Reduce the double integral

b6496b22-9c2b-4ad8-969a-a385756f9796 6 4 5 decimal

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

ode t y t diff t x ≡ t x t diff t x - ≡ 0 x 1 ≡ 1 y 0 ≡ 41 sys set convert t x t y 21 sys dsolve maple :

ode t - exp t - exp - 1 - exp + 21 mat

ode t y t diff t x ≡ t x t diff t x - ≡ 0 x a ≡ 1 y b ≡ 41 sys set convert t x t y 21 sys dsolve maple :

ode t - exp a * b a t - exp - 1 - exp + (* + 21 mat

x q x 2^1 - (x 2^1 + (- / :

x p 2 x * x 2^1 + (- / :

C x f x q x p x int exp * x int C + maple :

C2 x Φ C2 x q x p - (x int exp * maple :

C2 x Φ C2 x - 2 x atan * + 1 x 2^+ ln + exp *

C x f C 1 x 2^+ (* x + 1 x 2^+ /

0 x f 1 x Φ

n 2 :

x μ 1 n - (x p x int * exp maple : 1 x 2^+

C x new 1 C x f x μ * / : 1 C 1 x 2^+ (* x + /

1 x new

DE t φ t diff b m / t φ (2^* + :

cp 0 φ V.0 α cos * - :

t φ 2 DE cp 21 sys set convert t φ dsolve op maple :

where cosα is simply a parameter
correctly interpreted by maple t φ m V.0 * cosα * b t * V.0 * cosα * m + /

where cosα is simply a parameter
correctly interpreted by maple. m b V.0 t X ξ φ ξ 0 t int maple : m m t b * V.0 * cosα * + m / ln * b /

where cosα is simply a parameter
correctly interpreted by maple. m b V.0 t X m m t b * V.0 * cosα * + m / ln * b /

recast cosα => cos(α) m b V.0 α t X m m t b * V.0 * α cos * + m / ln * b / :

recast cosα => cos(α) m b V.0 α t Φ m V.0 * α cos * b t * V.0 * α cos * m + / :

m 1 : b 1 : V.0 1 : α 1 : 14 mat

m b V.0 α x X m b V.0 α x Φ 2 1 sys

x f 42 - 27 x * - 13 x 2^* + 5 x 3^* - x 4^+ :

Maple undocumented op

Expands/separates line arguments in vector.

May not be easy to handle on some function.

Usuful to manipulate "RootsOf"

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x expr x f op maple :

x expr 42 - 27 x * - 13 x 2^* 5 x 3^* - x 4^51 mat

f t 32 /^e t^- a t * sinh + (:

t expr f op maple :

t expr t 2 nthroot 3^1 exp t^- a t * sinh 31 mat

f β 1 el β 2 el 1 β 3 el x β 4 el - β 5 el / (2^* + / + :

t expr f op maple :

t expr β 1 el β 2 el β 5 el 2^* β 5 el 2^β 3 el x β 4 el - (2^* + / 21 mat

simplified
executable t expr 2 el β 2 el β 5 el 2^* β 5 el 2^β 3 el x β 4 el - (2^* + /

completely
expanded β 2 el β 5 el 2^* β 1 el β 5 el 2^β 3 el x β 4 el - (2^* + (* + β 5 el 2^β 3 el x β 4 el - (2^* + /

DE Verhulst [Logistic]

a k t f 1 y a y - (* / y int k t int ≡ y solve maple :

a k t f a k t * a * exp * 1 k t * a * exp + /

a k t f a k t * a * exp * 1 k t * a * exp + / :

a k y0 t f1 1 u a u - (* / u y0 y int k u 0 t int ≡ y solve maple :

a k y0 t f1 y0 a * k t * a * exp * a y0 1 - k t * a * exp + (* + /

a k y0 t f1 y0 a * k t * a * exp * a y0 1 - k t * a * exp + (* + / :

k y0 t0 t f2 1 u 1 u - (* / u y0 y int k u t0 t int ≡ y solve maple :

k y0 t0 t f2 y0 k t t0 - (* exp * 1 y0 1 - k t t0 - (* exp + (* + /

yo r t to - (* exp * 1 yo - yo r t to - (* exp * + / 111 yo / 1 - (r t to - (* - exp * + / ≡

