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Hi,

this code not answer:

> whit*evalf; tx := evalf((t1+t2)*(1/2));
> t1 := 326*40; t2 := 327*40;
> y1, y2 := eval([y1, y2], Ns(t1), Ns(t2));
>while (t1 < tx) and (tx < t2) do:
yx := rhs(Ns(tx)[c2]):
if yx > 0 then
y2 := yx: t2 := tx:
else
y1 := yx: t1 := tx:
fi;
od;
>Tx = evalf(tx);
> Hx := floor((1/3600)*tx):
> Mx := floor((tx-3600*Hx)*(1/60)):
> Sx := tx-3600*Hx-60*Mx:
> Hx, Mx, Sx;
> XS := [seq(X[i], i = 0 .. 328)]: YS := [seq(Y[i], i = 0 .. 328)]:
> VxS := [seq(Vx[i], i = 0 .. 328)]: VyS := [seq(Vy[i], i = 0 .. 328)]:
> save(G,Mz,Xs,VxS,VyS, 'orbit.sav'):

 can you help me?

can anybody help me? i want to check the consistency of my scheme. My equation is too long if i check manually, so i used maple 13 to simplify my equation. But it cannot simplify it because of length of output exceed limit 1000000

restart

eqn1 := u+(1-exp(-m))*u[t]+(1-exp(-m))^2*u[tt]/factorial(2)+(u-(1-exp(-m))*u[t]+(1-exp(-m))^2*u[tt]/factorial(2))-u-(1-exp(-m))*u[x]-(1-exp(-m))^2*u[xx]/factorial(2)-u+(1-exp(-m))*u[x]-(1-exp(-m))^2*u[xx]/factorial(2)+(1-exp(-m))^2*u+(1-exp(-m))^2*u^3-(1-exp(-m))^2*(4*(((x+1-exp(-m))^2-2)*cosh(x+1-exp(-m)+(t+1-exp(-m)))-(4*(x+1-exp(-m)))*sinh(x+1-exp(-m)+(t+1-exp(-m)))+(x+1-exp(-m))^6*cosh(x+1-exp(-m)+(t+1-exp(-m)))^3))*((x^2-2)*cosh(x+t)-4*x*sinh(x+t)+x^6*cosh(x+t)^3)*(((x+1-exp(-m))^2-2)*cosh(x+1-exp(-m)+t)-(4*(x+1-exp(-m)))*sinh(x+1-exp(-m)+t)+x^6*cosh(x+1-exp(-m)+t)^3)*((x^2-2)*cosh(x+1-exp(-m)+t)-4*x*sinh(x+1-exp(-m)+t)+x^6*cosh(x+1-exp(-m)+t)^3)/((((x+1-exp(-m))^2-2)*cosh(x+1-exp(-m)+(t+1-exp(-m)))-(4*(x+1-exp(-m)))*sinh(x+1-exp(-m)+(t+1-exp(-m)))+(x+1-exp(-m))^6*cosh(x+1-exp(-m)+(t+1-exp(-m)))^3)*((x^2-2)*cosh(x+t)-4*x*sinh(x+t)+x^6*cosh(x+t)^3)*(((x+1-exp(-m))^2-2)*cosh(x+1-exp(-m)+t)-(4*(x+1-exp(-m)))*sinh(x+1-exp(-m)+t)+(x+1-exp(-m))^6*cosh(x+1-exp(-m)+t)^3+(x^2-2)*cosh(x+1-exp(-m)+t)-4*x*sinh(x+1-exp(-m)+t)+x^6*cosh(x+1-exp(-m)+t)^3)+(((x+1-exp(-m))^2-2)*cosh(x+1-exp(-m)+t)-(4*(x+1-exp(-m)))*sinh(x+1-exp(-m)+t)+(x+1-exp(-m))^6*cosh(x+1-exp(-m)+t)^3)*((x^2-2)*cosh(x+1-exp(-m)+t)-4*x*sinh(x+1-exp(-m)+t)+x^6*cosh(x+1-exp(-m)+t)^3)*(((x+1-exp(-m))^2-2)*cosh(x+1-exp(-m)+(t+1-exp(-m)))-(4*(x+1-exp(-m)))*sinh(x+1-exp(-m)+(t+1-exp(-m)))+(x+1-exp(-m))^6*cosh(x+1-exp(-m)+(t+1-exp(-m)))^3+(x^2-2)*cosh(x+t)-4*x*sinh(x+t)+x^6*cosh(x+t)^3));

