96abaafc-363e-4806-be54-bc9a0c260b66 18 3 5 decimal

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

1. Reduce a double integral

r 2^x 2^- sqrt

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

r 2^x 2^- sqrt -

i px x * y py * + (* - exp y r 2^x 2^- sqrt - r 2^x 2^- sqrt int x r - r int maple i i px x * r 2^x 2^- (12 /^py * + (* - exp - i px x * r 2^x 2^- (12 /^py * - (* - exp + (* py / - x r..r - = int

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 * -

2. Introduce user system we kow nothing about ... yet

M lo hi Scale hi lo - (M M Min - M Max M Min - 106 -^+ / (* lo + :

asymmetric mapping x β f 11 x β^+ / :

r 0.375 : (pin hole radius ≡ Δ 31 : (size momentum ≡ N 31 : (canvas size ≡ N Δ ≥ (Δ>N not diffraction ≡ px py 12 mat position momentum ≡ N odd to escape divide zero j 0 N range : px Δ - 2 Δ * j * N / + : py Δ - 2 Δ * j * N / + : 12 mat 81 line

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 ≡

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) 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 / - x r - r int :

3. The particular application is ready to run. At this point: it is scalar wrt ∫(x)

t0 1 time :

P 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 Fo 21 line :

P P : x f x 0.125 f : fmap P f vectorize : CoC fmap - 0255 Scale : 41 sys

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

Circle of Confusion

CoC

1 time t0 - min 9

x f

Example 2 from maple pre-algorithm ================

x y F x 2^y 2^+ sqrt :

x c 0 x * :

x d x :

x y F y x c x d int x int

# # int # int

optimiz: Symbolic x y F y x c x d int x int maple x 2 sqrt x 2^sqrt * x x 2 sqrt x 2^sqrt * + ln * + x x ln * - (* 2 / x int

x integrand x 2 sqrt x 2^sqrt * x x 2 sqrt x 2^sqrt * + x / ln * + (* 2 / :

integrand is sanity checked

x integrand

u int x integrand x 1015 -^u int :

sanity checked 1 int 0.382598

x 2 sqrt x 2^sqrt * x x 2 sqrt x 2^sqrt * + ln * + x x ln * - (* 2 / x 2 sqrt x 2^sqrt * x x 2 sqrt x 2^sqrt * + x / ln * + (* 2 / ≡

Example 3 Revisit an easy arc-length ================

u f u u u sin * + sin :

optimiz NONE 1 u f u diff 2^+ sqrt u a b int 11 u sin + u u cos * + (u 1 u sin + (* cos * (2^+ sqrt u a b int

a 0 :

b 2 π * :

sensible to acc [1000] typical
reasonnable on that range [0,2π] 1 u f u diff (2^+ sqrt u a b int 12.062

u arcIntegrand 11 u sin + u u cos * + (u 1 u sin + (* cos * (2^+ sqrt :

x c 10 x ≤ (x 2 π * ≤ (& cases :

x arcIntegrand x c * 2 1 sys