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Error Solving numerical integral

Bell Na

Posted 3 years ago

Hello everybody, I'm trying to calculate an integral but it doesn't give a numerical result. Which mistake did I do? Any help would be appreciated! Thanks \[Mu]0 = 4\[Pi]10^-7; \[Rho]max = 0.22; q = {1}; i = 1; j = 1; n = {1}; aa = 0.130; m = 1; k = m; Sq[\[Rho]_, i_] := (Sin[(Indexed[q, i]\[Pi]\[Rho])/\[Rho]max]); dSq[\[Rho]_, i_] := (Indexed[q, i]\[Pi]/\[Rho]max)(Cos[(Indexed[q, i]\[Pi]\[Rho])/\[Rho]max]); \[Alpha][\[Rho]_] := ArcTan[\[Rho]/aa]; pn[\[Alpha]_, j_] := (N[LegendreP[Indexed[n, j], m, Cos[\[Alpha][\[Rho]]]], 20]); f[\[Rho]_] := (\[Rho]/ArcTan[\[Rho]/aa]); Cnm[j_, f_] := ((Indexed[n, j] - m)!/(Indexed[n, j] + m)!)(1/(f[\[Rho]])^(Indexed[n, j] + 2)); pn1[\[Alpha]_, j_] := (N[LegendreP[Indexed[n, j] + 1, m, Cos[\[Alpha][\[Rho]]]], 20]); Cn\[Rho][pn_, \[Alpha]_] := (m(pn[\[Alpha], j]/ Sin[\[Alpha][\[Rho]]])); Cnm\[Psi][j_, pn1_, pn_, \[Alpha]_] := ((1/ Sin[\[Alpha][\[Rho]]])((Indexed[n, j] + 1) pn[j, \[Alpha]] - (Indexed[n, j] - m + 1)* Cos[\[Alpha][\[Rho]]]pn1[j, \[Alpha]])); DD1 = Integrate[(\[Rho] Cnm[j, f] (Cn\[Rho][pn, \[Alpha]](k/\[Rho])Sq[\[Rho], i] - Cnm\[Psi][j, pn1, pn, \[Alpha]]* D[Sq[\[Rho], i], \[Rho]])), {\[Rho], 0, 0.1}]

Hello everybody,

I'm trying to calculate an integral but it doesn't give a numerical result. Which mistake did I do? Any help would be appreciated! Thanks

\[Mu]0 = 4*\[Pi]*10^-7;
\[Rho]max = 0.22;
q = {1};
i = 1;
j = 1;
n = {1};
aa = 0.130;
m = 1;
k = m;
Sq[\[Rho]_, i_] := (Sin[(Indexed[q, i]*\[Pi]*\[Rho])/\[Rho]max]);

dSq[\[Rho]_, 
   i_] := (Indexed[q, 
      i]*\[Pi]/\[Rho]max)*(Cos[(Indexed[q, 
         i]*\[Pi]*\[Rho])/\[Rho]max]);
\[Alpha][\[Rho]_] := ArcTan[\[Rho]/aa];


pn[\[Alpha]_, 
   j_] := (N[LegendreP[Indexed[n, j], m, Cos[\[Alpha][\[Rho]]]], 20]);

f[\[Rho]_] := (\[Rho]/ArcTan[\[Rho]/aa]);

Cnm[j_, f_] := ((Indexed[n, j] - m)!/(Indexed[n, j] + 
        m)!)*(1/(f[\[Rho]])^(Indexed[n, j] + 2));


pn1[\[Alpha]_, 
   j_] := (N[LegendreP[Indexed[n, j] + 1, m, Cos[\[Alpha][\[Rho]]]], 
    20]);

Cn\[Rho][pn_, \[Alpha]_] := (m*(pn[\[Alpha], j]/
      Sin[\[Alpha][\[Rho]]]));

Cnm\[Psi][j_, pn1_, 
   pn_, \[Alpha]_] := ((1/
      Sin[\[Alpha][\[Rho]]])*((Indexed[n, j] + 1)*
       pn[j, \[Alpha]] - (Indexed[n, j] - m + 1)*
       Cos[\[Alpha][\[Rho]]]*pn1[j, \[Alpha]]));
DD1 = Integrate[(\[Rho]*
    Cnm[j, 
     f] (Cn\[Rho][pn, \[Alpha]]*(k/\[Rho])*Sq[\[Rho], i] - 
      Cnm\[Psi][j, pn1, pn, \[Alpha]]*
       D[Sq[\[Rho], i], \[Rho]])), {\[Rho], 0, 0.1}]

POSTED BY: Bell Na

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Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 3 years ago