(1-exp(-m))^2*u[tt]-(1-exp(-m))^2*u[xx]+(1-exp(-m))^2*u+(1-exp(-m))^2*u^3-(1-exp(-m))^2*(4*((x+1-exp(-m))^2-2)*cosh(x+2-2*exp(-m)+t)-16*(x+1-exp(-m))*sinh(x+2-2*exp(-m)+t)+4*(x+1-exp(-m))^6*cosh(x+2-2*exp(-m)+t)^3)*((x^2-2)*cosh(x+t)-4*x*sinh(x+t)+x^6*cosh(x+t)^3)*(((x+1-exp(-m))^2-2)*cosh(x+1-exp(-m)+t)-4*(x+1-exp(-m))*sinh(x+1-exp(-m)+t)+x^6*cosh(x+1-exp(-m)+t)^3)*((x^2-2)*cosh(x+1-exp(-m)+t)-4*x*sinh(x+1-exp(-m)+t)+x^6*cosh(x+1-exp(-m)+t)^3)/((((x+1-exp(-m))^2-2)*cosh(x+2-2*exp(-m)+t)-4*(x+1-exp(-m))*sinh(x+2-2*exp(-m)+t)+(x+1-exp(-m))^6*cosh(x+2-2*exp(-m)+t)^3)*((x^2-2)*cosh(x+t)-4*x*sinh(x+t)+x^6*cosh(x+t)^3)*(((x+1-exp(-m))^2-2)*cosh(x+1-exp(-m)+t)-4*(x+1-exp(-m))*sinh(x+1-exp(-m)+t)+(x+1-exp(-m))^6*cosh(x+1-exp(-m)+t)^3+(x^2-2)*cosh(x+1-exp(-m)+t)-4*x*sinh(x+1-exp(-m)+t)+x^6*cosh(x+1-exp(-m)+t)^3)+(((x+1-exp(-m))^2-2)*cosh(x+1-exp(-m)+t)-4*(x+1-exp(-m))*sinh(x+1-exp(-m)+t)+(x+1-exp(-m))^6*cosh(x+1-exp(-m)+t)^3)*((x^2-2)*cosh(x+1-exp(-m)+t)-4*x*sinh(x+1-exp(-m)+t)+x^6*cosh(x+1-exp(-m)+t)^3)*(((x+1-exp(-m))^2-2)*cosh(x+2-2*exp(-m)+t)-4*(x+1-exp(-m))*sinh(x+2-2*exp(-m)+t)+(x+1-exp(-m))^6*cosh(x+2-2*exp(-m)+t)^3+(x^2-2)*cosh(x+t)-4*x*sinh(x+t)+x^6*cosh(x+t)^3))

(1)

a := simplify(eqn1);

`[Length of output exceeds limit of 1000000]`

(2)

``


Download consistency_expmle_4.mw

.


can anybody help me..? why my graph not come out? Is that any mistake in my coding?

restart

y := x^2-x*(exp(I*k*`&Delta;x`)+exp(-I*k*`&Delta;x`)-m^2+m^2*((4*((1^2-2)*cosh(1+1)-4*sinh(1+1)+1^6*cosh(1+1)^3))*((1^2-2)*cosh(1+1)-4*sinh(1+1)+1^6*cosh(1+1)^3)*((1^2-2)*cosh(1+1)-4*sinh(1+1)+1^6*cosh(1+1)^3))*((1^2-2)*cosh(1+1)-4*sinh(1+1)+1^6*cosh(1+1)^3)/(epsilon*((((1^2-2)*cosh(1+1)-4*sinh(1+1)+1^6*cosh(1+1)^3)*((1^2-2)*cosh(1+1)-4*sinh(1+1)+1^6*cosh(1+1)^3))*((1^2-2)*cosh(1+1)-4*sinh(1+1)+1^6*cosh(1+1)^3+((1^2-2)*cosh(1+1)-4*sinh(1+1)+1^6*cosh(1+1)^3))+(((1^2-2)*cosh(1+1)-4*sinh(1+1)+1^6*cosh(1+1)^3)*((1^2-2)*cosh(1+1)-4*sinh(1+1)+1^6*cosh(1+1)^3))*((1^2-2)*cosh(1+1)-4*sinh(1+1)+1^6*cosh(1+1)^3+((1^2-2)*cosh(1+1)-4*sinh(1+1)+1^6*cosh(1+1)^3)))))+1;

x^2-x*(exp(I*k*`&Delta;x`)+exp(-I*k*`&Delta;x`)-m^2+2*m^2*(-cosh(2)-4*sinh(2)+cosh(2)^3)^2/(epsilon*(-2*cosh(2)-8*sinh(2)+2*cosh(2)^3)))+1

