Any comment, idea or innovation to calculate this parametric implicit integral?

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Mehdi
Mehdi 2023-1-14
评论： Walter Roberson 2023-1-15
Ran in:
Any comment, idea or innovation to calculate this parametric implicit integral?
Note M, II, JJ are arbitrary positive integers (0<M, II, JJ<11).
F must be a function of Pm at the final!
clear
M = 3;
JJ = 5;
II = 5;
W = rand(II, JJ, M);
V = rand(II, JJ, M);
p = sym('p',[1 M]);
syms x y
w = sym('0');
v = sym('0');
L = sym('0');
for m=1:M
for i=1:II
for j=1:JJ
w =w+W(i, j, m)*legendreP(i-1, x)*legendreP(j-1, y)*p(m);
v =v+V(i, j, m)*legendreP(i-1, x)*legendreP(j-1, y)*p(m);
L = L+(legendreP(i-1, x)*legendreP(j-1, y))^2;
end
end
end
H = 1+tanh(w-v);
F = int(int(H*L,x,[-1 , 1]), y,[-1, 1])
F = 
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Mehdi
Mehdi 2023-1-14
编辑：Mehdi 2023-1-14
i am thiniking to replace 1+tanh(w-v) with other simpler continuous functions which help to solve this integral, since it acts like step function. What do you think?
Approximate solutions are also welcome.
Walter Roberson
Walter Roberson 2023-1-15
I am not clear as to what you are requesting?
The integral does not appear to be implicit, just not closed form.
If you are asking for a way to find a closed form expression for it, then I doubt that is possible.
You can use techniques such as taylor series, but that gets messy quickly and is going to be pretty inaccurate.

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