Showing an integral doesn't converge

[scigal89]

Homework Statement

Show that the following function is not square integrable, i.e. that it is not continuous.

<br /> <br /> \int_{-\infty}^{\infty} \left ( e^{ikx} \right )^{2}dx<br /> <br />

Homework Equations

See above. Also:

<br /> <br /> \int \left ( e^{ikx} \right )^{2}dx = -\frac{ie^{2ikx}}{2k}<br /> <br />

The Attempt at a Solution

<br /> <br /> =\lim_{A \rightarrow -\infty}\int_{A}^{C} e^{2ikx}dx+\lim_{B \rightarrow \infty}\int_{C}^{B} e^{2ikx}dx<br /> <br />

How do I go from there? What would I choose for C? Can it be anything?

 

[LCKurtz]

Homework Statement

Show that the following function is not square integrable, i.e. that it is not continuous.

Not being square integrable is not the same as not being continuous. f(x) = 1 is not square integrable on (-oo, oo) but is obviously continuous.

<br /> <br /> \int_{-\infty}^{\infty} \left ( e^{ikx} \right )^{2}dx<br /> <br />

Homework Equations

See above. Also:

<br /> <br /> \int \left ( e^{ikx} \right )^{2}dx = -\frac{ie^{2ikx}}{2k}<br /> <br />

The Attempt at a Solution

<br /> <br /> =\lim_{A \rightarrow -\infty}\int_{A}^{C} e^{2ikx}dx+\lim_{B \rightarrow \infty}\int_{C}^{B} e^{2ikx}dx<br /> <br />

How do I go from there? What would I choose for C? Can it be anything?

Yes, you can use anything for C. And if either integral diverges the whole thing does.