(1)

subs(m = 1-exp(-m), %);

x^2-x*(exp(I*k*`&Delta;x`)+exp(-I*k*`&Delta;x`)-(1-exp(-m))^2+2*(1-exp(-m))^2*(-cosh(2)-4*sinh(2)+cosh(2)^3)^2/(epsilon*(-2*cosh(2)-8*sinh(2)+2*cosh(2)^3)))+1

(2)

subs(epsilon = .17882484, %);

x^2-x*(exp(I*k*`&Delta;x`)+exp(-I*k*`&Delta;x`)-(1-exp(-m))^2+11.18412856*(1-exp(-m))^2*(-cosh(2)-4*sinh(2)+cosh(2)^3)^2/(-2*cosh(2)-8*sinh(2)+2*cosh(2)^3))+1

(3)

subs(k = n*Pi, %);

x^2-x*(exp(I*n*Pi*`&Delta;x`)+exp(-I*n*Pi*`&Delta;x`)-(1-exp(-m))^2+11.18412856*(1-exp(-m))^2*(-cosh(2)-4*sinh(2)+cosh(2)^3)^2/(-2*cosh(2)-8*sinh(2)+2*cosh(2)^3))+1

(4)

subs(`&Delta;x` = m, %);

x^2-x*(exp(I*n*Pi*m)+exp(-I*n*Pi*m)-(1-exp(-m))^2+11.18412856*(1-exp(-m))^2*(-cosh(2)-4*sinh(2)+cosh(2)^3)^2/(-2*cosh(2)-8*sinh(2)+2*cosh(2)^3))+1

(5)

subs(m = 0.1e-2, %);

x^2-x*(exp((0.1e-2*I)*n*Pi)+exp(-(0.1e-2*I)*n*Pi)-(1-exp(-0.1e-2))^2+11.18412856*(1-exp(-0.1e-2))^2*(-cosh(2)-4*sinh(2)+cosh(2)^3)^2/(-2*cosh(2)-8*sinh(2)+2*cosh(2)^3))+1

(6)

j := subs(n = 1, %);

x^2-x*(exp((0.1e-2*I)*Pi)+exp(-(0.1e-2*I)*Pi)-0.9990006498e-6+0.1117295170e-4*(-cosh(2)-4*sinh(2)+cosh(2)^3)^2/(-2*cosh(2)-8*sinh(2)+2*cosh(2)^3))+1

(7)

complexplot3d(x*j, x = -2-I .. 2+I);````

complexplot3d(x*(x^2-x*(exp((0.1e-2*I)*Pi)+exp(-(0.1e-2*I)*Pi)-0.9990006498e-6+0.1117295170e-4*(-cosh(2)-4*sinh(2)+cosh(2)^3)^2/(-2*cosh(2)-8*sinh(2)+2*cosh(2)^3))+1), x = -2-I .. 2+I)

(8)

a := fsolve(x*j, x);

0., .9865070072, 1.013677544

(9)

b := fsolve(x*j, x = 1);

0., .9865070072, 1.013677544

(10)

with(plots):

complexplot({a, b}, numpoints = 100, color = green, filled = true, title = "Stability Region");

 

``

``

``


Download Stability_1.mw

I'm new to mapleprimers and if I make some mistakes I apologize. 

The problem I ran and on which I ask for help is that the command of maple latex I results in latex only the result of an arithmetic expression and the expression does not translate into latex itself. for example, the command latex (3 + 2) I 5 and not as a result from the translation in latex of 3 + 2. I also tried using the command inetrt form but did not succeed, I tried it with the quotes '' and even succeed. I also tried it with double quotes and with simple expressions funzioa eg latex ("(3 + 2) * 5") from me as a result `` (3 + 2) * 5 '' but with more complex expressions: eg. latex("\""(2+1/(5))^(0)+(6/(9)*(((83-8)/(90))-15/(10)+(3/(9))/(2)) -6/(9))/(75/(100)*12/(9))"''") mi da 

Error, Got internal error in Typesetting:-Parse:-Postprocess : "internal error: invalid object ""

Thanking for the help, Best regards

 

can i use maple to get all the roots in a given expression?

