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eslamelidy

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These are questions asked by eslamelidy

please i want to known how to solve thi...

eslamelidy 20
plot integral sample
January 21 2022
0 0


 

``

> 

K := proc (x, t) options operator, arrow; 5*x+3*t end proc; 1; f := proc (x, t) options operator, arrow; 2*x+3*t^2 end proc

proc (x, t) options operator, arrow; 5*x+3*t end proc

  

proc (x, t) options operator, arrow; 2*x+3*t^2 end proc

 (1)
> 

"g(x,t):=f(x,t)+(∫)[0]^(t)K(x,t-u) ⅆW(u) "
where W(u) is wiener process   
how to solve this integeral at t=.5sec and plot sample paths for x=0..5

``


 

Download stochastic_integral.mw

how to export my data in sheet excel ...

eslamelidy 20 Maple 2021
export exceltools duplicate-question
October 11 2021
1 2

how to export my data in sheet excel for this file 

 

Nnew.mw  

how can i plot the relation between x an...

eslamelidy 20 Maple 18
plotting duplicate-question
July 17 2020
1 5

how can i plot the relation betwen x and B[k] from result of itterition in this work sheetx_phi.mw

> 

restart
> 

with(LinearAlgebra):
> 

lambda := 3.64*10^10

0.3640000000e11

 (1)
> 

mu := 5.46*10^10

0.5460000000e11

 (2)
> 

rho := 2330

2330

 (3)
> 

tau := 5*10^(-5)

1/20000

 (4)
> 

T[0] := 800

800

 (5)
> 

d[n] := -9*10^(-31)

-9/10000000000000000000000000000000

 (6)
> 

d[e] := 2.5*10^(-3)

0.2500000000e-2

 (7)
> 

E[g] := 1.11

1.11

 (8)
> 

C[e] := 695

695

 (9)
> 

alpha[T] := 4.14*10^(-6)

0.4140000000e-5

 (10)
> 

delta := (3*lambda+2*mu)*alpha[T];

904176.0000

 (11)
> 

r := 2

2

 (12)
> 

omega[0] := -.3

-.3

 (13)
> 

``
> 

epsilon[0] := 8.85*10^(-12)

0.8850000000e-11

 (14)
> 

k := 800

800

 (15)
> 

C[T] := sqrt((2*mu+lambda)/rho)

7905.015521

 (16)
> 

mu[0] := (4*3.17)*10^(-7)

0.1268000000e-5

 (17)
> 

t[1] := k/(rho*C[e]*C[T]^2)

0.7905763302e-11

 (18)
> 

q[2] := k*t[1]/(d[e]*rho*tau*C[e])

0.3124518178e-7

 (19)
> 

q[1] := k/(d[e]*rho*C[e])

.1976101522

 (20)
> 

a := .5

.5

 (21)
> 

mu := 5.46*10^10

0.5460000000e11

 (22)
> 

``
> 

q[3] := a/C[T]^2

0.8001373626e-8

 (23)
> 

epsilon[1] := delta^2*T[0]*t[1]/(k*rho)

0.2773919393e-2

 (24)
> 

epsilon[2] := alpha[T]*E[g]*t[1]/(d[n]*rho*tau*C[e])

-0.4985559321e12

 (25)
> 

kappa := 386

386

 (26)
> 

epsilon[3] := d[n]*k*kappa*t[1]/(alpha[T]*rho*C[e]*d[e])

-0.1310939149e-33

 (27)
> 

NULL
> 

NULL
> 

delta[n] := (3*lambda+2*mu)*d[n]

-0.1965600000e-18

 (28)
> 

H0 := 10^5

100000

 (29)
> 

R[H] := 1+epsilon[0]*((4*3.17)*10^(-7))^2*H0^2/rho

1.

