# Computing integrals on the half line

by hunt_mat
Tags: computing, integrals, line
 HW Helper P: 1,583 Hi, In my fluids work I have come to integrals of the type: $$\int_{0}^{\infty}\frac{e^{ikx}}{ak^{2}+bk+c}dk$$ I was thinking of evaluating this via residue calculus but I can't think of the right contour, any suggestions? Mat
 P: 3,014 Notice that $\vert e^{i \, k \, x} \vert = e^{-x \, \mathrm{Im}k}$. This means that the integral would diverge when we take the circle at infinity on the lower (upper) semicircle for positive (negative) x.
 P: 3,014 Computing integrals on the half line Notice that the inverse Fourier transform of the Heaviside step function: $$\int_{-\infty}^{\infty}{\frac{d k}{2\pi} \, \theta(k) \, e^{i \, k \, x}} = -\frac{1}{2\pi \, i \, x}, \ \mathrm{Im}x > 0$$ Thus, we may represent the Heaviside step function as: $$\theta(k) = -\frac{1}{2\pi \, i} \, {d t \, \frac{e^{-i \, k \, t}{t + i \, \eta}}, \ \eta \rightarrow +0$$ Why do we need it? Because your integral goes to: $$\int_{-\infty}^{\infty}{f(k) \, e^{i \, k \, x} \, \theta(k)}$$ If you substitute the integral representation for the step function and change the order of integration, you should get: $$-\frac{1}{2\pi \, i} \, \int_{-\infty}^{\infty}{\frac{d t}{t + i \, \eta} \, \int_{-\infty}{\infty}{f(k) \, e^{i \, k \, (x - t)}}}$$ Now, you may use the residue theorem for the integral over k, but you need to close the contour in different half-planes, depending on whetgher $x > t$ or $x < t$. The remaining integral over t is again over the whole real line, but , due to the above conditions, should be split into $-\infty$ to x, and from x to $\infty]. Then, making a sub  HW Helper P: 1,583 I should point out that [itex]x\in\mathbb{R}$ Dick, can you explain the substitution again, I don't quite get what you're doing here and you still haven't mentioned the contour you're integrating over.