 

for example: i have a expression "(2b+c)^(1/2)+c-(c-2b)^(1/3)"

and I want to get a list that contains "(2b+c)^(1/2)" and "(c-2b)^(1/3)"

 

is there any maple commands i can play with?

Hello maple users,

I want to compare 5 functions and see which one gives the best result (highest score). Unfortunatelly, I can't say one function always gives higher results than others. I have 8 variables and it makes the comparison complicated.

I assume 4 of them are constant and the other 4 or not.

c=0.1

p=1

w=0.5

wu=0.8

for Pb= 0:0.025:0.20                   Pb and Pa are probabilities, between 0-20, 0.025 increments
    for Pa= 0:0.025:(Pb-0.025)      ** Pa is always less than Pb
        for Ha= 0:50                      Ha takes values between 0-50
            for Hb= 0:(Ha-1)            ** Hb is always less than Ha

I attached the data file. I appreciate if someone tells me how to compare these functions. Should I put more assumptions to make it easier? Thank you.

data1.mw

Thanks guys for your help, I know I am asking sometimes some stupid questions, I spend hours trying to do that, but when I am stock.... I need some help... and when the problem is solved I can see how unintelligent am I .

I have document Maple.mw. I want to copy all text to mathtype. How do I copy and paste?

Link download: maple document

I find mapleform software   but I dont know how to use , and does anyone know another method to convert.  I know use 

> with(MmaTranslator);
> MmaToMaple();  

I can  automatically translated .nb files.  but this have  

Error, (in readline) file or directory does not exist
Error, (in readline) file or directory does not exist
Error, (in readline) file or directory does not exist
Error, (in readline) file or directory does not exist

another Error, missing operator or `;`

also I try to use 

> with(MmaTranslator);
> FromMmaNotebook(Mma_notebook_filename, options);

I still dont know how is works?  can you explains for me and show me some example  , here is my example files.plot.zip

 

hi friends

I encountered a problem and I can not draw the plot of this code

> sol := fsolve({diff(S, x) = 0, diff(S, y) = 0}, {x, y});



> with(VectorCalculus);
> with(linalg);
> s1 := evalf(subs(sol, linalg[grad](S, [x, y])));

> with(VectorCalculus);
> with(LinearAlgebra);
> s2 := evalf(subs(sol, linalg[hessian](S, [x, y]))); pmp0 := [x-subs(sol, x), y-subs(sol, y)]; sapprox := s0+evalm(`&*`(`&*`(transpose(pmp0), s2), pmp0));
> with(Statistics);
>
> with(stats); statevalf[icdf, chisquare[4]](.95);

> with(VectorCalculus);
> with(plottools);
> with(plots);
> with(linalg); ellips := {seq(stats*([statevalf[icdf, chisquare[4]]])(c) = sapprox, c = [.5, .95, .999])};
> plots(ellips(x, y), x = 950 .. 1000, y = 700 .. 750, grid = [50, 50], view = [950 .. 1000, 700 .. 750]);

 

 

can you helpe me?Thank you

Dear Friends,

I am solving 6 ODEs using maple15. then i got this error. anyone know abou this? thank you.

problem2.mw

 

 

restart:with (plots): B:=1:M:=1:Gr:=0.5:Pr:=3:w:=0.02:blt:=5:Bi:=10:

Eq1:=diff(f(eta),eta,eta,eta)-(diff(f(eta),eta))^(2)+f(eta)*diff(f(eta),eta,eta)+B*H(eta)*(F(eta)-diff(f(eta),eta))-M*diff(f(eta),eta)+Gr*theta(eta)=0;

diff(diff(diff(f(eta), eta), eta), eta)-(diff(f(eta), eta))^2+f(eta)*(diff(diff(f(eta), eta), eta))+H(eta)*(F(eta)-(diff(f(eta), eta)))-(diff(f(eta), eta))+.5*theta(eta) = 0

(1)