 (30)
> 

alpha[0] := 1+(4*3.17)*10^(-7)*H0^2

12681.00000

 (31)
> 

nu := 2

2

 (32)
> 

for y from 0 to 300 do x := 0+0.1e-1*y; t := .8; s := 4.7/t; A[1] := -(-s^4*R[H]-s^3*R[H]*q[3]-s^3*alpha[0]*q[1]-s^2*alpha[0]*q[1]*q[3]+s^2*epsilon[2]*q[1]*q[3]-s^3*alpha[0]+s^3*epsilon[2]-s^2*alpha[0]*q[2]-s*alpha[0]*q[2]*q[3]+s*epsilon[2]*epsilon[3]*q[3]+s*epsilon[2]*q[2]*q[3]+alpha[0]*epsilon[1]*epsilon[3]*q[3])/(s^2*alpha[0]+s*alpha[0]*q[3]-s*epsilon[2]*q[3]); A[2] := (s^5*R[H]*q[1]+s^4*R[H]*q[1]*q[3]+s^5*R[H]+s^4*R[H]*q[2]+s^4*alpha[0]*q[1]-s^4*epsilon[2]*q[1]+s^3*R[H]*q[2]*q[3]-s^2*R[H]*epsilon[1]*epsilon[3]*q[3]+s^3*alpha[0]*q[2]-s^3*epsilon[2]*epsilon[3]-s^3*epsilon[2]*q[2]-s^2*alpha[0]*epsilon[1]*epsilon[3])/(s^2*alpha[0]+s*alpha[0]*q[3]-s*epsilon[2]*q[3]); A[3] := (-s^6*R[H]*q[1]-s^5*R[H]*q[2]+s^4*R[H]*epsilon[1]*epsilon[3])/(-s^2*alpha[0]-s*alpha[0]*q[3]+s*epsilon[2]*q[3]); M[1] := (1/6)*sqrt(6)*sqrt((8*A[1]^3-36*A[2]*A[1]+108*A[3]+12*sqrt(12*A[1]^3*A[3]-3*A[1]^2*A[2]^2-54*A[1]*A[2]*A[3]+12*A[2]^3+81*A[3]^2))^(1/3)*((8*A[1]^3-36*A[2]*A[1]+108*A[3]+12*sqrt(12*A[1]^3*A[3]-3*A[1]^2*A[2]^2-54*A[1]*A[2]*A[3]+12*A[2]^3+81*A[3]^2))^(2/3)+2*A[1]*(8*A[1]^3-36*A[2]*A[1]+108*A[3]+12*sqrt(12*A[1]^3*A[3]-3*A[1]^2*A[2]^2-54*A[1]*A[2]*A[3]+12*A[2]^3+81*A[3]^2))^(1/3)+4*A[1]^2-12*A[2]))/(8*A[1]^3-36*A[2]*A[1]+108*A[3]+12*sqrt(12*A[1]^3*A[3]-3*A[1]^2*A[2]^2-54*A[1]*A[2]*A[3]+12*A[2]^3+81*A[3]^2))^(1/3); M[2] := (1/6)*sqrt(3)*sqrt((8*A[1]^3-36*A[2]*A[1]+108*A[3]+12*sqrt(12*A[1]^3*A[3]-3*A[1]^2*A[2]^2-54*A[1]*A[2]*A[3]+12*A[2]^3+81*A[3]^2))^(1/3)*(I*sqrt(3)*(8*A[1]^3-36*A[2]*A[1]+108*A[3]+12*sqrt(12*A[1]^3*A[3]-3*A[1]^2*A[2]^2-54*A[1]*A[2]*A[3]+12*A[2]^3+81*A[3]^2))^(2/3)-(4*I)*sqrt(3)*A[1]^2+(12*I)*sqrt(3)*A[2]-(8*A[1]^3-36*A[2]*A[1]+108*A[3]+12*sqrt(12*A[1]^3*A[3]-3*A[1]^2*A[2]^2-54*A[1]*A[2]*A[3]+12*A[2]^3+81*A[3]^2))^(2/3)+4*A[1]*(8*A[1]^3-36*A[2]*A[1]+108*A[3]+12*sqrt(12*A[1]^3*A[3]-3*A[1]^2*A[2]^2-54*A[1]*A[2]*A[3]+12*A[2]^3+81*A[3]^2))^(1/3)-4*A[1]^2+12*A[2]))/(8*A[1]^3-36*A[2]*A[1]+108*A[3]+12*sqrt(12*A[1]^3*A[3]-3*A[1]^2*A[2]^2-54*A[1]*A[2]*A[3]+12*A[2]^3+81*A[3]^2))^(1/3); M[3] := (1/6)*sqrt(-3*(8*A[1]^3-36*A[2]*A[1]+108*A[3]+12*sqrt(12*A[1]^3*A[3]-3*A[1]^2*A[2]^2-54*A[1]*A[2]*A[3]+12*A[2]^3+81*A[3]^2))^(1/3)*(I*sqrt(3)*(8*A[1]^3-36*A[2]*A[1]+108*A[3]+12*sqrt(12*A[1]^3*A[3]-3*A[1]^2*A[2]^2-54*A[1]*A[2]*A[3]+12*A[2]^3+81*A[3]^2))^(2/3)-(4*I)*sqrt(3)*A[1]^2+(12*I)*sqrt(3)*A[2]+(8*A[1]^3-36*A[2]*A[1]+108*A[3]+12*sqrt(12*A[1]^3*A[3]-3*A[1]^2*A[2]^2-54*A[1]*A[2]*A[3]+12*A[2]^3+81*A[3]^2))^(2/3)-4*A[1]*(8*A[1]^3-36*A[2]*A[1]+108*A[3]+12*sqrt(12*A[1]^3*A[3]-3*A[1]^2*A[2]^2-54*A[1]*A[2]*A[3]+12*A[2]^3+81*A[3]^2))^(1/3)+4*A[1]^2-12*A[2]))/(8*A[1]^3-36*A[2]*A[1]+108*A[3]+12*sqrt(12*A[1]^3*A[3]-3*A[1]^2*A[2]^2-54*A[1]*A[2]*A[3]+12*A[2]^3+81*A[3]^2))^(1/3); m[1, 1] := -(M[1]^2*q[3]-s^2)/s^2; m[1, 2] := -(M[2]^2*q[3]-s^2)/s^2; m[1, 3] := -(M[3]^2*q[3]-s^2)/s^2; m[2, 1] := epsilon[3]*(M[1]^2*q[3]-s^2)/(s^2*(-s*q[1]+M[1]^2-q[2])); m[2, 2] := epsilon[3]*(M[2]^2*q[3]-s^2)/(s^2*(-s*q[1]+M[2]^2-q[2])); m[2, 3] := epsilon[3]*(M[3]^2*q[3]-s^2)/(s^2*(-s*q[1]+M[3]^2-q[2])); m[3, 1] := (-M[1]*(M[1]^2*q[3]-s^2)*(-s*q[1]+M[1]^2-epsilon[3]-q[2])/(s^2*(-s*q[1]+M[1]^2-q[2])*(-s^2*R[H]+M[1]^2*alpha[0]))-epsilon[3]*(M[1]^2*q[3]-s^2)/(s^2*(-s*q[1]+M[1]^2-q[2]))+(M[1]^2*q[3]-s^2)/s^2)/mu; m[3, 2] := (-M[2]*(M[2]^2*q[3]-s^2)*(-s*q[1]+M[2]^2-epsilon[3]-q[2])/(s^2*(-s*q[1]+M[2]^2-q[2])*(-s^2*R[H]+M[2]^2*alpha[0]))-epsilon[3]*(M[2]^2*q[3]-s^2)/(s^2*(-s*q[1]+M[2]^2-q[2]))+(M[2]^2*q[3]-s^2)/s^2)/mu; m[3, 3] := (-M[3]*(M[3]^2*q[3]-s^2)*(-s*q[1]+M[3]^2-epsilon[3]-q[2])/(s^2*(-s*q[1]+M[3]^2-q[2])*(-s^2*R[H]+M[3]^2*alpha[0]))-epsilon[3]*(M[3]^2*q[3]-s^2)/(s^2*(-s*q[1]+M[3]^2-q[2]))+(M[3]^2*q[3]-s^2)/s^2)/mu; V[1] := (m[2, 2]*m[3, 3]-m[2, 3]*m[3, 2])*T[0]/((m[1, 1]*m[2, 2]*m[3, 3]-m[1, 1]*m[2, 3]*m[3, 2]-m[1, 2]*m[2, 1]*m[3, 3]+m[1, 2]*m[2, 3]*m[3, 1]+m[1, 3]*m[2, 1]*m[3, 2]-m[1, 3]*m[2, 2]*m[3, 1])*r)-(m[1, 2]*m[3, 3]-m[1, 3]*m[3, 2])*nu/((m[1, 1]*m[2, 2]*m[3, 3]-m[1, 1]*m[2, 3]*m[3, 2]-m[1, 2]*m[2, 1]*m[3, 3]+m[1, 2]*m[2, 3]*m[3, 1]+m[1, 3]*m[2, 1]*m[3, 2]-m[1, 3]*m[2, 2]*m[3, 1])*r*d[e]); V[2] := -(m[2, 1]*m[3, 3]-m[2, 3]*m[3, 1])*T[0]/((m[1, 1]*m[2, 2]*m[3, 3]-m[1, 1]*m[2, 3]*m[3, 2]-m[1, 2]*m[2, 1]*m[3, 3]+m[1, 2]*m[2, 3]*m[3, 1]+m[1, 3]*m[2, 1]*m[3, 2]-m[1, 3]*m[2, 2]*m[3, 1])*r)+(m[1, 1]*m[3, 3]-m[1, 3]*m[3, 1])*nu/((m[1, 1]*m[2, 2]*m[3, 3]-m[1, 1]*m[2, 3]*m[3, 2]-m[1, 2]*m[2, 1]*m[3, 3]+m[1, 2]*m[2, 3]*m[3, 1]+m[1, 3]*m[2, 1]*m[3, 2]-m[1, 3]*m[2, 2]*m[3, 1])*r*d[e]); V[3] := (m[2, 1]*m[3, 2]-m[2, 2]*m[3, 1])*T[0]/((m[1, 1]*m[2, 2]*m[3, 3]-m[1, 1]*m[2, 3]*m[3, 2]-m[1, 2]*m[2, 1]*m[3, 3]+m[1, 2]*m[2, 3]*m[3, 1]+m[1, 3]*m[2, 1]*m[3, 2]-m[1, 3]*m[2, 2]*m[3, 1])*r)-(m[1, 1]*m[3, 2]-m[1, 2]*m[3, 1])*nu/((m[1, 1]*m[2, 2]*m[3, 3]-m[1, 1]*m[2, 3]*m[3, 2]-m[1, 2]*m[2, 1]*m[3, 3]+m[1, 2]*m[2, 3]*m[3, 1]+m[1, 3]*m[2, 1]*m[3, 2]-m[1, 3]*m[2, 2]*m[3, 1])*r*d[e]); F[k] := sum(exp(-M[i]*x)*V[i], i = 1 .. 3); s := (4.7+I*m*Pi)/(.8); G[k] := sum((sum(exp(-M[i]*x)*V[i], i = 1 .. 3))(-1)^(4.7/t), m = 1 .. 1000); B[k] := exp(4.7)*Re((1/2)*F[k]+G[k])/t; print(x, B[k]/10^213) end do:

0., 0.1992794392e-4

  

0.1e-1, 405.1968646

  

0.2e-1, 390.9497022

  

0.3e-1, 376.7508308

  

0.4e-1, 362.6447807

  

0.5e-1, 348.6712262

  

0.6e-1, 334.8653546

  

0.7e-1, 321.2582106

  

0.8e-1, 307.8770196

  

0.9e-1, 294.7454880

  

.10, 281.8840866

  

.11, 269.3103112

  

.12, 257.0389286

  

.13, 245.0822062

  

.14, 233.4501244

  

.15, 222.1505743

  

.16, 211.1895453

  

.17, 200.5712947

  

.18, 190.2985093

  

.19, 177.1851378

  

.20, 163.5721140

  

.21, 150.7945764

  

.22, 138.8106908

  

.23, 127.5802985

  

.24, 117.0648679

  

.25, 107.2274480

  

.26, 98.03262160

  

.27, 89.44645840

  

.28, 81.43646938

  

.29, 73.97156169

  

.30, 67.02199266

  

.31, 60.55932687

  

.32, 54.55639165

  

.33, 48.98723509

  

.34, 43.82708326

  

.35, 39.05229999

  

.36, 34.64034546

  

.37, 30.56973756

  

.38, 26.82001293

  

.39, 23.37168921

  

.40, 20.20622852

  

.41, 17.30600112

  

.42, 14.65425098

  