NOT from Smath

a k y0 t0 t f3 1 u a u - (* / u y0 y int k u t0 t int ≡ y solve maple :

a k y0 t0 t f3 y0 a * k a * t t0 - (* exp * a y0 1 - k a * t t0 - (* exp + (* + /

r to yo t Logistic 111 yo / 1 - (r t to - (* - exp * + / :

yo yo - 1 + (r t to - (* - exp * + yo /

a k t f a k t * a * exp * 1 k t * a * exp + / :

a k y0 t f1 y0 a * k t * a * exp * a y0 1 - k t * a * exp + (* + / :

r 1 : yo 0.5 : to 10 : ε 0.75 : 14 mat

r to yo x Logistic r to ε - yo x Logistic r to ε + yo x Logistic to yo + 15 black 1 5 mat 4 1 sys 1.25 0.5 x f 1.25 0.75 x f 1.25 1 x f 3 1 sys 1.25 0.5 0.025 x f1 1.25 0.75 0.025 x f1 1.25 1 0.025 x f1 3 1 sys

So a P x f 1 u / u So S int : (a So * P / u 0 x int ≡ S solve maple So ln P * a So * x * + P / exp

So 0.1 : a 1 : P 1 : 13 mat

x Eiffel So ln P * x a * So * + P / exp :

x y axes x y / - :

x 24 + Eiffel x - 24 + Eiffel axes 3 1 sys

Read more . . . Inst_ChebyShev Centroid

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Solving wrt What ? ... wrt 't'

# # int

SYMBOLIC ... 1 a / - (1 ε 2^v 2^+ / v int (* 1 t int ≡ maple

Lambert solving functions

x W0 w w exp * x - 0 ≡ w 1 e / - roots : x W1 w w exp * x - 0 ≡ w 2 sqrt - roots : x W w w exp * x - 0 ≡ w 1 e / - roots : 31 sys

DE that has Lambert solution wrt 'y'

Lagrangian style dx x 2^/ y 2^dy * 1 y - / - 0 ≡

# # int

separation of variables dy y 2^1 y - / * 1 x 2^/ C + ≡

x W0 x W1 2 1 sys

x f 1 x 2^/ :

explicit the expressibles y g 1 y - y 2^/ :

integrate the expressibles x y C sol x f x int y g y int C + ≡ :

solving the expressibles
sol'n algebraic expression x y C sol x y C sol maple :

Lambert solution wrt 'y'
recast in Smath Lambert x y C sol y solve maple 11 C x * + x / - exp - LambertW / -

C x q 11 C x * + x / - exp - W / - :

0 0.999999999 q 1.000044723

α 0.625 :

Lambert[W0, W1]does not support if/otherwise

0 x q α x q 1 x q 1.125 x q 1 1 + 15 black 1 5 mat 1 1 0 x * + x / - exp - W1 / - 1 1 α x * + x / - exp - W1 / - 2.666666666 1 + 15 black 1 5 mat 8 1 sys

0 x q α x q 1 x q 1.125 x q 11 + 15 black 15 mat 110 x * + x / - exp - W1 / - 11 α x * + x / - exp - W1 / - 2.666666666 1 + 15 black 15 mat 81 sys

approx. couldn't solve

x 2.666666666 :

11 α x * + x / - exp - W1 / - 0.999986307

4 appVersion 0.98.6179.21440 ≡

I.11 I.22 21 mat I.11 R.0 R.1 + p L * + 1 p C * / + (* I.22 1 p C * / * - U.Co p / L I.Lo * + ≡ I.11 1 - p C * / (* I.22 1 p C * / R.2 + (* + E.m p / U.Co p / - ≡ 21 sys I.11 I.22 21 sys solve maple :

p I.C I.22 I.11 - simplify p collect maple :

p I.C C L E.m - R.2 I.Lo * + U.Co + (* p * U.Co R.2 R.0 + R.1 + (* + E.m R.0 R.1 + (* - (* R.0 R.1 + R.2 1 p 2^L * C * + (* + L C R.2 * R.0 R.1 + (* + (p * + /

p U.C p I.C p C * / :

C 2.2 106 -^* :

L 46103 -^* :

R.0 25 :

R.1 5 :

R.2 243 :

E.m 195 :

U.Co E.m R.1 * R.1 R.2 + / : 3.9315

I.Lo E.m R.1 R.2 + / : 0.7863

U.C1 E.m R.0 R.1 + (* R.0 R.1 + R.2 + / : 21.4286

! float is NOT optional ! t u.c p U.C p t invlaplace float convert maple :

x c 10 x ≤ cases :

x 0.001 * u.c x c *

t u.c 402715910446519651 - 126136567 t * 100000 / - exp 154193939 t * 50000 / cos * + (* 111085955131178126136567 t * 100000 / - exp * 154193939 t * 50000 / sin * + (* 620881716410319 /

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 ≡

x y r Clear 1

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

Symbolic

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^- 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 :

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 - 106 -^+ / (* 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 13.1

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