Sorry, I made a mistake, here is the correct code: \[Mu]0 = 4\[Pi]10^-7; \[Rho]max = 0.22; aa = 0.130; Sq[\[Rho]_, i_] := (Sin[(Indexed[q, i]\[Pi]\[Rho])/\[Rho]max]); dSq[\[Rho]_, i_] := (Indexed[q, i]\[Pi]/\[Rho]max)(Cos[(Indexed[q, i]\[Pi]\[Rho])/\[Rho]max]); \[Alpha][\[Rho]_] := ArcTan[\[Rho]/aa]; pn[\[Alpha]_, j_] := (N[LegendreP[Indexed[n, j], m, Cos[\[Alpha][\[Rho]]]], 20]); f[\[Rho]_] := (\[Rho]/ArcTan[\[Rho]/aa]); Cnm[j_, f_] := ((Indexed[n, j] - m)!/(Indexed[n, j] + m)!)(1/(f[\[Rho]])^(Indexed[n, j] + 2)); pn1[\[Alpha]_, j_] := (N[LegendreP[Indexed[n, j] + 1, m, Cos[\[Alpha][\[Rho]]]], 20]); Cn\[Rho][ pn_, \[Alpha]_] := (m(pn[\[Alpha], j]/Sin[\[Alpha][\[Rho]]])); Cnm\[Psi][j_, pn1_, pn_, \[Alpha]_] := ((1/ Sin[\[Alpha][\[Rho]]])((Indexed[n, j] + 1) pn[\[Alpha], j] - (Indexed[n, j] - m + 1)* Cos[\[Alpha][\[Rho]]]pn1[\[Alpha], j])); Block[{q = {1}, i = 1, j = 1, n = {1}, m = 1, k = 1}, NIntegrate[(\[Rho] Cnm[j, f] (Cn\[Rho][pn, \[Alpha]](k/\[Rho])Sq[\[Rho], i] - Cnm\[Psi][j, pn1, pn, \[Alpha]]* D[Sq[\[Rho], i], \[Rho]])), {\[Rho], 0, 0.1}]]

Sorry, I made a mistake, here is the correct code:

\[Mu]0 = 4*\[Pi]*10^-7;
\[Rho]max = 0.22;
aa = 0.130;
Sq[\[Rho]_, i_] := (Sin[(Indexed[q, i]*\[Pi]*\[Rho])/\[Rho]max]);
dSq[\[Rho]_, 
   i_] := (Indexed[q, 
      i]*\[Pi]/\[Rho]max)*(Cos[(Indexed[q, 
         i]*\[Pi]*\[Rho])/\[Rho]max]);
\[Alpha][\[Rho]_] := ArcTan[\[Rho]/aa];
pn[\[Alpha]_, 
   j_] := (N[LegendreP[Indexed[n, j], m, Cos[\[Alpha][\[Rho]]]], 
    20]);
f[\[Rho]_] := (\[Rho]/ArcTan[\[Rho]/aa]);
Cnm[j_, f_] := ((Indexed[n, j] - m)!/(Indexed[n, j] + 
        m)!)*(1/(f[\[Rho]])^(Indexed[n, j] + 2));
pn1[\[Alpha]_, 
   j_] := (N[LegendreP[Indexed[n, j] + 1, m, Cos[\[Alpha][\[Rho]]]], 
    20]);
Cn\[Rho][
   pn_, \[Alpha]_] := (m*(pn[\[Alpha], j]/Sin[\[Alpha][\[Rho]]]));
Cnm\[Psi][j_, pn1_, 
   pn_, \[Alpha]_] := ((1/
      Sin[\[Alpha][\[Rho]]])*((Indexed[n, j] + 1)*
       pn[\[Alpha], j] - (Indexed[n, j] - m + 1)*
       Cos[\[Alpha][\[Rho]]]*pn1[\[Alpha], j]));
Block[{q = {1}, i = 1, j = 1, n = {1}, m = 1, k = 1}, 
 NIntegrate[(\[Rho]*
    Cnm[j, 
     f] (Cn\[Rho][pn, \[Alpha]]*(k/\[Rho])*Sq[\[Rho], i] - 
      Cnm\[Psi][j, pn1, pn, \[Alpha]]*
       D[Sq[\[Rho], i], \[Rho]])), {\[Rho], 0, 0.1}]]