.43, 12.23506127

  

.44, 10.03332140

  

.45, 8.034694822

  

.46, 6.225587898

  

.47, 4.593118969

  

.48, 3.125089577

  

.49, 1.809955352

  

.50, .6367985224

  

.51, -.4046990616

  

.52, -1.324281757

  

.53, -2.131145339

  

.54, -2.833961366

  

.55, -3.440900557

  

.56, -3.959655544

  

.57, -4.397462807

  

.58, -4.761123780

  

.59, -5.057025096

  

.60, -5.291158522

  

.61, -5.469139620

  

.62, -5.596226100

  

.63, -5.677335363

  

.64, -5.717061286

  

.65, -5.719690555

  

.66, -5.689218249

  

.67, -5.629362675

  

.68, -5.543579981

  

.69, -5.435077780

  

.70, -5.306828427

  

.71, -5.161581784

  

.72, -5.001877308

  

.73, -4.830055721

  

.74, -4.648270190

  

.75, -4.458496938

  

.76, -4.262545543

  

.77, -4.062068598

  

.78, -3.858571095

  

.79, -3.653419299

  

.80, -3.447849233

  

.81, -3.242974818

  

.82, -3.039795550

  

.83, -2.839203889

  

.84, -2.641992256

  

.85, -2.448859703

  

.86, -2.260418251

  

.87, -2.077198928

  

.88, -1.899657470

  

.89, -1.728179795

  

.90, -1.563087142

  

.91, -1.404640926

  

.92, -1.253047446

  

.93, -1.108462174

  

.94, -.9709939859

  

.95, -.8407090153

  

.96, -.7176343927

  

.97, -.6017617225

  

.98, -.4930503695

  

.99, -.3914305768

  

1.00, -.2968063580

  

1.01, -.2090582474

  

1.02, -.1280458807

  

1.03, -0.5361039903e-1

  

1.04, 0.1442329617e-1

  

1.05, 0.7624440011e-1

  

1.06, .1320542742

  

1.07, .1820645715

  

1.08, .2264955283

  

1.09, .2655743625

  

1.10, .2995338094

  

1.11, .3286107310

  

1.12, .3530448641

  

1.13, .3730776218

  

1.14, .3889510461

  

1.15, .4009068006

  

1.16, .4091852736

  

1.17, .4140247375

  

1.18, .4156606158

  

1.19, .4143247938

  

1.20, .4102450006

  

1.21, .4036442692

  

1.22, .3947404353

  

1.23, .3837457124

  

1.24, .3708663038

  

1.25, .3563020723

  

1.26, .3402462557

  

1.27, .3228852236

  

1.28, .3043982786

  

1.29, .2849574921

  

1.30, .2647275751

  

1.31, .2438657908

  

1.32, .2225218850

  

1.33, .2008380578

  

1.34, .1789489531

  

1.35, .1569816739

  

1.36, .1350558249

  

1.37, .1132835796

  

1.38, 0.9176974401e-1

  

1.39, 0.7061187116e-1

  

1.40, 0.4990036538e-1

  

1.41, 0.2971860988e-1

  

1.42, 0.1014311718e-1

  

1.43, -0.8756330759e-2

  

1.44, -0.2691651136e-1

  

1.45, -0.4428059057e-1

  

1.46, -0.6079793867e-1

  

1.47, -0.7642394939e-1

  

1.48, -0.9111983792e-1

  

1.49, -.1048524503

  

1.50, -.1175940457

  

1.51, -.1293221053

  

1.52, -.1400191110

  

1.53, -.1496723398

  

1.54, -.1582736461

  

1.55, -.1658192483

  

1.56, -.1723095185

  

1.57, -.1777487660

  

1.58, -.1821450266

  

1.59, -.1855098472

  

1.60, -.1878580843

  

1.61, -.1892076902

  

1.62, -.1895795127

  

1.63, -.1889970940

  

1.64, -.1874864724

  

1.65, -.1850759882

  

1.66, -.1817960960

  

1.67, -.1776791744

  

1.68, -.1727593506

  

1.69, -.1670723178

  

1.70, -.1606551675

  

1.71, -.1535462209

  

1.72, -.1457848648

  

1.73, -.1374113982

  

1.74, -.1284668778

  

1.75, -.1189929708

  

1.76, -.1090318171

  

1.77, -0.9862588936e-1

  

1.78, -0.8781786615e-1

  

1.79, -0.7665050397e-1

  

1.80, -0.6516651986e-1

  

1.81, -0.5340847477e-1

  

1.82, -0.4141866772e-1

  

1.83, -0.2923903059e-1

  

1.84, -0.1691103000e-1

  

1.85, -0.4475574873e-2

  

1.86, 0.8027071258e-2

  

1.87, 0.2055737256e-1

  

1.88, 0.3307659754e-1

  

1.89, 0.4554689214e-1

  

1.90, 0.5793134849e-1

  

1.91, 0.7019406310e-1

  

1.92, 0.8230019855e-1

  

1.93, 0.9421603540e-1

  

1.94, .1059090207

  

1.95, .1173478117

  

1.96, .1285023130

  

1.97, .1393437184

  

1.98, .1498445350

  

1.99, .1599786176

  

2.00, .1697211843

  

2.01, .1790488398

  

2.02, .1879395952

  

2.03, .1963728739

  

2.04, .2043295246

  

2.05, .2117918242

  

2.06, .2187434816

  

2.07, .2251696396

  

2.08, .2310568678

  

2.09, .2363931559

  

2.10, .2411679087

  

2.11, .2453719318

  

2.12, .2489974201

  

