Reduce the double integral

3daf5f85-42b2-4171-822e-42749feab939 58 4 5 decimal

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

4 appVersion 0.98.6179.21440 ≡

set 'n' the length of expansion

n 12 :

Optimiz OFF [None]

Thiele continued fraction x g x sin confrac x n convert maple : x 1 x 2^6 x 2^107 / - x 2^68611 / - x 2^21783857 / 1331 x 2^* 607202 / + / + / + / + / + /

Thiele continued fraction x g x cos confrac x n convert maple : 1 x 2^2 - x 2^6 - x 2^103 / x 2^37813 / 177 x 2^* 338 / - / + / + / + / +

Thiele continued fraction x g x tan confrac x n convert maple : x 1 x 2^3 - x 2^5 x 2^7 - x 2^9 x 2^11 / - / + / + / + / + /

Thiele continued fraction x g x exp confrac x n convert maple : 1 x 1 x 2 - x 3 - x 2 x 5 x 2 - x 7 - x 2 x 9 x 2 - x 11 / - / + / + / + / + / + / + / + / + / + / +

x Clear 1

Thiele continued fraction x g x atan confrac x n convert maple : x 1 x 2^3 x 2^54 / x 2^289 / x 2^8164 / 225 x 2^* 704 / + / + / + / + / + /

Thiele continued fraction x g x asin confrac x n convert maple : x 1 x 2^6 - x 2^1017 / x 2^4046549 / - x 2^6028021173833 / 12889271025 x 2^* 104890816822 / - / + / + / + / + /

Thiele continued fraction x g x acos confrac x n convert maple : π 2 / x 1 - x 2^6 x 2^1017 / - x 2^4046549 / x 2^6028021173833 / - 12889271025 x 2^* 104890816822 / + / + / + / + / + / +

Thiele continued fraction x Ln 1 x + ln confrac x n convert maple : x 1 x 2 x 3 x 1 x 5 x 23 / x 7 x 12 / x 9 x 25 / x 11 / + / + / + / + / + / + / + / + / + / + /

x f x sqrt :

Thiele continued fraction x SQRT 1 x + f confrac x n convert maple : 1 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 / + / + / + / + / + / + / + / + / + / + / + / + / +

x λ 10 x ≤ (x 4 ≤ (& cases :

x 1 - Ln x λ * x ln 2 1 sys x 1 - SQRT x λ * x sqrt 2 1 sys

set 'n' the length of expansion

n 10 :

Optimiz OFF [None]

Thiele continued fraction x g x exp confrac x n convert maple : 1 x 1 x 2 - x 3 - x 2 x 5 x 2 - x 7 - x 2 x 9 / + / + / + / + / + / + / + / + / +

Optimiz NONE

Padé rational fraction x g confrac x convert maple x 5 / 495 / + 12245 x 85351 / - 1876212601 x 711455195279 / - 41417793614661241 x 37686194931943489359 / - 3461343788889257628320041 x 858879507571 / - (* / + (* / + (* / + (* / +

Maple does not expand in continued fraction "ladder style" Expansion of p(x) is in advanced numerical maths J_Fraction

x p 3 x 3^* 10 x 2^* + 12 + 3 x 2^* 2 x 2^* - 12 - / :

x p confrac x convert maple 3 x * 10 + 36 x 113 / - 139 x 113 / + (* / + / +

x S 3 x 3^* 10 x 2^* + 12 + 3 x 2^* 2 x 2^* - 12 - / parfrac x convert maple : 3 x * 10 + 13236 x * + x 2^12 - / +

x S x 3^x + x 2^3 x * - 2 + / parfrac x convert maple : x 3 + 2 x 1 - / - 10 x 2 - / +

t f t a i b * + (* exp :

t f trig convert maple t a * cosh t a * sinh + (i t b * sin * t b * cos + (*

x sin x cos * exp convert maple i 1 - i x * exp 2^+ (* 1 i x * exp 2^+ (* 4 i x * exp 2^* / -

x 1 + (x 2 + (* expand maple 2 x 3 x + (* +

x 1 + (x 2 + (* expand maple x 2^3 x * + 2 +

Optimiz SYMBOLIC x 1 + x 2 + / expand maple 2 x + x 2 x + (* + 2 x + (2^/

Optimiz NONE x 1 + x 2 + / expand maple 1 x * x 2 + / 1 x 2 + / +

Assume expr(x) comes from source and ovelaps several pages ...
maple will compact c/w respective paramaters from calc suite. x expr β 1 el β 5 el x 2^* β 4 el x * + 1 + / β 2 el x * β 5 el x 2^* β 4 el x * + 1 + / + β 3 el x 2^* β 5 el x 2^* β 4 el x * + 1 + / + :

x expr simplify maple x β 3 el x * β 2 el + (* β 1 el + 1 x β 5 el x * β 4 el + (* + /

f 2 c * y * 3 x 2^* y 2^* y 4^- c 2^x 2^y 2^+ (* - 4 c 4^* + (* x cos * y sinh * 2 c * y * x 4^3 y 2^* x 2^* - c 2^x 2^y 2^+ (* - 4 c 4^* - (* x sin * y cosh * x y * x 2^y 2^+ (y 2^x 2^- 2 c 2^* + (* (* 2 c * sin * 31 mat :

x y f f 1 el f 2 el + f 3 el + :

x sin (2^exp convert maple 1 - i x * exp 2^+ (2^4 i x * exp 2^* / -

x asin ln convert maple i 1 x 2^- sqrt i x * + ln * -

z H z i ρ * - (z i ρ * + (* z ρ + (* z 3^/ :

z H parfrac z convert maple 1 ρ 3^z 3^/ + ρ 2^z 2^/ + ρ z / +

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

Explode for exploring the term by term composition.

x expr x f op maple :

x collect x expr 1 el x expr 2 el x expr 3 el x expr 4 el x f 51 sys :

x collect

x expr x 1 x 2^+ 11 - x + / 12 - x + / 41 mat

If you have a true partial fraction from source it's wise to simplify first to reduce the otherwise long expansion. For this univariate 'x' example: maple(,) simplify = factor. As a rule,, start assume (x>0) in Lapalce .

x 3 + 2 x 1 - / - 10 x 2 - / + simplify maple x 1 x 2^+ (* 1 - x + (2 - x + (* /

x x 2^1 + x 1 - (x 2 - (* / * x 0 > assume x s laplace maple x 1 x 2^+ (* 1 - x + (2 - x + (* s * /

x 3 + 2 x 1 - / - 10 x 2 - / + (x 0 > assume x s laplace factor maple x 1 x 2^+ (* 1 - x + (2 - x + (* s * /

x 3 + 2 x 1 - / - 10 x 2 - / + (x 0 > assume x s laplace simplify maple x 1 x 2^+ (* 1 - x + (2 - x + (* s * /

0 x < x 3 + 2 x 1 - / - 10 x 2 - / + Unwanted if 0 x < x x 2 ^ 1 + x 1 - ( x 2 - ( * / * Unwanted if

Sanity check

t sqrt laplace s maple t sqrt laplace s maple

t q t sqrt :

s L t q t s laplace maple : π sqrt 2 s 2 nthroot 3^* /

x acos x sin * exp convert maple i x acos * i x * exp 1 i x * exp / - (* 2 / -

x sin x acos * exp convert maple i i x * exp 1 i x * exp / - (* x acos * 2 / -

x sin x atan * exp convert maple i i x * exp 1 i x * exp / - (* x atan * 2 / -

x sin exp convert maple i i x * exp 1 i x * exp / - (* 2 / -

x cos exp convert maple i x * exp 2 / 12 i x * exp * / +

n 12 :

Thiele continued fraction x g x exp confrac x n convert maple :

Optimiz NONE

J_Frac piecewise x collect x g confrac x convert maple :

Optimiz NONE

J_Frac expanded x J_Frac x collect expand maple :

x g 1 x 1 x 2 - x 3 - x 2 x 5 x 2 - x 7 - x 2 x 9 x 2 - x 11 / - / + / + / + / + / + / + / + / + / + / +

x collect x 6 / - 716 / - 25556 x 175573 / - 7918405329 x 91506291806385 / - 35540459019612315025 x 12083495748749577127145 / - 1276917665261004014096397441 x 643629406036131369242178451835 / - 57162792361422133133965249672323225 x 315657529184296635 / - (* / + (* / + (* / + (* / + (* / -

x J_Frac x 6 / - 716 / - 25556 x * 1053073 / - 47510405329 x * 66799591724745 / - 189395106112251612315025 x * 29900610230158152003521 / - 781875872127230846525004014096397441 x * 1289525031210915217251184296635 / - 229456958885652497704151601397133965249672323225 x * 58174620407114915 - / + / + / + / + / -

x parfrac x g parfrac x convert maple :

x parfrac 716 / - x 6 / - 1552320 x * 378000 x 2^* - 28000 x 3^* + 2555 x 4^* - 4268880 - 332640 - 151200 x * + 30240 x 2^* - 3360 x 3^* + 210 x 4^* - 6 x 5^* + / +

x J_Frac x 6 / - 716 / - 25556 x * 1053073 / - 47510405329 x * 66799591724745 / - 189395106112251612315025 x * 29900610230158152003521 / - 781875872127230846525004014096397441 x * 1289525031210915217251184296635 / - 229456958885652497704151601397133965249672323225 x * 58174620407114915 - / + / + / + / + / - :

x J_Frac simplify maple 6329240 x 5^+ (* 35 x 4^* + 2520 x 2^* + (* 1120 x 3^* + 30240 x * + (* x 6^+ 6351699 x 3^- 9 x 2^* + (* x 4^+ 720 x * - (* - x 5^+ (* / -

n 11 :

optimiz NONE x Taylor x cos x n series maple : 1 x 2^2 / - x 4^24 / + x 6^720 / - x 8^40320 / + x 10^3628800 / -

u 0.25 :

u cos u Taylor 21 mat 0.968912421710645 0.968912421710645 21 mat

x cos x 13 taylor polynom convert maple 1 x 2^2 / - x 4^24 / + x 6^720 / - x 8^40320 / + x 10^3628800 / - x 12^479001600 / +

e x^x 7 taylor polynom convert maple 1 x + x 2^2 / + x 3^6 / + x 4^24 / + x 5^120 / + x 6^720 / +

x sqrt x 3 taylor polynom convert maple Error, does not have a taylor expansion, try series() ans

x f x 0.25^x n series polynom maple : x 0.25^x 11 series polynom maple

traditional code e x^x 7 taylor confrac convert maple 1 x 1 x 2 - x 3 - x 2 x 5 x 2 / - / + / + / + / + / +

x f x sqrt :

traditional code x f confrac x 7 convert maple Error, (in convert/confrac) wrong number (or type) of parameters in function ser

x f x sqrt :

Thiele continued fraction x SQRT 1 x + f confrac x n convert maple : 1 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 / + / + / + / + / + / + / + / + / + / + / + / +

Thiele continued fraction x Ln 1 x + ln confrac x 15 convert maple : x 1 x 2 x 3 x 1 x 5 x 23 / x 7 x 12 / x 9 x 25 / x 11 x 13 / x 137 x * 2 / + / + / + / + / + / + / + / + / + / + / + / + / + /

Laplace algebra in control systems p 1 0.2 s * + (1 0.5 s * + (* 0.8 10 s * + (1 0.2 s * + (* 1 0.5 s * + (* 1 0.05 / + / :

allvalues expands Maple kernel RootOf ≡ Maple stacks 3 solutions t sol 1 el t sol 2 el t sol 3 el 31 mat ≡ 21 mat

t sol p s t invlaplace allvalues maple :

collect the complete solution t Sol t sol 1 el t sol 2 el + t sol 3 el + :

t F t sol 2 el :

x damped 0 x ≤ x F Re Undefined if :

x enveloppe 0 x ≤ x F abs Undefined if :

x damped x enveloppe x enveloppe - 3 1 sys

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

x c 11 x ≤ (x 16 ≤ (& cases :

x ch 10 x ≤ (x 16 ≤ (& cases :

n 116 range :

r 2 :

z H z 0.5 + z 2^z - 0.5 + / :

