Computing integrals on the half line

hunt_mat

Hi,

In my fluids work I have come to integrals of the type:
[tex]
\int_{0}^{\infty}\frac{e^{ikx}}{ak^{2}+bk+c}dk
[/tex]
I was thinking of evaluating this via residue calculus but I can't think of the right contour, any suggestions?

Mat

Some Pig

Try the punctured disc with boundary ##C_{\epsilon}\cup[\epsilon,R]\cup C_R.##

Attached Thumbnails

 

Dickfore

Notice that [itex]\vert e^{i \, k \, x} \vert = e^{-x \, \mathrm{Im}k}[/itex]. This means that the integral would diverge when we take the circle at infinity on the lower (upper) semicircle for positive (negative) x.

Dickfore

Notice that the inverse Fourier transform of the Heaviside step function:
[tex]
\int_{-\infty}^{\infty}{\frac{d k}{2\pi} \, \theta(k) \, e^{i \, k \, x}} = -\frac{1}{2\pi \, i \, x}, \ \mathrm{Im}x > 0
[/tex]
Thus, we may represent the Heaviside step function as:
[tex]
\theta(k) = -\frac{1}{2\pi \, i} \, {d t \, \frac{e^{-i \, k \, t}{t + i \, \eta}}, \ \eta \rightarrow +0
[/tex]

Why do we need it? Because your integral goes to:
[tex]
\int_{-\infty}^{\infty}{f(k) \, e^{i \, k \, x} \, \theta(k)}
[/tex]
If you substitute the integral representation for the step function and change the order of integration, you should get:
[tex]
-\frac{1}{2\pi \, i} \, \int_{-\infty}^{\infty}{\frac{d t}{t + i \, \eta} \, \int_{-\infty}{\infty}{f(k) \, e^{i \, k \, (x - t)}}}
[/tex]
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 [itex]x > t[/itex] or [itex]x < t[/itex]. The remaining integral over t is again over the whole real line, but , due to the above conditions, should be split into [itex]-\infty[/itex] to x, and from x to [itex]\infty]. Then, making a sub

hunt_mat

I should point out that [itex]x\in\mathbb{R}[/itex]

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.