2.13, .2520379316

  

2.14, .2544883873

  

2.15, .2563450336

  

2.16, .2576054280

  

2.17, .2582684148

  

2.18, .2583340904

  

2.19, .2578037910

  

2.20, .2566800472

  

2.21, .2549665668

  

2.22, .2526681898

  

2.23, .2497908663

  

2.24, .2463416160

  

2.25, .2423284949

  

2.26, .2377605558

  

2.27, .2326478187

  

2.28, .2270012215

  

2.29, .2208325951

  

2.30, .2141546081

  

2.31, .2069807476

  

2.32, .1993252583

  

2.33, .1912031162

  

2.34, .1826299809

  

2.35, .1736221655

  

2.36, .1641965755

  

2.37, .1543706937

  

2.38, .1441625153

  

2.39, .1335905282

  

2.40, .1226736599

  

2.41, .1114312327

  

2.42, 0.9988294660e-1

  

2.43, 0.8804880801e-1

  

2.44, 0.7594912341e-1

  

2.45, 0.6360442901e-1

  

2.46, 0.5103548172e-1

  

2.47, 0.3826319666e-1

  

2.48, 0.2530863187e-1

  

2.49, 0.1219293308e-1

  

2.50, -0.1062687499e-2

  

2.51, -0.1443699931e-1

  

2.52, -0.2790877250e-1

  

2.53, -0.4145683033e-1

  

2.54, -0.5506005799e-1

  

2.55, -0.6869746087e-1

  

2.56, -0.8234817347e-1

  

2.57, -0.9599150501e-1

  

2.58, -.1096069557

  

2.59, -.1231742615

  

2.60, -.1366734005

  

2.61, -.1500846438

  

2.62, -.1633885522

  

2.63, -.1765660349

  

2.64, -.1895983404

  

2.65, -.2024670969

  

2.66, -.2151543339

  

2.67, -.2276425010

  

2.68, -.2399144754

  

2.69, -.2519536023

  

2.70, -.2637436967

  

2.71, -.2752690610

  

2.72, -.2865145057

  

2.73, -.2974653604

  

2.74, -.3081074905

  

2.75, -.3184273102

  

2.76, -.3284117843

  

2.77, -.3380484505

  

2.78, -.3473254268

  

2.79, -.3562314068

  

2.80, -.3647556901

  

2.81, -.3728881654

  

2.82, -.3806193364

  

2.83, -.3879403119

  

2.84, -.3948428203

  

2.85, -.4013191999

  

2.86, -.4073624159

  

2.87, -.4129660473

  

2.88, -.4181242979

  

2.89, -.4228319938

  

2.90, -.4270845748

  

2.91, -.4308781034

  

2.92, -.4342092550

  

2.93, -.4370753080

  

2.94, -.4394741607

  

2.95, -.4414043009

  

2.96, -.4428648179

  

2.97, -.4438553898

  

2.98, -.4443762751

  

2.99, -.4444283083

  

3.00, -.4440128867

 (33)
> 

;
> 

``


 

Download x_phi.mw

 

system of diff equations i want to solve...

eslamelidy 20 Maple 2018
differential-equations
August 16 2019
1 2

i want to solve the systems of diff equation what's the problem   0.mw

initial vaule problem...

eslamelidy 20 Maple
integral dsolve
January 05 2019
1 15


i want someone hlep me in this worksheet the diff eq of complex i want to sovle it with any numeric method 
 

> 

restart
> 

with(Physics):
> 

with(IntegrationTools):
> 

v := 1;

1

  

-500

  

.1

  

.5

  

.5

 (1)

``

> 

M[1] := Int(-Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2)))), tt = -500 .. z)

Int(-(2*I)*exp(-(.5*I)*tt)/((0.1e-1+tt^2)^(1/2)*exp((0.1e-1+tt^2)^(1/2)))+(2*I)*exp((.5*I)*tt)/((0.1e-1+tt^2)^(1/2)*exp((0.1e-1+tt^2)^(1/2))), tt = -500 .. z)

 (2)
> 

M[2] := Int(Physics:-`*`(Physics:-`*`(Physics:-`*`(Physics:-`*`(4, I), tt), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2))))-Physics:-`*`(Physics:-`*`(Physics:-`*`(Physics:-`*`(4, I), tt), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2)))), tt = -500 .. z):
> 

M[3] := Int(-Physics:-`*`(Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), tt^2), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), tt^2), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2)))), tt = -500 .. z):
> 

M[4] := Int(-Physics:-`*`(Physics:-`*`(2, exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(v, exp(sqrt(b^2+tt^2))))-Physics:-`*`(Physics:-`*`(2, exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(v, exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(2, tt), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(v, exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(2, tt), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(v, exp(sqrt(b^2+tt^2)))), tt = -500 .. z):
> 

M := Physics:-`*`(z^2, M[1])+Physics:-`*`(z, M[2])+Physics:-`*`(z, M[3])+M[4]:
> 

Mc[1] := Physics:-`*`(z^2, Int(-Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2)))), tt = -500 .. z)):
> 

Mc[2] := Physics:-`*`(z, Int(Physics:-`*`(Physics:-`*`(Physics:-`*`(Physics:-`*`(4, I), tt), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2))))-Physics:-`*`(Physics:-`*`(Physics:-`*`(Physics:-`*`(4, I), tt), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2)))), tt = -500 .. z)):
> 

Mc[3] := Physics:-`*`(z, Int(-Physics:-`*`(Physics:-`*`(2, exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(v, exp(sqrt(b^2+tt^2))))-Physics:-`*`(Physics:-`*`(2, exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(v, exp(sqrt(b^2+tt^2)))), tt = -500 .. z)):
> 

