Computing real integrals with complex analysis

Fractional functions

$$ I = \int_{- \infty}^{+ \infty} \frac{P_m(x)}{Q_k(x)} dx $$

  1. Check convergence
  2. Compute complex-integral real-integrals-with-complex-analysis-contour.png

Note: in $\int_{-R}^{R} F(x)dx \rightarrow I$ we use the fact that the particular integral converges (1.). Generally, we estimate a cauchy-principal-value.

$$ I = I_{\gamma_R} + I_{C_R} $$

  1. Prove that $$\int_{C_R} F(z) dz \rightarrow 0$$

Fractional functions with $\sin$ and $\cos$

$$ Q_{\sin} = \int_{- \infty}^{+ \infty} \frac{P_m(x)}{Q_k(x)} sin(ax + b) dx $$

  1. Compute instead $$ Q = \int_{- \infty}^{+ \infty} \frac{P_m(x)}{Q_k(x)} e^{i(ax + b)} dx = Q_{\cos} + i Q_{\sin} $$

Note1: The original function must be real. To solve problem with complex function, we can use the fact that $\cos(x) = \frac{1}{2}(e^{ix} + e^{-ix})$

Note2: To prove convergence to zero we can use Jordan Lemmacomplex-jordan-lemma.png

  1. Then $$ Q_{\sin} = \operatorname{Im}(Q) $$