solving a complex fourier transformed equation

michielm · **Joined:** Sun, 14 Mar 2010 14:12:02 UTC **Posts:** 35

Hi,
I am working on the reproduction of a calculation I found in a paper and I am stuck halfway. I have a function:
$z(x,y)=\theta y - \frac{\theta}{2 \pi}\int^{\infty}_{-\infty}\tilde{\eta} e^{ikx} e^{-|k|y} dk$

Where $\tilde{\eta}$ is the fourier transform of $\eta$ i.e. $\eta(x)=\frac{1}{2\pi}\int \tilde{\eta}(k) e^{ikx} dk$

Now I want to calculate $U_{cap}$ which is defined as $U_{cap}=\frac{1}{2}\gamma \int \int \left[\frac{\partial z}{\partial x}^2 +\frac{\partial z}{\partial y}^2 - \theta^2 \right]$

Solving the part between square brackets first I end up with:

$U_{cap}=\frac{1}{2} \frac{\theta^2}{\pi} \gamma \int \int \int^{\infty}_{-\infty} \tilde{\eta} |k| e^{ikx} e^{-|k|y} dk$

Which then results in: $U_{cap}=\frac{1}{2}\gamma \frac{\theta^2}{\pi} \int^{\infty}_{-\infty} \frac{i\tilde{\eta}}{|k|}e^{ikx} e^{-|k|y} dk$

However, in the paper the solution reads: $U_{cap}=\frac{1}{2}\gamma \frac{\theta^2}{\pi} \int^{\infty}_{-\infty} \frac{1}{2}|\tilde{\eta}|^2 |k| dk$

My question is twofold: (i) is there any mistake in my derivation? (ii) is my answer somehow equivalent to the answer of the paper and, if so, why?

Thanks

outermeasure · **Posted:** Thu, 17 Nov 2011 13:18:07 UTC

michielm wrote:

Hi,
I am working on the reproduction of a calculation I found in a paper and I am stuck halfway. I have a function:
$z(x,y)=\theta y - \frac{\theta}{2 \pi}\int^{\infty}_{-\infty}\tilde{\eta} e^{ikx} e^{-|k|y} dk$

Where $\tilde{\eta}$ is the fourier transform of $\eta$ i.e. $\eta(x)=\frac{1}{2\pi}\int \tilde{\eta}(k) e^{ikx} dk$

Now I want to calculate $U_{cap}$ which is defined as $U_{cap}=\frac{1}{2}\gamma \int \int \left[\frac{\partial z}{\partial x}^2 +\frac{\partial z}{\partial y}^2 - \theta^2 \right]$

Solving the part between square brackets first I end up with:

$U_{cap}=\frac{1}{2} \frac{\theta^2}{\pi} \gamma \int \int \int^{\infty}_{-\infty} \tilde{\eta} |k| e^{ikx} e^{-|k|y} dk$

Which then results in: $U_{cap}=\frac{1}{2}\gamma \frac{\theta^2}{\pi} \int^{\infty}_{-\infty} \frac{i\tilde{\eta}}{|k|}e^{ikx} e^{-|k|y} dk$

However, in the paper the solution reads: $U_{cap}=\frac{1}{2}\gamma \frac{\theta^2}{\pi} \int^{\infty}_{-\infty} \frac{1}{2}|\tilde{\eta}|^2 |k| dk$

My question is twofold: (i) is there any mistake in my derivation? (ii) is my answer somehow equivalent to the answer of the paper and, if so, why?

Thanks

What answer of the paper?

michielm · **Joined:** Sun, 14 Mar 2010 14:12:02 UTC **Posts:** 35

My result: $U_{cap}=\frac{1}{2}\gamma \frac{\theta^2}{\pi} \int^{\infty}_{-\infty} \frac{i\tilde{\eta}}{|k|}e^{ikx} e^{-|k|y} dk$

However, in the paper the solution reads: $U_{cap}=\frac{1}{2}\gamma \frac{\theta^2}{\pi} \int^{\infty}_{-\infty} \frac{1}{2}|\tilde{\eta}|^2 |k| dk$

outermeasure · **Posted:** Thu, 17 Nov 2011 14:22:44 UTC

michielm wrote:

My result: $U_{cap}=\frac{1}{2}\gamma \frac{\theta^2}{\pi} \int^{\infty}_{-\infty} \frac{i\tilde{\eta}}{|k|}e^{ikx} e^{-|k|y} dk$

However, in the paper the solution reads: $U_{cap}=\frac{1}{2}\gamma \frac{\theta^2}{\pi} \int^{\infty}_{-\infty} \frac{1}{2}|\tilde{\eta}|^2 |k| dk$

I'm not too sure how you get your answer, but the paper's solution looks like a case of missing some modulus signs.

michielm · **Joined:** Sun, 14 Mar 2010 14:12:02 UTC **Posts:** 35

I get my answer in the following way:
$U_{cap}$ is defined as $U_{cap}=\frac{1}{2}\gamma \int \int \left[ \left( \frac{\partial z}{\partial x} \right) ^2 +\left(\frac{\partial z}{\partial y}\right)^2 - \theta^2 \right] dx dy$

Let's first calculate the partial derivates:
$\frac{\partial z}{\partial x}=\frac{-\theta}{2\pi}\int^{\infty}_{-\infty} \tilde{\eta} i k e^{ikx} e^{-|k|y} dk$
and
$\frac{\partial z}{\partial y}=\theta+\frac{\theta}{2\pi}\int^{\infty}_{-\infty} \tilde{\eta} |k| e^{ikx} e^{-|k|y} dk$

