# Fourier transform integration using well-known result

#### binbagsss

Problem

F denotes a forward fourier transform, the variables I'm transforming between are x and k
- See attachment

Relevant equations

So first of all I note I am given a result for a forward fourier transform and need to use it for the inverse one.

The result I am given to use, written out is :

$\int e^{-ikx} e^{-\alpha x^{2}} dx = \frac{1}{2\alpha} e^{-k^{2}/4\alpha}$

I note that $F(k) C(t)$ gives me a function of k, so I apply $F^{-1}$ on this I get a function of x.

Attempt:

My thoughts are I'm looking to change signs in my exponential terms so that it is , effectively a forward transform and then use the result, so just thinking of it as a integration result rather than a particular fourier transform.

However if I do this my exponential terms are:

$e^{-(-ikx+\alpha k^{2})}$

, I don't know how I can then apply the result, completing the square looks like the only candidate to me , but this seems like it will be too scrappy, in particular with 'i' terms, how do I deal with this $e^{ikx}$ term, if I'm on the right lines, is completion of the square necessary or is there some other approach to being able to use this result ?

Many thanks in advance.

#### Attachments

• 6.1 KB Views: 202
Related Calculus and Beyond Homework Help News on Phys.org

#### blue_leaf77

Science Advisor
Homework Helper
but this seems like it will be too scrappy,
Not too scrappy, it will only give you an extra constant phase factor.
Anyway, you can just use the forward transform as a "template" with which you do the back transform. Note that in the RHS of the equation you are given, the expression is real so the sign of i should not matter.

#### binbagsss

No sorry I'm still totally stuck, I think I see the logic in your arguement that the sign of i shouldn't matter but I don't see how I can apply the result given if the sign is wrong.

Any more hints anyone?

Cheers

#### vela

Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
How about making the change of variables $k' = -k$?

#### lurflurf

Homework Helper
From your given transform you should see at one something reminiscent of the quadratic formula for Re(a)>0
Which as you say relates to completing the square
$$\int_{-\infty}^\infty\! \exp\left(-a x^2+2b x+c\right) \,\mathrm{d}x=\\ \exp\left(\frac{b^2}{a}+c\right)\int_{-\infty}^\infty\! \exp\left(-a (x-b/a)^2\right) \,\mathrm{d}x=\\ \exp\left(\frac{b^2}{a}+c\right)\sqrt{\frac{\pi}{a}}$$
Notice the first integral cares what b is, but the second does not since the effect of b (and c) has been moved outside.
See that since b is squared in the result -(-i)^2= -i^2=1 and the sign does not matter

Last edited:

#### blue_leaf77

Science Advisor
Homework Helper
No sorry I'm still totally stuck, I think I see the logic in your arguement that the sign of i shouldn't matter but I don't see how I can apply the result given if the sign is wrong.

Any more hints anyone?

Cheers
Even if you were to change the sign of $i$ or equivalently the sign of $k$ as suggested by vela above, you will get
$$\int e^{ikx} e^{-\alpha x^{2}} dx = \frac{1}{2\alpha} e^{-k^{2}/4\alpha}$$
If you are still wondering why this is true, try expanding $e^{ikx}$ into Cartesian form. The imaginary part of the integral that contains $\sin kx$ vanishes because the product with $e^{-\alpha x^{2}}$ results in an odd function and you integrate it over entire axis.

#### binbagsss

From your given transform you should see at one something reminiscent of the quadratic formula for Re(a)>0
Which as you say relates to completing the square
$$\int_{-\infty}^\infty\! \exp\left(-a x^2+2b x+c\right) \,\mathrm{d}x=\\ \exp\left(\frac{b^2}{a}+c\right)\int_{-\infty}^\infty\! \exp\left(-a (x-b/a)^2\right) \,\mathrm{d}x=\\ \exp\left(\frac{b^2}{a}+c\right)\sqrt{\frac{\pi}{a}}$$
Notice the first integral cares what b is, but the second does not since the effect of b (and c) has been moved outside.
See that since b is squared in the result -(-i)^2= -i^2=1 and the sign does not matter
yeh that's all fine.

I don't know what result you've used in the last equality though, it wasn't using the result quoted in the OP which I'm trying to get at.
Anyway so completing the square I get:

$e^{(-1/D)\frac{x^{2}D^{2}}{4}} \int e^{-1/D(k-ixD/2)^{2}} dk$

In order to then use the result given I need to write it in a similar complete square form in order to make a comparison...pretty sure this isn't what my lecturer was hinting at...

#### blue_leaf77

Science Advisor
Homework Helper
In order to then use the result given I need to write it in a similar complete square form in order to make a comparison...pretty sure this isn't what my lecturer was hinting at...
What about changing the integration variable?

#### lurflurf

Homework Helper
Member warned about providing too much help
I don't know what Fourier convention is in use looks like a typo
I have
$$F(k)\left[e^{-\alpha x^2}\right]= \int_{-\infty}^\infty\! \exp\left(-i k x-\alpha x^2\right) \,\mathrm{d}x= \sqrt{\frac{\pi}{\alpha}}\exp\left(\frac{-k^2}{4\alpha}\right)$$
let 1/alpha=4D t
to see
$$\frac{1}{\sqrt{4D \pi t}}F(k)\left[e^{-\frac{ x^2}{4D t}}\right]= \frac{1}{\sqrt{4D \pi t}}\int_{-\infty}^\infty\! \exp\left(-i k x-\frac{ x^2}{4D t}\right) \,\mathrm{d}x= \exp\left(-D k^2t\right)$$
substitute it in
$$\frac{Q}{2\pi}\int_{-\infty}^\infty\! \exp\left(i k x-D k^2t\right) \,\mathrm{d}x=QF^{-1}(x)\left[e^{-D k^2t}\right]=QF^{-1}(x)\left[\frac{1}{\sqrt{4D \pi t}}F(k)\left[e^{-\frac{ x^2}{4D t}}\right]\right]$$
cancel the forward and inverse transforms to obtain the result

• binbagsss

#### binbagsss

I don't know what Fourier convention is in use looks like a typo
I have
$$F(k)\left[e^{-\alpha x^2}\right]= \int_{-\infty}^\infty\! \exp\left(-i k x-\alpha x^2\right) \,\mathrm{d}x= \sqrt{\frac{\pi}{\alpha}}\exp\left(\frac{-k^2}{4\alpha}\right)$$
let 1/alpha=4D t
to see
$$\frac{1}{\sqrt{4D \pi t}}F(k)\left[e^{-\frac{ x^2}{4D t}}\right]= \frac{1}{\sqrt{4D \pi t}}\int_{-\infty}^\infty\! \exp\left(-i k x-\frac{ x^2}{4D t}\right) \,\mathrm{d}x= \exp\left(-D k^2t\right)$$
substitute it in
$$\frac{Q}{2\pi}\int_{-\infty}^\infty\! \exp\left(i k x-D k^2t\right) \,\mathrm{d}x=QF^{-1}(x)\left[e^{-D k^2t}\right]=QF^{-1}(x)\left[\frac{1}{\sqrt{4D \pi t}}F(k)\left[e^{-\frac{ x^2}{4D t}}\right]\right]$$
cancel the forward and inverse transforms to obtain the result
exactly what i was looking for,- thank you.

### Want to reply to this thread?

"Fourier transform integration using well-known result"

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving