Solving an Integral with Wave Packet: Find \varphi(k)

[12504106]

Homework Statement

Consider the wave packet \varphi(k) = \int B(k)cos(kx) dk from 0 to infinity and B(k) = exp(^{-a^{2}k^{2}}). Find \varphi(k)

Homework Equations

The Attempt at a Solution

After looking up integral tables, i got an expression involving error function (erf) and imaginary error function (erfi) which i don't know how to continue.

 

[vela]

The integrand is an even function, so you can write

\int_0^\infty B(k)\cos(kx)\,dk = \frac{1}{2}\int_{-\infty}^\infty B(k)\cos(kx)\,dk

Does that help?

 

[12504106]

The integrand is an even function, so you can write

\int_0^\infty B(k)\cos(kx)\,dk = \frac{1}{2}\int_{-\infty}^\infty B(k)\cos(kx)\,dk

Does that help?

But i will still end up with those error function which i can't evaluate. Is there any method which does not involve the error function?

 

[12504106]

But i will still end up with those error function. Is there any method which will not involve the error functions?

 

[vela]

Not really, but you can evaluate the integral exactly because of the limits.

There is another approach you can try: Write cos kx as the real part of e^ikx. Then the integral is a Fourier transform, which you can look up in a table.