Mc[4] := Int(-Physics:-`*`(Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), tt^2), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), tt^2), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v^2), exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(2, tt), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(v, exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(2, tt), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(v, exp(sqrt(b^2+tt^2)))), tt = -500 .. z):
> 

Mc := Mc[1]+Mc[2]+Mc[3]+Mc[4]:
> 

N[1] := Int(Physics:-`*`(Physics:-`*`(Physics:-`*`(2, tt), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v), exp(sqrt(b^2+tt^2))))-Physics:-`*`(Physics:-`*`(Physics:-`*`(2, tt), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v), exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(2, I), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/exp(sqrt(b^2+tt^2))), tt = -500 .. z):
> 

N[2] := Physics:-`*`(z, Int(-Physics:-`*`(Physics:-`*`(2, exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v), exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(2, exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v), exp(sqrt(b^2+tt^2)))), tt = -500 .. z)):
> 

N := N[1]+N[2]:
> 

Nc[1] := Int(-Physics:-`*`(Physics:-`*`(Physics:-`*`(2., tt), exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v), exp(sqrt(b^2+tt^2))))+Physics:-`*`(Physics:-`*`(Physics:-`*`(2., tt), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v), exp(sqrt(b^2+tt^2))))-Physics:-`*`(Physics:-`*`(Physics:-`*`(2., I), exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/exp(sqrt(b^2+tt^2))), tt = -500 .. z):
> 

Nc[2] := Physics:-`*`(z, Int(Physics:-`*`(Physics:-`*`(2., exp(Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v), exp(sqrt(b^2+tt^2))))-Physics:-`*`(Physics:-`*`(2., exp(-Physics:-`*`(Physics:-`*`(Physics:-`*`(.5, I), v), tt))), 1/Physics:-`*`(Physics:-`*`(sqrt(b^2+tt^2), v), exp(sqrt(b^2+tt^2)))), tt = -500 .. z)):
> 

Nc := Nc[1]+Nc[2]:
> 

V := Physics:-`*`(Physics:-`*`(1/Physics:-`*`(4, Pi^2), 1/sqrt(b^2+z^2)), Physics:-`*`(exp(-Physics:-`*`(2, sqrt(b^2+z^2))), Physics:-`*`(2, sqrt(b^2+z^2))+2)-2):
> 

Vc := Physics:-`*`(Physics:-`*`(Physics:-`*`(-1, 1/Physics:-`*`(4, Pi^2)), 1/sqrt(b^2+z^2)), Physics:-`*`(exp(-Physics:-`*`(2, sqrt(b^2+z^2))), Physics:-`*`(2, sqrt(b^2+z^2))+2)-2):
> 

``
> 

H := proc (z) local t; if not z::numeric then return ('procname')(args) end if; evalf(-I*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(-1/2+(1/2)*p^2*v^2+1/sqrt(b^2+z^2)+(1/4)*(exp(-2*sqrt(b^2+z^2))*(2*sqrt(b^2+z^2)+2)-2)/(Pi^2*sqrt(b^2+z^2)))/(((z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))-1)*v)+I*((-1/2+(1/2)*p^2*v^2+1/sqrt(b^2+z^2))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int((-1)*2.*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+2.*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+(-1)*2.*I*exp((-1)*.5*I*v*tt)/exp(sqrt(b^2+tt^2)), tt = -500 .. z)+z*(Int(2.*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+(-1)*2.*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2))), tt = -500 .. z)))/(((z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))-1)*v)) end proc:
> 

H(500)

-6.287499768+0.1713975e-19*I

 (3)
> 

NULL
> 

L := proc (z) local t; if not z::numeric then return ('procname')(args) end if; evalf(-I*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*((-1/2+(1/2)*q^2*v^2+1/sqrt(b^2+z^2))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(2*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))-2*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+(2*I)*exp((-1)*.5*I*v*tt)/exp(sqrt(b^2+tt^2)), tt = -500 .. z)+z*(Int(-2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2))), tt = -500 .. z)))/(((z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))-1)*v)+I*(-1/2+(1/2)*q^2*v^2+1/sqrt(b^2+z^2)-(1/4)*(exp(-2*sqrt(b^2+z^2))*(2*sqrt(b^2+z^2)+2)-2)/(Pi^2*sqrt(b^2+z^2)))/(((z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))-1)*v)) end proc:
> 

``
> 

G := proc (z) local t; if not z::numeric then return ('procname')(args) end if; evalf(I*(-1/2+(1/2)*p^2*v^2+1/sqrt(b^2+z^2)+(1/4)*(exp(-2*sqrt(b^2+z^2))*(2*sqrt(b^2+z^2)+2)-2)/(Pi^2*sqrt(b^2+z^2)))/(((z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))-1)*v)-I*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*((-1/2+(1/2)*p^2*v^2+1/sqrt(b^2+z^2))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int((-1)*2.*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+2.*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+(-1)*2.*I*exp((-1)*.5*I*v*tt)/exp(sqrt(b^2+tt^2)), tt = -500 .. z)+z*(Int(2.*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+(-1)*2.*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2))), tt = -500 .. z)))/(((z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))-1)*v)) end proc:
> 

