How to calculation this integral ?

30 Jan 2021
1 Answer
Updated 3 Mar 2021
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L0 = 0:20
M0 = -20:20
N0 = 0:20
C = 10 ,
b = 0 ,
V = 2/3 ,
V1 = 2.0001/3 ,
K = 2*pi ,
"EpsilonN : E(n=0) = 1 and for E(n>=1)=2 ",
EpsilonN = [1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2]
%-------------------------------------------------------------------------------------------------------
for L1 = 0:length(L0)
for M1 = 0:length(M0)
for N1 = 0:length(N0)
if (L1/V1)+M1 ~= (N1/V0) %SA1 ~= abs(SA2)
Clmn = (EpsilonN/2*V0*pi)*besselj(M1,K*B)*((sin(((L1/V1)+M1+(N1/V0))*V0*pi)/(L1/V1)+M1+(N1/V0))+...
(sin(((L1/V1)+M1-(N1/V0))*V0*pi)/(L1/V1)+M1-(N1/V0))); %<part 1>
else N1 ~= 0 %n ~= 0 ,
Clmn = (EpsilonN/2)*besselj(M1,K*B) ; %<part 3>
if (L1/V1)+M1 == (N1/V0) %SA1 == SA2 ,
Clmn = EpsilonN*besselj(M1,K*B); %<part 2>
end
end
end
end
end
Clmn

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Answers (1)

Walter Roberson
Walter Roberson on 3 Feb 2021
Ran in:

1 vote

Ran in:
format long g
L0 = 0:20;
M0 = -20:20;
N0 = 0:20;
C = 10;
b = 0;
V = 2/3;
V1 = 2.0001/3;
K = 2*pi;
%"EpsilonN : E(n=0) = 1 and for E(n>=1)=2 ",
EpsilonN = 2 * ones(size(N0));
EpsilonN(N0 == 0) = 1;
%-------------------------------------------------------------------------------------------------------
for L1 = 1:length(L0)
L = L0(L1);
for M1 = 1:length(M0)
M = M0(M1);
for N1 = 1:length(N0)
N = N0(N1);
EpsN = EpsilonN(N1);
if abs((L/V1)+M) ~= abs(N/V)
C = (EpsN/2*V*pi)*besselj(M,K*b)*((sin(((L/V1)+M+(N/V))*V*pi)/(L/V1)+M+(N/V))+...
(sin(((L/V1)+M-(N/V))*V*pi)/(L/V1)+M-(N/V))); %<part 1>
elseif N ~= 0
C = (EpsN/2)*besselj(M,K*b) ; %<part 3>
else
C = EpsN*besselj(M,K*b); %<part 2>
end
Clmn(L1, M1, N1) = C;
end
end
end
Clmn
Clmn =
Clmn(:,:,1) = NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 1 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000219324541345203 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000219324538639666 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000219324534130437 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000219324527817516 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000219324519700904 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0002193245097806 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000219324498056251 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.00021932448452892 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000219324469197819 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000219324452062476 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000219324433124846 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000219324412381271 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000219324389834943 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000219324365484953 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000219324339331957 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000219324311374587 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000219324281613548 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000219324250049389 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000219324216680976 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000219324181508891 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Clmn(:,:,2) = NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000438649082691512 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0004386490772802 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000438649068260572 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000438649055634953 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000438649039401948 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000438649019561557 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.00043864899611285 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000438648969058151 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000438648938395602 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000438648904126131 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000438648866249274 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000438648824761776 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000438648779670612 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000438648730970667 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000438648678662871 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000438648622750013 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000438648563227445 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000438648500098885 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000438648433362474 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000438648363017747 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Clmn(:,:,3) = NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 0 NaN NaN NaN NaN NaN 0 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000438649082690117 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000438649077279735 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000438649068261502 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000438649055636348 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000438649039402413 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000438649019561557 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000438648996111919 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000438648969057221 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000438648938394672 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000438648904125201 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000438648866249739 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.000438648824761776 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.000438648779670612 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 Comments
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Arash Shahpasand
Arash Shahpasand on 4 Feb 2021
Dear Walter Roberson,
I hope you are doing well and healthy,
thank you very very very much for the quick response.
when get calculation of summation , i get error "Error using symengine Singularity."
syms l m
AS1 = symsum(symsum(Clmn.*besselj((l/V1)+m,K*C),m,-20,20),l,0,20)
%--------------------------------------------------------------------------------------------------------------
syms l m B
AS2 = symsum(B*symsum(Clmn.*besselj((l/V1)+m,K*C),m,-20,20),l,0,20)
and for this calculation to find An and Bl, need your help, Plz .
"linear algebra equations of the unknown coefficient A(n) and B(l)"
an = (2/V).*EpsilonN.*exp((-1i.*N0.*pi)/(2.*V)).*cos(N0.*gama/V)
AX=B ,
X = [Bl ; An]
A = [A11 A12 ; A21 A22]
B = [R11; R22]
X = linsolve(A,B)
or use "X = inv(A)*B "
Regards
Walter Roberson
Walter Roberson on 5 Feb 2021
Do not use symsum() for that purpose. Calculate an array of besselj((l/V1)+m,K*C) values that is length(L0) by length(M0), and use .* against Clmn, and sum() that across the second dimension. Construct a vector of B values that is a column length(L0) tall, and .* that by the result of the sum, and sum() the result of the multiplication across the first dimension. The result should be 1 x 1 x length(N0) .
Walter Roberson
Walter Roberson on 5 Feb 2021
I do not want to get involved in solving infinite numbers of linear equations. Any such a proposal is numeric nonsense, and needs to be approached through theoretical techniques, which might include:
  • "renormalization" of infinities -- the sort of mathematics that "proves" that 1-1+1+1+1... infinity "equals" 1/12
  • limit processes... which would not give you the individual values anyhow, since there are an infinite number of individual values
  • switching from summation to integration (which still will not give you all infinite number of results.)
The determinant of an infinite matrix is going to be either 0 or infinite or exactly 1. In the first two cases, you cannot find the solution for the equations; in the third case, the solution is the matrix times the marginal vector and no simultaneous equation work needs to be done.

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Asked:

Arash Shahpasand
Arash Shahpasand
on 30 Jan 2021

Commented:

Walter Roberson
Walter Roberson
on 3 Mar 2021

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