Reduce the double integral

0af5e881-3d45-493a-9fc7-d6b55a3afad8 14 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

This Smath Fourier Quantum is adapted from Mathcad 11.

I was given writen permission from Frank Rioux to publish

the Mathcad 11 work sheet in my former Mathcad web site.

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Fourier Quantum CoC [Circle of Confusion]

jmGiraud: 2019_05_22

Optimiz => SYMBOLIC M lo hi Scale hi lo - (M M Min - M Max M Min - / (* lo + :

Reduce the double integral

Fourier transform of position wave function into 
momentum representation to get diffraction pattern 
@ the detection screen ... recast Φ(px,py) . px py Φ i px x * y py * + (* - exp y r 2^x 2^- sqrt - r 2^x 2^- sqrt int x r - r int ≡

1. Reduce a double integral

pass the double integrand to maple.
evaluate maple SYMBOLIC i px x * y py * + (* - exp y r 2^x 2^- sqrt - r 2^x 2^- sqrt int x r - r int

evaluate symbolic/recast i px x * y py * + (* - exp y r 2^x 2^- sqrt - r 2^x 2^- sqrt int x r - r int maple

Symbolic

i i px x * r 2^x 2^- sqrt py * + (* - exp - i px x * r 2^x 2^- sqrt py * - (* - exp + (* py / - x r..r - = int

isolate/recast the integrand i py / i r 2^x 2^- sqrt py * x px * + (* - exp * i py / i r 2^x 2^- sqrt - (py * x px * + (* - exp * -

recast/insert the limits i i px x * r 2^x 2^- sqrt py * + (* - exp - i px x * r 2^x 2^- sqrt py * - (* - exp + (* py / - x r - r int

assign the integral
wrt the dependents
Φ(px,py) ... acc 50
.... SYMBOLIC .... px py Φ i i px x * r 2^x 2^- sqrt py * + (* - exp - i px x * r 2^x 2^- sqrt py * - (* - exp + (* py / - vectorize x r - r int :

Fourier transform of position wave function into 
momentum representation to get diffraction pattern 
@ the detection screen ... recast Φ(px,py) . px py Φ i px x * y py * + (* - exp y r 2^x 2^- sqrt - r 2^x 2^- sqrt int x r - r int ≡

β 0.125 :

x q 11 x abs β^+ / :

The mapping function

x q 0.275 0.625 0.275 0.25 2 2 mat 2 1 sys

Observe . . .

For the double integration, this Simpson is 'n' dependent,

perfectly symmetric vs the built-in integrator.

Set 'n=24' representative demo of the CoC project.

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inner integrand borrowed from collapsed ...  extracted from Maple Φ(px,py) x px py f i i px x * r 2^x 2^- sqrt py * + (* - exp - i px x * r 2^x 2^- sqrt py * - (* - exp + (* py / - :

Numeric

f a b n ∫∫ h b a - n / : 13 / h 2 / a px py f b px py f + (* h a k h * + px py f k 1 n 1 - sum * + (* 23 / h * a h 2 / - k h * + px py f k 1 n sum * + eval 21 line :

[n] 24 * 15 = 360° a r - : b r : n 24 : 13 mat

recast Simpson double integral px py Φ f a b n ∫∫ :

r 0.275 : (pin hole radius ≡ Δ 51 : (size momentum ≡ N 51 : (canvas size ≡ N Δ ≥ (N<Δ not diffraction ≡ px py 12 mat position momentum ≡ N odd escape divide zero j 0 N range : px Δ - 2 Δ * j * N / + : py Δ - 2 Δ * j * N / + : 12 mat 81 line

..... NUMERIC ..... CoC m 1 N range n 1 N range Fo m 1 - n 1 - el px m el py n el Φ abs (2^: for for fmap Fo q vectorize : fmap - 0255 Scale 31 line :

t0 1 time :

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1 time t0 - min 12.1

Fo m 1 - n 1 - returned ORIGIN 0 ≡ el

hatch

Erzeugt eine Matrix für die Darstellung vertikaler Linien zwischen den Funktionen f(x) und 
g(x) im Bereich von x1 bis x2 mit 2*n Linien. Generates a matrix for 2*n vertical lines (hatching) between functions f(x) and g(x)
in x-range from x1 to x2 f g x.1 x.2 N hatch dx x.2 x.1 - N / (: h x.1 x.1 f x.1 x.1 g 22 mat : x x.1 x.2 dx - x.1 dx + range h h x dx 2 / + x dx 2 / + g x dx 2 / + x dx 2 / + f x dx + x dx + f x dx + x dx + g 42 mat stack : for h 41 line :

x r Ψ 12 r * sqrt / x r - ≥ (x r ≤ (* 0 cases :

x f x 2 Ψ abs (2^:

x g 0 :

coordinate-space prob. density

f g 2 - 2 20 hatch 2 1 sys

jmG: 20230416