Eq2:=(1+Nr)*diff(theta(eta),eta,eta)+Pr*f(eta)*diff(theta(eta),eta)+(2/3)*H(eta)*B*(theta1(eta)-theta(eta))=0;

(1+Nr)*(diff(diff(theta(eta), eta), eta))+3*f(eta)*(diff(theta(eta), eta))+(2/3)*H(eta)*(theta1(eta)-theta(eta)) = 0

(2)

Eq3:=H(eta)*F(eta)+H(eta)*diff(G(eta),eta)+G(eta)*diff(H(eta),eta)=0;

H(eta)*F(eta)+H(eta)*(diff(G(eta), eta))+G(eta)*(diff(H(eta), eta)) = 0

(3)

Eq4:=F(eta)^2+G(eta)*diff(F(eta),eta)+B*(F(eta)-diff(f(eta),eta))=0;

F(eta)^2+G(eta)*(diff(F(eta), eta))+F(eta)-(diff(f(eta), eta)) = 0

(4)

Eq5:=G(eta)*diff(G(eta),eta)+B*(f(eta)+G(eta))=0;

G(eta)*(diff(G(eta), eta))+f(eta)+G(eta) = 0

(5)

Eq6:=G(eta)*diff(theta1(eta),eta)+l*B*(theta1(eta)-theta(eta))=0;

G(eta)*(diff(theta1(eta), eta))+l*(theta1(eta)-theta(eta)) = 0

(6)

bcs:=f(0)=0,(D(f))(0)=1,(D(theta))(0)=-Bi*(1-theta(0)),(D(f))(blt)=0,F(blt)=0,G(blt)=-f(blt),H(eta)=w,theta(blt)=0,theta1(blt)=0;

f(0) = 0, (D(f))(0) = 1, (D(theta))(0) = -10+10*theta(0), (D(f))(5) = 0, F(5) = 0, G(5) = -f(5), H(eta) = 0.2e-1, theta(5) = 0, theta1(5) = 0

(7)

L:=[0.5,1,1.5,2];

[.5, 1, 1.5, 2]

(8)

for k from 1 to 4 do p:=dsolve(eval({Eq1,Eq2,Eq3,Eq4,Eq5,Eq6,bcs},Nr=L[k]),[f(eta),F(eta),G(eta),H(eta),theta(eta),theta1(eta)],numeric,output=listprocedure);end do:

Error, (in dsolve/numeric/bvp) unevaluated names in system not allowed: {Y[9], Y[10]}

 

``

``

``

 

Download problem2.mw

I would like to pick out specific christoffel symbols once maple calculates them. Right now I am using CoefficientList, but I would like a more direct way, as I am working with someone using Maple 17, and they are put in a different order in the CoefficientList. Here is the bit of my code I would like help with.

hi friends

After this cods i see very error

 > restart;

read(orbit.sav ): whit(plots):
ax := -G*Mz*x/(x^2+y^2)^(3/2);
ay := -G*Mz*y/(x^2+y^2)^(3/2);
i := 'i'; j := i+1;
for k from 0 to 3 do
x := 7*10^6; Vx := 0;
y := 0; Vy := 9000;
dt := evalf(1/2^k);
for i from 0 to 328 do
X[i] := evalf(x); Y[i] := evalf(y);
for n to 40*2^k do
x := evalf((1/2)*ax*dt^2+Vx*dt+x); y := evalf((1/2)*ay*dt^2+Vy*dt+y);
Vx := evalf(ax*dt+Vx); Vy := evalf(ay*dt+Vy)
od;
if i mod 41= 0 then
dX[k, i] := X[i]-XS[j]; dY[k, i] := Y[i]-YS[j]
fi
od;
p[k] := plot([seq([(X[i]-XS[j])*(1/1000), (Y[i]-YS[j])*(1/1000)], i = 0 .. 328)], color = green) end do;
p1 := display({seq(p[k], k = 0 .. 3)}, thickness = 3)
SI := [seq(41*i, i = 0 .. 8)]
p2 := plot({seq([seq([(1/1000)*dX[k, i], (1/1000)*dY[k, i]], k = 0 .. 1), [0, 0]], i = SI)}, color = black)
display({p1, p2}, scaling = constrained, labels = ['dx', 'dy'])
display({p1, p2}, view = [-.1 .. .5, -.4 .. .2], scaling = constrained, labels = ['dx', 'dy'])

can you help me Please?

