Complex analysis(fourier transformation):

Please include any axioms/lemmas needed.

Need help ONLY on problem 2, 3, and 5.

Reference:

Let f : R - C be a function. For a real number t, the Fourier transform of f is defined to be (according to one particular convention): F(f) (t ) = / f(x)e-istar. We usually assume that If(x)|dr op(3;) where this sum is over j such that the imaginary part of z, is positive. Similarly to the previous problem set, you may assume that we may write the integral around the semicircle as a sum of integrals . for z, inside the semicircle, where % is a closed curve enclosing z, and no other z;. You may also use Part 2.2 of the previous problem set without proof. Now we assume that y is negative. We integrate s along the semicircle going from - R to R along the real axis, and then along the semicircular arc from R to - R in the lower half-plane 3. The integral along the arc goes to zero as R -+ co similarly to the first part of the question (you do not need to prove this). Explain why it is important that we are in the lower half-plane (not the upper half-plane) for this deduction to follow (note: here we assume y 0). 4. Explain why if y