``
> 

K := proc (z) local t; if not z::numeric then return ('procname')(args) end if; evalf(I*((-1/2+(1/2)*q^2*v^2+1/sqrt(b^2+z^2))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(2*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))-2*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+(2*I)*exp((-1)*.5*I*v*tt)/exp(sqrt(b^2+tt^2)), tt = -500 .. z)+z*(Int(-2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2)))+2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v*exp(sqrt(b^2+tt^2))), tt = -500 .. z)))/(((z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))-1)*v)-I*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(-1/2+(1/2)*q^2*v^2+1/sqrt(b^2+z^2)-(1/4)*(exp(-2*sqrt(b^2+z^2))*(2*sqrt(b^2+z^2)+2)-2)/(Pi^2*sqrt(b^2+z^2)))/(((z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))*(z^2*(Int(-(2*I)*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int((4*I)*tt*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))-(4*I)*tt*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+z*(Int(-2*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))-2*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))+Int(-(2*I)*tt^2*exp((-1)*.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+(2*I)*tt^2*exp(.5*I*v*tt)/(sqrt(b^2+tt^2)*v^2*exp(sqrt(b^2+tt^2)))+2*tt*exp((-1)*.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2)))+2*tt*exp(.5*I*v*tt)/(v*exp(sqrt(b^2+tt^2))), tt = -500 .. z))-1)*v)) end proc:
> 

NULL
> 

NULL
> 

sys := {diff(X(z), z) = Physics:-`*`(H(z), Y(z))+Physics:-`*`(L(z), X(z)), diff(Y(z), z) = Physics:-`*`(G(z), Y(z))+Physics:-`*`(K(z), X(z))}

{diff(X(z), z) = H(z)*Y(z)+L(z)*X(z), diff(Y(z), z) = G(z)*Y(z)+K(z)*X(z)}

 (4)
> 

IC_1 := {X(-500) = 0, Y(-500) = 1}

{X(-500) = 0, Y(-500) = 1}

 (5)
> 

dsol3 := dsolve(`union`(sys, IC_1), numeric, method = dverk78, output = procedurelist, known = [H, L, G, K])

