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In addition I tried to calculate the integral by definin $A(c)$ using A[c_?Qnumeric]. In this case the integral did not give results at all.

In addition I tried to calculate the integral by definin $A(c)$ using A[c_?NumericQ]. In this case the integral did not give results at all.

I get many warnings which are probably due to the definition of $A(x,y)$, but in other cases I did this I got the correct results despite these warnings. In this case I get that the integral fail to converge, and indeed the result I get here is $0.605531$ (plus an imagenary contribution) is probablt too far from the approximately expected $2.467$. I tried many integration methods using the Methods option, but non of them helped. Monte-Carlo methods also gave results of about $0.605$.

I get many warnings which are probably due to the definition of $A(c)$, but in other cases I did this I got the correct results despite these warnings. In this case I get that the integral fail to converge, and indeed the result I get here is $0.605531$ (plus an imagenary contribution) is probablt too far from the approximately expected $2.467$. I tried many integration methods using the Methods option, but non of them helped. Monte-Carlo methods also gave results of about $0.605$.

The function $\phi(x,y)$ is defined using a function $A(x,y)$ which is given by an parametric numerical integral. The definition is

A[c_?NumericQ] := Evaluate[2 NIntegrate[(2 ArcCos[Sqrt[1 - c]/Sqrt[(c + Cos[π x]) Sec[(πx)/2]^2]])/π,
{x, -((2 ArcCos[Sqrt[1 - c]/Sqrt[1 + c]])/π), (2 ArcCos[Sqrt[1 - c]/Sqrt[1 + c]])/π}]]

r[x_, y_] := 1/Sqrt[π] Sqrt[A[1 - 2/(Cos[π/2 x]^-2 + Cos[π/2 y]^-2)]]

a = Sqrt[π]/2;
ϕ[x_, y_] := 2 ArcSin[1/Sqrt[2] a r[x/a, y/a]];

I get many warnings which are probably due to the definition of $A(x,y)$, but in other cases I did this I got the correct results despite these warnings. In this case I get that the integral fail to converge, and indeed the result I get here is $0.605531$ (plus an imagenary contribution) is probablt too far from the approximately expected $2.467$. I tried many integration methods using the Methods option, but non of them helped. Monte-Carlo methods also gave results of about $0.605$.

The function $\phi(x,y)$ is defined using a function $A(c)$ which is given by an parametric numerical integral. The definition is

A[c_] := Evaluate[2 NIntegrate[(2 ArcCos[Sqrt[1 - c]/Sqrt[(c + Cos[π x]) Sec[(πx)/2]^2]])/π,
{x, -((2 ArcCos[Sqrt[1 - c]/Sqrt[1 + c]])/π), (2 ArcCos[Sqrt[1 - c]/Sqrt[1 + c]])/π}]]

r[x_, y_] := 1/Sqrt[π] Sqrt[A[1 - 2/(Cos[π/2 x]^-2 + Cos[π/2 y]^-2)]]

a = Sqrt[π]/2;
ϕ[x_, y_] := 2 ArcSin[1/Sqrt[2] a r[x/a, y/a]];

I get many warnings which are probably due to the definition of $A(x,y)$, but in other cases I did this I got the correct results despite these warnings. In this case I get that the integral fail to converge, and indeed the result I get here is $0.605531$ (plus an imagenary contribution) is probablt too far from the approximately expected $2.467$. I tried many integration methods using the Methods option, but non of them helped. Monte-Carlo methods also gave results of about $0.605$.

In addition I tried to calculate the integral by definin $A(c)$ using A[c_?Qnumeric]. In this case the integral did not give results at all.