Thank you

 

 

 

 

     Maple is seriously used in my article Approximation of subharmonic functions in the half-plane by the logarithm of the modulus of an analytic function. Math. Notes 78, No 4, 447-455 in two places. The purpose of this post is to present these applications.                                                                                                 First, I needed to prove the elementary inequality (related to the properties of the minimal harmonic majorant of the function 1/Im z in a certain strip)                                                                                                    2R+sqrt(R)-R(R+sqrt(R))y - 1/y   1/4                                                                                                  for    y ≥ 1/(R+sqrt(R)) and  y ≤ 1/R, the parameter R is greater than or equal to 1.   The artless attemt                                                                          
restart; `assuming`([maximize(2*R+sqrt(R)-R*(R+sqrt(R))*y-1/y, y = 1/(R+sqrt(R)) .. 1/R)], [R >= 1])

maximize(2*R+R^(1/2)-R*(R+R^(1/2))*y-1/y, y = 1/(R+R^(1/2)) .. 1/R)

(1)

fails. The second (and successful) try consists in the use of optimizers:

F := proc (R) options operator, arrow; evalf(maximize(2*R+sqrt(R)-R*(R+sqrt(R))*y-1/y, y = 1/(R+sqrt(R)) .. 1/R)) end proc:

F(1)

.171572876

(2)

 

Optimization:-Minimize('F(R)', {R >= 1})

[.171572875253809986, [R = HFloat(1.0)]]

(3)

To be sure ,
DirectSearch:-Search(proc (R) options operator, arrow; F(R) end proc, {R >= 1})
;

[.171572875745665, Vector(1, {(1) = 1.0000000195752754}, datatype = float[8]), 11]

(4)

Because 0.17
"158 < 0.25, the inequality is  proved.   "
Now we establish this  by the use of the derivative. 

solve(diff(2*R+sqrt(R)-R*(R+sqrt(R))*y-1/y, y) = 0, y, explicit)

1/(R^(3/2)+R^2)^(1/2), -1/(R^(3/2)+R^2)^(1/2)

(5)

maximize(1/sqrt(R^(3/2)+R^2)-1/(R+sqrt(R)), R = 1 .. infinity, location)

(1/2)*2^(1/2)-1/2, {[{R = 1}, (1/2)*2^(1/2)-1/2]}

(6)

minimize(eval(2*R+sqrt(R)-R*(R+sqrt(R))*y-1/y, y = 1/sqrt(R^(3/2)+R^2)), R = 1 .. infinity, location)

3-2*2^(1/2), {[{R = 1}, 3-2*2^(1/2)]}

(7)

evalf(3-2*sqrt(2))

.171572876

(8)

The second use of Maple was the calculation of the asymptotics of the following integral (This is the double integral of the Laplacian of 1/Im z over the domain {z: |z-iR/2| < R/2} \ {z: |z| ≤ 1}.). That place is the key point of the proof. Its direct calculation in the polar coordinates fails.

`assuming`([(int(int(2/(r^2*sin(phi)^3), r = 1 .. R*sin(phi)), phi = arcsin(1/R) .. Pi-arcsin(1/R)))/(2*Pi)], [R >= 1])

(1/2)*(int(int(2/(r^2*sin(phi)^3), r = 1 .. R*sin(phi)), phi = arcsin(1/R) .. Pi-arcsin(1/R)))/Pi

(9)

In order to overcome the difficulty, we find the inner integral

`assuming`([(int(2/(r^2*sin(phi)^3), r = 1 .. R*sin(phi)))/(2*Pi)], [R*sin(phi) >= 1])

(R*sin(phi)-1)/(sin(phi)^4*R*Pi)

(10)

and then we find the outer integral. Because
`assuming`([int((R*sin(phi)-1)/(sin(phi)^4*R*Pi), phi = arcsin(1/R) .. Pi-arcsin(1/R))], [R >= 1])

int((R*sin(phi)-1)/(sin(phi)^4*R*Pi), phi = arcsin(1/R) .. Pi-arcsin(1/R))

(11)

is not successful, we find the indefinite integral  

J := int((R*sin(phi)-1)/(sin(phi)^4*R*Pi), phi)

