= Solution
In the <Fraunhofer approximation>, contributions from positions $\xi$ across the slit have relative phase $k\xi\sin\theta$. With uniform illumination, the field relative to its axial value is
$$
\frac{E(\theta)}{E(0)}=\frac1w\int_{-w/2}^{w/2}e^{ik\xi\sin\theta}\,d\xi
=\frac{\sin\beta}{\beta},\qquad\beta=\frac{\pi w\sin\theta}{\lambda}.
$$
Thus the <single-slit diffraction> intensity is
$$
\boxed{I(\theta)=I(0)\left(\frac{\sin\beta}{\beta}\right)^2.}
$$
On the detector $\sin\theta=y/\sqrt{d^2+y^2}$; near the axis, $\beta\simeq\pi wy/(\lambda d)$. The central maximum takes the continuous limit $\sin\beta/\beta\to1$, and the minima satisfy $w\sin\theta=m\lambda$ for nonzero integers $m$. The stated far-field condition justifies adding the aperture contributions with a linear phase.
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