H(z) under unit discrete impulse
Y(z) algo input to  ... Inv(n,Y) z Y z 0.5 + z 2^z - 0.5 + / (z z 1 - / (* :

0 H 1

The inverse Z_transform in the discrete domain n Y Inv r n^2 π * / r i θ * exp * Y n θ * i * exp * θ 02 π * int * :

j 1 n rows range y j el n j el Y Inv : for

R n y abs vectorize augment : H R . 14 red plot : 21 sys

1/1 sanity Mathcad source code

x H x ch * H 2 1 sys

Z_system for inverse transform

x H x Y x c * 2 1 sys

R 112 2.5 3 3.5 4 3.75 5 3.5 6 3.125 7 2.875 8 2.813 9 2.875 10 2.969 11 3.031 12 3.047 13 3.031 14 3.008 15 2.992 16 2.988 162 mat

Maple double integration

x y n r Clear 1

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

Symbolic

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

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 :

Scale/Map

Optimiz => SYMBOLIC Mat lo hi Scale hi lo - (Mat Mat Min - Mat Max Mat Min - 1012 -^+ / (* lo + :

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

Fourier transform of position wave function 
into momentum representation to get diffraction 
pattern @ the detection screen . 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 ≡

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

Numeric

Simpson Duoble Integration

fo a b n ∫∫ Simpson double integration of 'fo' in range [a,b] over 'n' steps acc h b a - n / : 13 / h 2 / a px py fo b px py fo + (* h a k h * + px py fo k 1 n 1 - sum * + (* 23 / h * a h 2 / - k h * + px py fo k 1 n sum * + eval 31 line :

... n = 12 convenient demo a r - : b r : n 12 : 13 mat

recast Simpson double integral px py Φ fo 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 5.7

Maple solve-inverse a function

maple will NOT solve any and all model.
Gaussian fnct ... piecewise. x μ colony f colony μ 1 el 11 μ 2 el / 1 - (μ 3 el x μ 4 el - (* - exp * + / + :

y Inverse x μ colony f y ≡ x solve maple : μ 3 el μ 4 el * μ 2 el colony - μ 1 el - y + (* colony colony μ 2 el * - y - y μ 2 el * + / ln - μ 3 el /

Smath undocumented "root" dual solver

t Wr Z K.p 1 α β α - (/ t 259.3467 + (- α K.p * / exp * + β β α - (/ t 259.3467 + (- β K.p * / exp * - (* + :

β Clear 1

undocumented native Smath  "0.98.6179.21440" w sol t Wr w - t root :

w sol Z 18 α * - (* K.p 18 α * 1 - 259346710000 t * + 10000 α * K.p * / - exp + (* + 259346710000 t * + 1250 K.p * / - exp - (* + w 18 α * - (* - 18 α * - / t root

simplify the arguent

w t argument β α - (Z w - (* K.p β 1259346710000 t * + 10000 β * K.p * / - exp - (* α 1 - 259346710000 t * + 10000 α * K.p * / - exp + (* + (* + β α - / :

Solving for 't' Given 'w' ... range wrt 't' w t Sol w t argument 0 ≡ t # # solve :

Solving for 'w' Given 't' ... range wrt 'w' w t Sol w t argument 0 ≡ w # # solve :

Maple RootOf . . .

applied for solving PDF in statistics..

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Fréchet distribution
α > 0 ... β > 0 α β x Fr α β / β x / (α 1 +^* β x / (α^- exp * :

w α β x Fr ≡ x solve maple w β * α β_Z / (1 α +^* β_Z / (α^- exp * - RootOf

recast RootOf

Fr 51 x Fr 11 x Fr 21 x Fr 0.5 1 x Fr 41 sys :

plot Fr 1 - Y 1.5 Y 22 mat X 1 el Y . 14 black 15 mat X 2 el Y . 14 black 15 mat 41 sys :

α 5 : β 1 : Y 1 : 13 mat w t ipdf w β * α β t / (1 α +^* β t / (α^- exp * - : w t Sol w t ipdf 0 ≡ t 1 - 10 solve : X Y t Sol (: 41 sys

'X' blue peak
'Y' blue peak 0.96419 1.87428 21 sys

x c 1 0.001 x ≤ (x 1.874278 ≤ (& cases :

t0 1 time :

Fréchet distribution ... α >0 ... β > 0

plot x t Sol 1 el x c * x t Sol 2 el x c * 1 1.5 1 0.5 2 2 mat 3 1 sys

1 time t0 - min 1.8

X α β X Fr vectorize 21 mat 0.809 1.234 21 mat 1121 mat 21 mat

By inspection ... Sol(■ ,t)produces two sols Forcibly each canvas branch each indexed on the plot.