-(1/2)*cos(phi)/(Pi*sin(phi)^2)+(1/2)*ln(csc(phi)-cot(phi))/Pi+(1/3)*cos(phi)/(R*Pi*sin(phi)^3)+(2/3)*cos(phi)/(R*Pi*sin(phi))

(12)

We verify that  the domain of the antiderivative includes the range of the integration.
plot(-cos(phi)/sin(phi)^2+ln(csc(phi)-cot(phi)), phi = 0 .. Pi)

 

plot((2/3)*cos(phi)/sin(phi)^3+(4/3)*cos(phi)/sin(phi), phi = 0 .. Pi)

 

    That's all right. By the Newton-Leibnitz formula,

``
eval(J, phi = Pi-arcsin(1/R))-(eval(J, phi = arcsin(1/R)));

(1/3)*(1-1/R^2)^(1/2)*R^2/Pi+(1/2)*ln((1-1/R^2)^(1/2)*R+R)/Pi-(4/3)*(1-1/R^2)^(1/2)/Pi-(1/2)*ln(R-(1-1/R^2)^(1/2)*R)/Pi

(13)

Finally, the*asymptotics*is found by

asympt(eval(J, phi = Pi-arcsin(1/R))-(eval(J, phi = arcsin(1/R))), R, 3)

(1/3)*R^2/Pi-(3/2)/Pi+(1/2)*(ln(2)+ln(R))/Pi-(1/2)*(-ln(2)-ln(R))/Pi+O(1/R^2)

(14)

      It should be noted that a somewhat different expression is written in the article. My inaccuracy, as far as I remember it, consisted in the integration over the whole disk {z: |z-iR/2| < R/2} instead of {z: |z-iR/2| < R/2} \ {z: |z| ≤ 1}. Because only the form of the asymptotics const*R^2 + remainder is used in the article, the exact value of this non-zero constant is of no importance.

       It would be nice if somebody else presents similar examples here or elsewhere.

 

Download Discovery_with_Maple.mw

hi friends

i have a problem in maple with an error

dsnumsort:=proc(numpr::list,coor::list)
local i,j,n;
global C1,C2,C3,V1,V2,V3;
n:=nops(coor):
print("Order of the variables:");
for i from 2 to 2*n +1 do
for j from 1 to n do
if[numpr[i]]=
select(has,numpr,diff(coor[j](t),t)) then
C[j]:=i-1;V[j]:=i;
print(coor[j],C[j]," ",diff(coor[j](t),t),V[j]);
fi;
od;
od;
end :

Us:=subs(G=1,m=1,L=1,U):
D2r:=[diff(z(t),t,t),diff(x(t),t,t),diff(y(t),t,t)]:
g:=subs(z=z(t),x=x(t),y=y(t),grad(Us,[z,x,y])):
IniC:=z(0)=0.75,D(z)(0)=0, x(0)=1,D(x)(0)=0,y(0)=0,D(y)(0)=1:

Ns:=dsolve({seq(D2r[i]=g[i],i=1..3),IniC},{z(t),x(t),y(t)},numeric);

dsnumsort(Ns(0),[z,x,y]):

for i from 0 to 1000 do ;
T:=i/25;
NsT:=Ns(T):

X[i]:=rhs(NsT[C1]); Vx[i]:=rhs(NsT[V1]);
Y[i]:=rhs(NsT[C2]); Vy[i]:=rhs(NsT[V2]);
Z[i]:=rhs(NsT[C3]); Vz[i]:=rhs(NsT[V3]);

KepVec[i]:=convert(crossprod([X[i],Y[i], Z[i]],[Vx[i],Vy[i], Vz[i]]),list);
KepAbs[i]:=norm(KepVec[i],2);
od:

but i see this error and I can't draw PLOT:

Error, invalid input: rhs received Ns(0)[C1], which is not valid for its 1st argument, expr

this cods is for draw plot:

spacecurve({[seq([X[i], Y[i], Z[i]], i = 0 .. 1000)], [[-1/2, 0, 0], [1/2, 0, 0]]}, labels = ['x', 'y', 'z']);

spacecurve([seq(KepVec[i], i = 0 .. 1000)], orientation = [0, 90], labels = ['x', 'y', 'z'])

plot([seq([(1/25)*i, KepAbs[i]], i = 0 .. 1000)], labels = ['t', 'MofI'])

can you helpe me?Thank you

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