I am trying to evaluate an integral of type
$$ I = \int_{-\infty}^{\infty} \frac{e^{ikx}P(x)}{Q(x)} \,dx$$
where
- $P(x), Q(x)$ are polynomials;
- $Q(x)$ has no zeros on the real line;
- $\mathoperator{Order}(P(x)) + 2 > \mathoperator{Order}(Q(x)).$
Now according to theory, If I want the contribution of the curved portion of the contour to be zero I have to choose a semi-circular contour lying on top of the Real axis if $k > 0,$ and choose a semi-circular contour lying below the Real axis if $k < 0.$ These contours are shown below:
Fig 1: Upper Semi-Circular Contour ( for k > 0 )
Fig 2: Lower semicircular contour ( for k < 0 )
Now to prove that the curved section of the contour does not contribute I need to compute an ML estimate.Now let c = $\left|k\right|$.Regardless of which countour we pick in each of the cases:
Case 1 ( k > 0 ) :
$$
\left| \int_{\gamma _R} f(z) dz \right| \leq ML
$$
$$
\left| \int_{\gamma _R} f(z) dz \right| \leq
\lim_{R – > \infty} \left [
\frac{\left|e^{icz}\right|
\left| P(z) \right|}
{\left| Q(z) \right|} \right] * \pi R
$$
Case 2 ( k < 0 ) :
$$
\left| \int_{\gamma _R} f(z) dz \right| \leq ML
$$
$$
\left| \int_{\gamma _R} f(z) dz \right| \leq
\lim_{R – > \infty} \left [
\frac{\left|e^{-icz}\right|
\left| P(z) \right|}
{\left| Q(z) \right|} \right] * \pi R
$$
Where $z = Re^{i\theta}$ in both cases.Which seems to imply that the choice of contour does not matter because $|e^{icx}|$ =
$|e^{-icx}|$ = 1, so the two ML estimates will always go to the same value.How is this reasoning wrong? Is there a special reason why the ML estimate over the curved portion will only go to zero if I pick the appropriate semi-circular contour?
Best Answer
Assuming $k$ is real : $|\exp(ikz)|=|\exp(ik(x+iy))|=\exp(-ky)$. This expression is less than $1$ if $k>0$ and $y>0$ or if $k<0$ and $y<0$. Thus you want to use a semicircle in the upper halfplane when $k>0$ and a semicricle in the lower halfplane when $k<0$.
If $k$ is not real, your integral won't converge.