Evaluate the following integrals

a) .
b) .

If  is entire such that  in  then show that f has the form  where  are constants with .

Which of the following binary codes are linear?

i)   

ii)   

Justify your answer. 

Let  and  be two binary codes with generator matrices

respectively.

i) Find the minimum distance of both the codes.

Table 1: Table for F₁₆.

For each of the linear codes, find the degree, a generator matrix and a parity check matrix. 

 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 |b_n| < n² |a_n| 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.

 Evaluate  where c is the circle .