= Solution
The expression is a <scalar diffraction theory> integral for a <monochromatic wave>. The complex quantity $\psi_P$ represents a field <complex amplitude>, with physical field obtained by taking the real part after multiplication by $e^{-i\omega t}$. The <light intensity> is proportional to $|\psi_P|^2$. The coordinates $x,y$ label transverse positions in the aperture plane, $\Sigma$ is its transmitting region, and $h(x,y)$ is the incident field multiplied by the aperture's complex transmission. The <wavenumber> is $k=2\pi/\lambda$ in the propagation medium, and $R$ is the axial propagation distance if $h$ already includes the incident field.
To derive the <Fresnel diffraction>, start from <Huygens-Fresnel principle> contributions with propagation factor $e^{iks}/s$. With the coordinate origin at the projection of $P$ into the aperture plane,
$$
s=\sqrt{R^2+x^2+y^2}=R+\frac{x^2+y^2}{2R}-\frac{(x^2+y^2)^2}{8R^3}+\cdots.
$$
The <paraxial approximation> requires $\rho/R\ll1$, where $\rho^2=x^2+y^2$. Replace the slowly varying $1/s$ and the obliquity factor by their axial values, but retain the quadratic variation of <wave phase>. Neglect of the next <wave phase> term requires $k\rho^4/(8R^3)\ll1$ over the relevant aperture. Thus, apart from a common constant,
$$
\psi_P\simeq\frac{e^{ikR}}{i\lambda R}\iint_\Sigma h(x,y)e^{ik(x^2+y^2)/(2R)}\,dx\,dy.
$$
The quadratic <wave phase> may be of order one or much larger; setting it to zero would require the stronger conditions for <Fraunhofer diffraction>. The scalar treatment also neglects the detailed vector field at aperture edges and assumes dimensions large enough compared with <wavelength> for this approximation to be useful.
For a finite-distance axial point source at distance $p$ and an observer at distance $q$, the incident field contributes its own quadratic <wave phase>. If $h$ instead denotes only the aperture transmission, the combined curvature is described by an effective distance
$$
\boxed{\frac1{R_{\rm eff}}=\frac1p+\frac1q,\qquad R_{\rm eff}=\frac{pq}{p+q}.}
$$
These are two equivalent conventions for distributing the incident <wave phase> between $h$ and the exponential. Both distances must be included in the later calculation.
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