0.25 t Sol 0.5 t Sol 1 t Sol 31 mat 0.728 1.623 21 mat 0.76 1.427 21 mat 0.809 1.234 21 mat 31 mat

NOTE ... Canvas Sol(x,t) is wrt [α, β]. For oter Frechet [α, β], consider ranging the solve bloc to suit the PDF as well as finding optimum c(x). This can be done using Golden ratio.

∫kernel(f,a,b,n) ... Simpson/Romberg Kernel [Jean]

Basic Romberg-Simpson ... integration acc from options ... NUMERIC f a b Romberg w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : 32 / b a - (* w G w 01 int * 31 line eval :

f a b n ∫kernel

Simpson Integrator in the Romberg Transform Domain [0,1] <= [a,b] ...  
... integrates  Smath ainterp(X,Y,x), cinterp(X,Y,x), cubic(x) from  l_p_c_spline, Y^X 
Integration accuracy from user command line 'n' ... NUMERIC.
The integration variate is in Romberg [0,1] within 'n' ... 'n' independent of work sheet option. 
Thus, on long range of the integration variate ∫kernel(f,a,b,n) improves the convergence. f a b n ∫kernel w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : ... α 0 : β 1 : h β α - n / : 15 mat Simpson 13 / h 2 / α G β G + (* h α k h * + G k 1 n 1 - sum * + (* 23 / h * α h 2 / - k h * + G k 1 n sum * + eval : 32 / b a - (* Simpson * 41 line :

Some Fréchet PDF will not integrate Simpson 
Under the hood, Fr is the Power fnct X^Y.
Recast to suit ∫kernel(f,a,b,n) α β x Fr α β / β x / (α 1 +^* β x / (α^- exp * :

α 5 : β 1 : 12 mat

recast to suit integrator x f α β x Fr :

f 0.01 550 ∫kernel 0.9997

Maple Undocumented . . .

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The matrix function x y φ x y * :

... NUMERIC ...
The times table you can expand at will
there is simpler method direct from Smath. 55 φ matrix maple 1234524681036912154812162051015202555 mat

αγ 1 ω - (* π + x uu + ≡ uu solve maple αγ 1 ω - (* π + x -

x f 2010 x * + x 2^+ :

datum solution(s) X x f fsolve maple : 7.2361 - 2.7639 - 21 mat

elevated solution(s) x f 16 ≡ fsolve maple 9.5826 - 0.4174 - 21 mat

x f X 1 el 0 + 20 red 1 5 mat X 2 el 0 + 20 red 1 5 mat 3 1 sys

x sol x sin 14 / ≡ x fsolve maple x sin 12 / ≡ x fsolve maple x sin 34 / ≡ x fsolve maple x sin 1 ≡ x fsolve maple 41 mat :

_sol 1 el_sol 2 el_sol 3 el_sol 4 el 41 mat 0.2527 0.5236 0.8481 1.5708 41 mat

x sin 0.5 ≡ fsolve maple ° 30

x ≤ 1 x sol x sin 1 ≡ x fsolve maple : ° 90

x sin ln 0 ≡ x fsolve maple 1.5708

x sin ln 1.5708 0 + 50 red 1 5 mat 2 1 sys

sol x sin 12 / ≡ x solve maple x sin 12 / ≡ x fsolve maple 21 mat : 0.5236 0.5236 21 mat

multiple x sin 12 / ≡ x 010 solve : 0.5236 2.618 6.8068 8.9012 41 mat

j 2 :

X multiple j el :

x sin 0.5 X 1 X 0.5 - 2 2 mat 3 1 sys