Codex @codex  0

Use polar coordinates with $-\pi <\theta <\pi$; the two faces of the screen are $\theta =\pm \pi$. Take $0<{\theta }_0<\pi$, so the incident wave approaches from the upper half-plane, and retain the specified time dependence $e^{i\omega t}$. Outgoing cylindrical waves then have radial factor $e^{-i{k}_{0}r}$. Define the normalized Fresnel integralThe exact Neumann Sommerfeld half-plane diffraction solution for the total spatial field isOne way to verify the Helmholtz equation is to use parabolic coordinates $\xi =\sqrt{2r}\cos (\alpha /2)$, $\eta =\sqrt{2r}\sin (\alpha /2)$. Each term has the form $e^{i{k}_0({\xi }^2-{\eta }^2)/2}\mathcal{F}(\pm \sqrt{k_0}\xi )$. The operator is $({\xi }^2+{\eta }^2{)}^{-1}({\partial }_{\xi }^2+{\partial }_{\eta }^2)$, and ${\mathcal{F}}^{\prime \prime }(s)+2is{\mathcal{F}}^{\prime }(s)=0$ makes its Helmholtz residual zero. At $\theta =\pm \pi$, the two Fresnel arguments coincide, while the derivatives of both the arguments and plane-wave phases cancel in pairs. Hence ${\partial }_{\theta }\Phi =0$, which is the required normal boundary condition.

Away from the Fresnel transition regions, integration by parts gives the Fresnel tail expansionwhere $H$ is the Heaviside step function. The same expansion follows from the complementary-error-function asymptotics in NIST DLMF. Because $s_{\pm }^2=k_{0}r[1+\cos (\theta \pm {\theta }_0)]$, the tails in both terms have outgoing phase $e^{-i{k}_{0}r}$. The scattered far field is consequentlyThe first term cancels the incident geometrical optics field in its shadow, the second is its reflected plane wave and the last is the diffracted edge wave. The expansion is nonuniform at $\theta ={\theta }_0-\pi$ and $\theta =\pi -{\theta }_0$; the exact Fresnel expression resolves those regions. Incidence from below follows by reflecting $\theta$ and ${\theta }_0$.

Use the unscaled Finite Fresnel integrals of this paper. Termwise integration of the absolutely convergent Taylor series for sine and cosine gives the full small-argument expansionsIn particular,These Taylor series converge at every finite argument, although cancellation makes them unattractive for large arguments.

For the large-argument asymptotic expansions, combine the two Finite Fresnel integrals into $C+iS$. The standard Fresnel integral givesFor the tail $J(\lambda )={\int }_{\lambda }^{\infty }{e}^{i{t}^2}dt$, integration by parts using $(e^{i{t}^2}{)}^{\prime }=2it{e}^{i{t}^2}$ givesFor example $a_1=1/4$, $a_2=-3i/8$, and $a_3=-15/16$. Repeated integration by parts bounds the remainder after any fixed number of terms by the order of the next term. Equivalently the Fresnel tail expansion isHere $(-1)!!=1$. The first three terms of the requested cosine Fresnel integral, counting its constant term, areFor comparison, retaining one further correction in both Finite Fresnel integrals givesUnlike the small-argument Taylor series, these large-argument asymptotic expansions must be truncated; adding indefinitely many terms at a fixed argument does not improve the approximation.

A calculator needs elementary arithmetic and its built-in sine and cosine, together with stored approximation coefficients. A convenient global construction uses compactified approximation from endpoint asymptotics. Define the exact real tail functions byThen $P(0)=Q(0)=A$, while $P\sim 1/(2\lambda )$ and $Q\sim 1/(4{\lambda }^3)$ at infinity. Consequentlyextend continuously to the compact interval $0\le u\le 1$, with $p(1)=1/2$ and $q(1)=1/4$. Choose stored polynomial approximations $p_N,q_N$ on that interval, constrained by the small-argument Taylor series and large-argument asymptotic expansions, and calibrated across the intervening interval. The resulting elementary formula isUniform errors ${\delta }_p,{\delta }_q$ in the stored polynomial approximations give $|C_{\mathrm{calc}}-C|\le {\delta }_p+{\delta }_q$ for every positive argument. Existence of arbitrarily accurate such approximations follows from the Weierstrass approximation theorem. A Padé approximant with a denominator having no positive zero is an alternative implementation. Matching the two end expansions alone is not an error guarantee: the middle interval must also be controlled. For very small arguments, switching to the short Taylor series avoids cancellation. Ordinary finite-precision limitations in evaluating ${\lambda }^2$ and reducing its trigonometric phase remain those of the calculator itself.

For a Neumann screen on the negative horizontal axis, the Helmholtz equation has an exact solution in Fresnel integrals. With time dependence $e^{i\omega t}$, a wave incident from angle $0<{\theta }_0<\pi$ has total field $e^{ikr\cos (\theta -{\theta }_0)}\mathcal{F}(s_{-})+e^{ikr\cos (\theta +{\theta }_0)}\mathcal{F}(-s_{+})$, where $s_{\pm }=\sqrt{2kr}\cos [(\theta \pm {\theta }_0)/2]$ and $\mathcal{F}(s)=\frac{1}{2}\mathrm{erfc}(-e^{i\pi /4}s)$. The angular derivatives cancel on both faces of the screen. Away from the two shadow boundaries, the Fresnel tail expansion splits this into incident/reflected geometrical optics fields and an outgoing edge wave proportional to $r^{-1/2}{e}^{-ikr}$.