Please, help to find a mistake in the code for double integral
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I have a double integral (a picture of formula is attached) and a code. But this integral should be equal 1 at s = 0 at any parameters n, t, k. Thus, all curves should start from the point (0,1). But, I obtain at vpa the ans = 0.35331 at s = 0, and curves start from different points at different parameters. I suspect that there is an error somewhere in the code entry itself, but I can't see it now. Please help me to find it.
n = 1 ;
t = 1;
k = 1; %(k is ksi in formula)
s = 0:0.5:3;
fun = @(x,q,p) ((x.*exp(2*n*t.*(x.^2)))./sqrt(1-x.^2)).*(exp(-2*t.*(q.^2)).*besselj(0,q.*p.*x).*(1+((sqrt(-pi)./(q*k)).*(1+((q*k).^2)/2).*erfi(q*k*sqrt(-1)/2).*exp(-((q*k).^2)/4))));
f3 = arrayfun(@(p)integral2(@(x,q)fun(x,q,p),0,1,0,Inf),s);
Cor = -((2*k*sqrt(2*n*t)*exp(-2*n*t))/(pi*erf(sqrt(2*n*t))*(atan(k/(2*sqrt(2*t)))-sqrt(2*t)/(2*k*(0.25+(2*t)/(k^2)))))).*f3;
plot(Cor,'b-')
%vpa(Cor,5)
4 Comments
Torsten
on 24 Feb 2023
And if "erf" does not work, you take another function that sounds similarily and hope you get the correct result ?
Google
error function for complex argument & matlab
and you will find some functions from the file exchange to compute the error function with complex argument (if this is really what is meant in the formula).
Answers (2)
Torsten
on 24 Feb 2023
Moved: Image Analyst
on 24 Feb 2023
You can use
erf(i*z) = i*erfi(z)
Thus in your case
erf(i*q*k/2) = i*erfi(q*k/2)
But you will come into trouble here since you integrate from 0 to Inf with respect to q.
And erfi(x) -> Inf very fast as x-> Inf.
You will have to evaluate
i*erfi(q*k/2)*exp(-(q*k/2).^2)
as one expression to avoid Inf * 0 here.
Torsten
on 24 Feb 2023
n = 1 ;
t = 1;
k = 1; %(k is ksi in formula)
s = 0:0.5:20;
fun = @(x,q,p) ((x.*exp(2*n*t.*(x.^2)))./sqrt(1-x.^2)).*(exp(-2*t.*(q.^2)).*besselj(0,q.*p.*x).*(1+((sqrt(-pi)./(q*k)).*(1+((q*k).^2)/2).*func(q,k))));
f3 = arrayfun(@(p)integral2(@(x,q)fun(x,q,p),0,1,0,Inf),s);
Cor = -((2*k*sqrt(2*n*t)*exp(-2*n*t))/(pi*erf(sqrt(2*n*t))*(atan(k/(2*sqrt(2*t)))-sqrt(2*t)/(2*k*(0.25+(2*t)/(k^2)))))).*f3;
plot(s,Cor,'b-')
grid on
function values = func(q,k)
values = arrayfun(@(q)1i*2/sqrt(pi)*integral(@(t)exp(t.^2-((q*k)/2)^2),0,q*k/2),q);
end
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