Fourier transform in the complex plane

[Tspirit]

Homework Statement

I am reading the book of Gerry and Knight "Introductory Quantum Optics" (2004). In page 60, Chapter 3.7, there is two equation referring Fourier Transformation in the complex plane as follows:
$$g(u)=\int f(\alpha)e^{\alpha^{*}u-\alpha u^{*}}d^{2}\alpha, (3.94a)$$
$$f(\alpha)=\frac{1}{\pi^{2}}\int g(u)e^{u^{*}\alpha-u\alpha*}d^{2}u.(3.94b)$$
I have never seen this form in other textbook. My question is, Can it be derived from traditional Fourier transform in the real domain,
$$g(u)=\int f(\alpha)e^{-2\pi i\alpha u}d^{2}\alpha$$
$$f(\alpha)=\int g(u)e^{2\pi i\alpha u}d^{2}u$$
where $\alpha$ and $u$ are both real?

Homework Equations

Refer the proleam statement.

The Attempt at a Solution

I have no idea expanding it from real domain to complex domain. Who can derive it or give me a reference? Thank you.

 

[tnich]

Homework Statement

I am reading the book of Gerry and Knight "Introductory Quantum Optics" (2004). In page 60, Chapter 3.7, there is two equation referring Fourier Transformation in the complex plane as follows:
$$g(u)=\int f(\alpha)e^{\alpha^{*}u-\alpha u^{*}}d^{2}\alpha, (3.94a)$$
$$f(\alpha)=\frac{1}{\pi^{2}}\int g(u)e^{u^{*}\alpha-u\alpha*}d^{2}u.(3.94b)$$
I have never seen this form in other textbook. My question is, Can it be derived from traditional Fourier transform in the real domain,
$$g(u)=\int f(\alpha)e^{-2\pi i\alpha u}d^{2}\alpha$$
$$f(\alpha)=\int g(u)e^{2\pi i\alpha u}d^{2}u$$
where $\alpha$ and $u$ are both real?

Homework Equations

Refer the proleam statement.

The Attempt at a Solution

What I suspect (from the $d^{2}\alpha$ and from what you have said) is that $g(u)=\int f(\alpha)e^{\alpha^{*}u-\alpha u^{*}}d^{2}\alpha$ is meant to represent a two-dimensional transform.
If $α \equiv |α| e^{iθ_α}$ and $u \equiv |u| e^{iθ_u}$, then $\alpha^{*}u-\alpha u^{*} = 2i|α||u|sin(θ_u-θ_α)$.
I am not sure how to interpret that. Maybe it makes sense in context.