If I square both I get:
$\left(\frac{\partial z}{\partial x}\right)^2=\frac{-\theta^2}{4\pi^2}\int^{\infty}_{-\infty} \tilde{\eta}^2 k^2 e^{2ikx} e^{-2|k|y} dk$
and
$\left(\frac{\partial z}{\partial y}\right)^2=\theta^2+\frac{\theta^2}{\pi}\int^{\infty}_{-\infty} \tilde{\eta} |k| e^{ikx} e^{-|k|y} dk + \frac{\theta^2}{4\pi^2}\int^{\infty}_{-\infty} \tilde{\eta}^2 k^2 e^{2ikx} e^{-2|k|y} dk$

The part between square brackets then becomes:
$\left[\left(\frac{\partial z}{\partial x}\right)^2 +\left(\frac{\partial z}{\partial y}\right)^2 - \theta^2 \right]=\frac{\theta^2}{\pi}\int^{\infty}_{-\infty} \tilde{\eta} k e^{ikx} e^{-|k|y} dk$
(the first and the last term in $\left(\frac{\partial z}{\partial y}\right)^2$ cancel against $-\theta^2$ and $\left(\frac{\partial z}{\partial x}\right)^2$ respectively)

Plugging this into $U_{cap}$ I end up with:

$U_{cap}=\frac{1}{2}\gamma \frac{\theta^2}{\pi} \int \int \left[\int^{\infty}_{-\infty} \tilde{\eta} |k| e^{ikx} e^{-|k|y} dk\right] dx dy$

Which then results in: $U_{cap}=\frac{1}{2}\gamma \frac{\theta^2}{\pi} \int^{\infty}_{-\infty} \frac{i\tilde{\eta}}{|k|}e^{ikx} e^{-|k|y} dk$

I hope this is more clear. Is there any mistake in here (or in my thinking)?

outermeasure · **Posted:** Thu, 17 Nov 2011 16:44:30 UTC

michielm wrote:

I get my answer in the following way:
$U_{cap}$ is defined as $U_{cap}=\frac{1}{2}\gamma \int \int \left[ \left( \frac{\partial z}{\partial x} \right) ^2 +\left(\frac{\partial z}{\partial y}\right)^2 - \theta^2 \right] dx dy$

Let's first calculate the partial derivates:
$\frac{\partial z}{\partial x}=\frac{-\theta}{2\pi}\int^{\infty}_{-\infty} \tilde{\eta} i k e^{ikx} e^{-|k|y} dk$
and
$\frac{\partial z}{\partial y}=\theta+\frac{\theta}{2\pi}\int^{\infty}_{-\infty} \tilde{\eta} |k| e^{ikx} e^{-|k|y} dk$

If I square both I get:
$\left(\frac{\partial z}{\partial x}\right)^2=\frac{-\theta^2}{4\pi^2}\int^{\infty}_{-\infty} \tilde{\eta}^2 k^2 e^{2ikx} e^{-2|k|y} dk$
and
$\left(\frac{\partial z}{\partial y}\right)^2=\theta^2+\frac{\theta^2}{\pi}\int^{\infty}_{-\infty} \tilde{\eta} |k| e^{ikx} e^{-|k|y} dk + \frac{\theta^2}{4\pi^2}\int^{\infty}_{-\infty} \tilde{\eta}^2 k^2 e^{2ikx} e^{-2|k|y} dk$

The part between square brackets then becomes:
$\left[\left(\frac{\partial z}{\partial x}\right)^2 +\left(\frac{\partial z}{\partial y}\right)^2 - \theta^2 \right]=\frac{\theta^2}{\pi}\int^{\infty}_{-\infty} \tilde{\eta} k e^{ikx} e^{-|k|y} dk$
(the first and the last term in $\left(\frac{\partial z}{\partial y}\right)^2$ cancel against $-\theta^2$ and $\left(\frac{\partial z}{\partial x}\right)^2$ respectively)

Plugging this into $U_{cap}$ I end up with:

$U_{cap}=\frac{1}{2}\gamma \frac{\theta^2}{\pi} \int \int \left[\int^{\infty}_{-\infty} \tilde{\eta} |k| e^{ikx} e^{-|k|y} dk\right] dx dy$

Which then results in: $U_{cap}=\frac{1}{2}\gamma \frac{\theta^2}{\pi} \int^{\infty}_{-\infty} \frac{i\tilde{\eta}}{|k|}e^{ikx} e^{-|k|y} dk$

I hope this is more clear. Is there any mistake in here (or in my thinking)?

Why does
$\displaystyle\left(\frac{\partial z}{\partial x}\right)^2=\frac{-\theta^2}{4\pi^2}\int_{-\infty}^\infty \tilde{\eta}^2k^2 e^{2ikx}e^{-2\lvert k\rvert y}\,\mathrm{d}k$ ?
You only know it is $\displaystyle\frac{-\theta^2}{4\pi^2}\left(\int_{-\infty}^\infty \tilde{\eta}k e^{ikx}e^{-\lvert k\rvert y}\,\mathrm{d}k\right)^2$ . Ditto for $\left(\dfrac{\partial z}{\partial y}\right)^2$ . But they are inconsequential.

What I am worry about is the y-integral, because it doesn't necessarily converge. Indeed, it does not converge if we are working with $y\in\mathbb{R}$ . Maybe you are working with $y\in\mathbb{R}^+$ ? But if that is the case, what about x? If you do not work on the whole of $\mathbb{R}$ then you need something extras in your Fourier inversion formula for $\eta$ .

michielm · **Joined:** Sun, 14 Mar 2010 14:12:02 UTC **Posts:** 35

That is a very good point. Can't believe I missed that! Thanks!

About the convergence of the y-integral, indeed the boundaries from $-\infty$ to $\infty$ are only formal at the moment. The physics of the problem allow me to put a lower and upper boundary to avoid singularities.

I will see if I can get to the paper's solution now.

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solving a complex fourier transformed equation

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