proc (x_dverk78) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_dverk78) else _xout := evalf(x_dverk78) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _fcn, _i, _octl, _ctl, _y0, _yini, _ycur, _reinit, _pars, _n, _ysav, _ini, _par; option `Copyright (c) 1993 by the University of Waterloo. All rights reserved.`; Digits := max(15, Digits); _xout := _xin; _octl := array( 1 .. 32, [( 1 ) = (3), ( 2 ) = (1), ( 3 ) = (0), ( 4 ) = (0), ( 5 ) = (0), ( 6 ) = (0), ( 7 ) = (0), ( 9 ) = (1), ( 8 ) = (0), ( 11 ) = (-1), ( 10 ) = (-1), ( 13 ) = (-1), ( 12 ) = (-1), ( 15 ) = (-1), ( 14 ) = (-1), ( 18 ) = (-1), ( 19 ) = (-1), ( 16 ) = (-1), ( 17 ) = (-500.), ( 22 ) = (-1), ( 23 ) = (-1), ( 20 ) = (-500.), ( 21 ) = (-1), ( 27 ) = (-499.), ( 26 ) = (2), ( 25 ) = (-500.), ( 24 ) = (0), ( 31 ) = (-500.), ( 30 ) = (1), ( 29 ) = (2), ( 28 ) = (0.1e-7), ( 32 ) = (0) ] ); _yini := Array(0..2, {(1) = -500., (2) = 0.}); _y0 := Array(0..2, {(1) = -500., (2) = 0.}); _ycur := array( 1 .. 2, [ ] ); _ctl := array( 1 .. 32, [( 1 ) = (3), ( 2 ) = (1), ( 3 ) = (0), ( 4 ) = (0), ( 5 ) = (0), ( 6 ) = (0), ( 7 ) = (0), ( 9 ) = (1), ( 8 ) = (0), ( 11 ) = (-1), ( 10 ) = (-1), ( 13 ) = (-1), ( 12 ) = (-1), ( 15 ) = (-1), ( 14 ) = (-1), ( 18 ) = (-1), ( 19 ) = (-1), ( 16 ) = (-1), ( 17 ) = (-500.), ( 22 ) = (-1), ( 23 ) = (-1), ( 20 ) = (-500.), ( 21 ) = (-1), ( 27 ) = (-499.), ( 26 ) = (2), ( 25 ) = (-500.), ( 24 ) = (0), ( 31 ) = (-500.), ( 30 ) = (1), ( 29 ) = (2), ( 28 ) = (0.1e-7), ( 32 ) = (0) ] ); _fcn := proc (N, X, Y, YP) option `[Y[1] = X(z), Y[2] = Y(z)]`; YP[1] := H(X)*Y[2]+L(X)*Y[1]; YP[2] := G(X)*Y[2]+K(X)*Y[1]; 0 end proc; _pars := []; _n := 2; _ysav := Array(1..2, {(1) = 0., (2) = 1.}); if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then return _y0[0] elif _xout = "method" then return "dverk78" elif _xout = "numfun" then return round(_ctl[24]) elif _xout = "initial" then return [seq(_yini[_i], _i = 0 .. _n)] elif _xout = "parameters" then return [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_yini[_i], _i = 0 .. _n)], [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _ctl[17]-_y0[0] = 0. then error "no information is available on last computed point" else _xout := _ctl[17] end if elif _xout = "enginedata" then return eval(_octl, 1) elif _xout = "function" then return eval(_fcn, 1) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); if _par <> [] then `dsolve/numeric/process_parameters`(_n, _pars, _par, _yini) end if; if _ini <> [] then `dsolve/numeric/process_initial`(_n, _ini, _yini, _pars) end if; if _pars <> [] then _par := {seq(rhs(_pars[_i]) = _yini[_n+_i], _i = 1 .. nops(_par))}; for _i from 0 to _n do _y0[_i] := subs(_par, _yini[_i]) end do; for _i from _n+1 to _n+nops(_pars) do _y0[_i] := _yini[_i] end do else for _i from 0 to _n do _y0[_i] := _yini[_i] end do end if; _octl[25] := _y0[0]; _octl[20] := _y0[0]; _octl[17] := _y0[0]; _octl[31] := _y0[0]; for _i to op(2, op(2, op(_octl))) do _ctl[_i] := _octl[_i] end do; for _i to _n+nops(_pars) do _ysav[_i] := _y0[_i] end do; if _xout = "initial" then return [seq(_yini[_i], _i = 0 .. _n)] elif _xout = "parameters" then return [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] else return [seq(_yini[_i], _i = 0 .. _n)], [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] end if else return "procname" end if end if; if _y0[0]-_xout = 0. then return [seq(_y0[_i], _i = 0 .. _n)] elif _octl[31]-_y0[0] = 0. then _octl[31] := _y0[0]-sign(_xout-_y0[0]) end if; _reinit := false; if _xin <> "last" then if 0 < 0 and `dsolve/numeric/checkglobals`(0, table( [ ] ), _pars, _n, _yini) then _reinit := true; if _pars <> [] then _par := {seq(rhs(_pars[_i]) = _yini[_n+_i], _i = 1 .. nops(_par))}; for _i from 0 to _n do _y0[_i] := subs(_par, _yini[_i]) end do; for _i from _n+1 to _n+nops(_pars) do _y0[_i] := _yini[_i] end do else for _i from 0 to _n do _y0[_i] := _yini[_i] end do end if end if; if _pars <> [] and select(type, {seq(_yini[_n+_i], _i = 1 .. nops(_pars))}, 'undefined') <> {} then error "parameters must be initialized before solution can be computed" end if end if; if _reinit or _ctl[17]-_xout <> 0. then if _reinit or 0 < _ctl[18] and _xout < _ctl[31] or _ctl[18] < 0 and _ctl[31] < _xout then for _i to op(2, op(2, op(_octl))) do _ctl[_i] := _octl[_i] end do; for _i to _n+nops(_pars) do _ysav[_i] := _y0[_i] end do else _ctl[29] := 2 end if; _ctl[9] := 2; _ctl[27] := _xout; if Digits <= trunc(evalhf(Digits)) then try evalhf(`dsolve/numeric/dverk78_engine`(_fcn, var(_ysav), var(_ycur), `dsolve/numeric/dverk78_aa`, `dsolve/numeric/dverk78_cc`, `dsolve/numeric/dverk78_dd`, var(_ctl), var(array( 1 .. 2, [ ] )), var(array( 1 .. 2, [ ] )), var(array( 1 .. 2, 1 .. 23, [ ] )))) catch: if searchtext('evalhf', lastexception[2]) <> 0 or searchtext('real', lastexception[2]) <> 0 or searchtext('hardware', lastexception[2]) <> 0 then `dsolve/numeric/dverk78_engine`(_fcn, _ysav, _ycur, `dsolve/numeric/dverk78_aa`, `dsolve/numeric/dverk78_cc`, `dsolve/numeric/dverk78_dd`, _ctl, array( 1 .. 2, [ ] ), array( 1 .. 2, [ ] ), array( 1 .. 2, 1 .. 23, [ ] )) else error end if end try else `dsolve/numeric/dverk78_engine`(_fcn, _ysav, _ycur, `dsolve/numeric/dverk78_aa`, `dsolve/numeric/dverk78_cc`, `dsolve/numeric/dverk78_dd`, _ctl, array( 1 .. 2, [ ] ), array( 1 .. 2, [ ] ), array( 1 .. 2, 1 .. 23, [ ] )) end if; if _ctl[29]-3 <> 0 then Rounding := `if`(_y0[0] < _xout, -infinity, infinity); if _ctl[29]+1 = 0 then error "cannot evaluate the solution past %1, maxfun limit exceeded (see <a href='http://www.maplesoft.com/support/help/search.aspx?term=dsolve,maxfun' target='_new'>?dsolve,maxfun</a> for details)", evalf[8](_ctl[20]) elif _ctl[29]+2 = 0 then error "cannot evaluate the solution past %1, hmin > hmax, maybe error tolerance is too small", evalf[8](_ctl[20]) elif _ctl[29]+3 = 0 then error "cannot evaluate the solution past %1, step size < hmin, problem may be singular or error tolerance may be too small", evalf[8](_ctl[20]) else error "cannot evaluate the solution past %1, unknown error code returned from dverk78: %2", evalf[8](_ctl[20]), _ctl[29] end if end if; if _Env_smart_dsolve_numeric = true then if _y0[0] < _ctl[17] and procname("right") < _ctl[17] then procname("right") := _ctl[17] elif _ctl[17] < _y0[0] and _ctl[17] < procname("left") then procname("left") := _ctl[17] end if end if end if; [_ctl[25], seq(_ycur[_i], _i = 1 .. _n)] end proc, (2) = Array(0..0, {}), (3) = [z, X(z), Y(z)], (4) = []}); _vars := _dat[3]; _pars := map(rhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_dverk78, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_dverk78, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_dverk78, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_dverk78, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_dverk78), 'string') = rhs(x_dverk78); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_dverk78), 'string') = rhs(x_dverk78)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_dverk78) else _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_dverk78) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error end try end proc

 (6)
> 

dsol3(500)

Warning,  computation interrupted

  
> 

NULL

NULL

``


 

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