POSTED BY: Gianluca Gorni

Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 3 years ago

This version gives a number: \[Mu]0 = 4\[Pi]10^-7; \[Rho]max = 0.22; aa = 0.130; Sq[\[Rho]_, i_] := (Sin[(Indexed[q, i]\[Pi]\[Rho])/\[Rho]max]); dSq[\[Rho]_, i_] := (Indexed[q, i]\[Pi]/\[Rho]max)(Cos[(Indexed[q, i]\[Pi]\[Rho])/\[Rho]max]); \[Alpha][\[Rho]_] := ArcTan[\[Rho]/aa]; pn[j_] := (N[LegendreP[Indexed[n, j], m, Cos[\[Alpha][\[Rho]]]], 20]); f[\[Rho]_] := (\[Rho]/ArcTan[\[Rho]/aa]); Cnm[j_, f_] := ((Indexed[n, j] - m)!/(Indexed[n, j] + m)!)(1/(f[\[Rho]])^(Indexed[n, j] + 2)); pn1[\[Alpha]_, j_] := (N[LegendreP[Indexed[n, j] + 1, m, Cos[\[Alpha][\[Rho]]]], 20]); Cn\[Rho][ pn_, \[Alpha]_] := (m(pn[\[Alpha], j]/Sin[\[Alpha][\[Rho]]])); Cnm\[Psi][j_, pn1_, pn_, \[Alpha]_] := ((1/ Sin[\[Alpha][\[Rho]]])((Indexed[n, j] + 1) pn[\[Alpha], j] - (Indexed[n, j] - m + 1)* Cos[\[Alpha][\[Rho]]]pn1[\[Alpha], j])); With[{q = {1}, i = 1, j = 1, n = {1}, m = 1, k = 1}, NIntegrate[(\[Rho] Cnm[j, f] (Cn\[Rho][pn, \[Alpha]](k/\[Rho])Sq[\[Rho], i] - Cnm\[Psi][j, pn1, pn, \[Alpha]]* D[Sq[\[Rho], i], \[Rho]])), {\[Rho], 0, 0.1}]] but it may not be what you are after. Your code is very difficult to read because you use the same symbols for both constants and variables, values and functions.

This version gives a number:

\[Mu]0 = 4*\[Pi]*10^-7;
\[Rho]max = 0.22;
aa = 0.130;
Sq[\[Rho]_, i_] := (Sin[(Indexed[q, i]*\[Pi]*\[Rho])/\[Rho]max]);
dSq[\[Rho]_, 
   i_] := (Indexed[q, 
      i]*\[Pi]/\[Rho]max)*(Cos[(Indexed[q, 
         i]*\[Pi]*\[Rho])/\[Rho]max]);
\[Alpha][\[Rho]_] := ArcTan[\[Rho]/aa];
pn[j_] := (N[LegendreP[Indexed[n, j], m, Cos[\[Alpha][\[Rho]]]], 
    20]);
f[\[Rho]_] := (\[Rho]/ArcTan[\[Rho]/aa]);
Cnm[j_, f_] := ((Indexed[n, j] - m)!/(Indexed[n, j] + 
        m)!)*(1/(f[\[Rho]])^(Indexed[n, j] + 2));
pn1[\[Alpha]_, 
   j_] := (N[LegendreP[Indexed[n, j] + 1, m, Cos[\[Alpha][\[Rho]]]], 
    20]);
Cn\[Rho][
   pn_, \[Alpha]_] := (m*(pn[\[Alpha], j]/Sin[\[Alpha][\[Rho]]]));
Cnm\[Psi][j_, pn1_, 
   pn_, \[Alpha]_] := ((1/
      Sin[\[Alpha][\[Rho]]])*((Indexed[n, j] + 1)*
       pn[\[Alpha], j] - (Indexed[n, j] - m + 1)*
       Cos[\[Alpha][\[Rho]]]*pn1[\[Alpha], j]));
With[{q = {1}, i = 1, j = 1, n = {1}, m = 1, k = 1},
 NIntegrate[(\[Rho]*
    Cnm[j, 
     f] (Cn\[Rho][pn, \[Alpha]]*(k/\[Rho])*Sq[\[Rho], i] - 
      Cnm\[Psi][j, pn1, pn, \[Alpha]]*
       D[Sq[\[Rho], i], \[Rho]])), {\[Rho], 0, 0.1}]]

but it may not be what you are after. Your code is very difficult to read because you use the same symbols for both constants and variables, values and functions.

POSTED BY: Gianluca Gorni

Bell Na

Posted 3 years ago

Hi Gianluca, thanks for the comment. There are two functions, pn and pn1. Alpha is a function of \rho. I've updated the script up there but still I cannot get the result.

POSTED BY: Bell Na

Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 3 years ago

Sometimes you write `pn, pn1` and some others `Pn, Pn1`: if they mean the same, use the same capitalization. Then you must give a value to `k` and `\[Alpha]`.

POSTED BY: Gianluca Gorni

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