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These are replies submitted by vv

@Rouben Rostamian  
Yes, it works, not very intuitive (without a plot) though.

a:=1/3: h:=1/20: d:=1/4:
M:= # Moebius
<(1+u*a * cos(t/2))*cos(t), (1+u*a * cos(t/2))*sin(t), a*u * sin(t/2)>:
M_u, M_t := <diff(M,u)>, <diff(M,t)>:
p1:=plot3d( M+h*N, t=0..2*Pi, u=-1..1, scaling=constrained, color=orange):
p2:=plot3d( M-h*N, t=0..2*Pi, u=-1..1, scaling=constrained, color=green):
q1:=seq( plots:-arrow(eval(M+h*N, [u=0,t=k*Pi/narr]),eval(d*N, [u=0,t=k*Pi/narr]), width=1/20, color=red), k=0..2*narr-1):
q2:=seq( plots:-arrow(eval(M-h*N, [u=0,t=k*Pi/narr]),-eval(d*N, [u=0,t=k*Pi/narr]), width=1/20, color=blue), k=0..2*narr-1):

My intuition refuses to work about the normals for the Mobius strip.
Mobius strip is not orientable; the normal vector is not continuous. I cannot imagine the "correct" exterior normals for the two "translated" strips.  Isn't the 3D printer going to be confused?


It works in Maple 2019. But not properly for Digits>15 (try 25 and 20). Probably Carl will fix this.

Unfortunately you have not included a corresponding Your_NormInv to compare the accuracy.


It works in newer versions.
For Maple 2015 just replace z1=z  with  abs(z1-z)<10^(-Digits+1)   (or something similar).


AFAIK the Risch algorithm is not fully implemented in Maple.
You can see the methods which are tried using

BTW, rewriting just a bit the integrand, Maple integrates easily:

int(sin(x)^(1/3)*(1-sin(x)^2)*cos(x), x);



ode:=y(x)*diff(y(x),x) - y(x) = 0:

   y(x) = 0, y(x) = x + _C1

This is obviously the correct answer (two solutions; actually there are other solutions by combining these on intervals).

If we solve ode wrt y'  ==>  diff(y(x),x) = 1
[obtained  dividing by y (supposed to be <> 0; that is how solve works)].

Of course, dsolve( diff(y(x),x) = 1 ) ==>  y(x) = x + _C1,
so, y=0 disappears. But y(x)=0 satisfies (of course) the ode.

@Rouben Rostamian  

OP wants an asymptotic expansion for G0(s). He should use asympt(G0(s),s)

The question does not seem to have much sense.


# K[r, s]
for p from 1 to 3 do  
for  s from 1 to 3  do
ff := L[p, s](r, theta, phi)*F[p, 1](r, theta):
# print(indets=Aij);
g1:=expand(ff, Aij, distributed);
g2:=coeffs(g1, Aij, 'T'):
C1:=int~([g2], theta = 0 .. 2*Pi, r = 0.5 .. 1, epsilon=1e-8, numeric):
kk:=add(C1 .~ T);
kkk := evalf( int(F[p, 1](r, theta)^2, theta = 0 .. 2*Pi, r = 0.5 .. 1) );
k[p, s] := kk/kkk;
print([p, s] = %);


@Carl Love 

Unfortunately, as I see, sw_zgeevx_  also fails sometimes.

The bug seems to occur only for multiple (repeated) eigenvalues in the non-symmetric case, so, in practice should be extremely rare.
The examples were found using e.g.

LinearAlgebra:-CompanionMatrix((x-15)^3, x)^+;






f[1,1] is not defined.

2. You must rename my f  to ff  (because you have introduced a f)

ff := L[p, s](r, theta, phi)*F[p, 1](r, theta):
g1:=expand(ff, Aij, distributed);

3. Now you have B[i,j] etc

Sorry, but I cannot continue with so many changes.



With series(...,r=r0) you may approximate only near r0.
In your example you have a huge r. Use asympt instead.
[Note that you have not actually used the power of slode; only a few terms, just like the standard series]

R:=370; Order:=6:
ode:= diff(g(r),r$2)- r/R*g(r)=0:
ic:=g(2*R)=0, D[1](g)(0)=R:





asympt(G, r):




evalf[50](eval([G,Gapp], r=400));

[-0.40702431977939449083274686710027843869975497791315e-117, 0.27836877438958079551079281963767137932752590618428e-482]


evalf[50](eval([G,Gapp], r=1000));

[0.81928332104566902988587362191002268450537068088687e-127, 0.81928332104563623792440214614327162747404927396426e-127]


evalf[50](eval([G,Gapp], r=10^10));

[0.16056840058550626226900357319617922990909434242673e15051930008898, 0.16056840058550626226900357319617922981745765695502e15051930008898]



The integral

J:=Int( exp(I*m*omega + I*b*cos(omega) ), omega=0..2*Pi) ;#          (m integer,  b positive constant)

Int(exp(I*m*omega+I*b*cos(omega)), omega = 0 .. 2*Pi)


value(J);  # wrong!




f:=expand(op(1,J)) assuming m::integer;



Prepare the change of variables

g := expand(z^m * exp(I*b*(z+1/z)/2));  z = exp(I*omega);



z = exp(I*omega)


G:=simplify(g, [exp(I*omega)=z]) assuming m::integer



answer = 2*Pi*residue(G/z, z=0); # Residue theorem, z=0 essential singularity

answer = 2*Pi*residue(z^m*exp(((1/2)*I)*b*z)*exp(((1/2)*I)*b/z)/z, z = 0)


fii:=convert(G, Sum, dummy=i);  # we need the constant (coeff of z^0):

z^m*(Sum((((1/2)*I)*b*z)^i/factorial(i), i = 0 .. infinity))*(Sum((((1/2)*I)*b/z)^i/factorial(i), i = 0 .. infinity))


ai:=eval(op([2,1], fii), z=1); ai_bis:=eval(op([3,1], fii), z=1);





Sum(ai * subs(i=i+m, ai), i=0..infinity);  # coeff of z^(-m) in fii

Sum((((1/2)*I)*b)^i*(((1/2)*I)*b)^(i+m)/(factorial(i)*factorial(i+m)), i = 0 .. infinity)


answer := simplify(2*Pi*value(%)) assuming m::nonnegint;

2*Pi*BesselJ(m, b)*exp(((1/2)*I)*m*Pi)


evalf(eval([J,answer],[m=3,b=7/3])); #check1

[-0.2869900475e-12-1.168527755*I, -1.168527755*I]


evalf(eval([J,answer],[m=2,b=5]));   #check2

[-.2925772544-0.3974126502e-12*I, -.2925772544]


evalf(eval([J,answer],[m=0,b=1]));   #check3

[4.807878861+0.3488081236e-12*I, 4.807878862]


evalf(eval([J,answer],[m=7,b=4]));   #check4

[-0.2695364704e-12-0.9535405641e-1*I, -0.9535405642e-1*I]


@Carl Love 

I assumed OP knew what he was doing. The original problem was not stated.


Your constants are not assigned. After giving them numeric values, you may try to compute the integrals, but the integrands being very "odd", you may need to fine-tune int as above.

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