into partial fractions: [ \frac{x^{2}+2}{(x^{2}+1)(x^{2}+4)}=\frac{A}{x^{2}+1}+\frac{B}{x^{2}+4}. ] Thus [ x^{2}+2=A(x^{2}+4)+B(x^{2}+1)=(A+B)x^{2}+(4A+B). ] Equating coefficients gives [ _____ __________ ____ ___ ___ ____ ______.
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Find the radius of convergence of the following series.
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) 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 image of the circle under the mapping
. What happens when
?
Evaluate , where c is the eight like figure shown in Fig. 1.
Find the maximum modulus of on the closed circular region defined by
.
b) Consider and the closed circular region
. Find points in R where |f(z)| has its maximum and minimum values.
Find all solutions to the equation .