I’m in troubles to find the zeros of the next equation of two variables:

(w - k)A + B = 0

The issue is that the terms ‘*A*’ and ‘*B*’ are not numbers but are integral of certain functions ‘*f(k,w,p)*’ and ‘*g(k,w,p)*’ involving the three variables in a very complicate way (since the integration is only on p, it eventually must disappear),

A(k,w) = \int_0^{\infty} f(k,w,p) dp

B(k,w) = \int_0^{\infty} g(k,w,p) dp

I have tried to use a double subsequent *for loop* to the variables k and w, so that I can integrate numerically over p only. Then I check which pairs (k, w) satisfy the equation by setting a range close to zero. The next is a simple case that works, to illustrate my method

```
using QuadGK
eps = 0.01
T = 100
c = 0.0008/(8*pi^2)
nF(p) = 1/(exp(p/T) + 1)
nB(p) = 1/(exp(p/T) - 1)
for w in 0.01:0.01:5
for k in 0.01:0.01:w
if w > k
L1(p) = log( (p + 0.5*(w+k))/(p + 0.5*(w-k)) ) - log( (p - 0.5*(w+k))/(p - 0.5*(w-k)) )
L2(p) = log( (p + 0.5*(w+k))/(p + 0.5*(w-k)) ) + log( (p - 0.5*(w+k))/(p - 0.5*(w-k)) )
A(p) = (c/k^2)*( (4*p + (w^2 - k^2)*L1(p)/(2*k) - 2*p*w*log((w+k)/(w-k))/k + w*p*L2(p)/k)*nF(p) + (4*p + (w^2 - k^2)*L1(p)/(2*k) - 2*p*w*log((w+k)/(w-k))/k + p*w*L2(p)/k )*nB(p) )
B(p) = (c/k^2)*( (2*p*(w^2 - k^2)*log((w+k)/(w-k))/k - p*(w^2-k^2)*L2(p)/k - 4*w*p - w*(w^2-k^2)*L1(p)/(2*k))*nF(p) + (2*p*(w^2-k^2)*log((w+k)/(w-k))/k - p*(w^2-k^2)*L2(p)/k - w*(w^2-k^2)*L1(p)/(2*k) - 4*w*p)*nB(p) )
IA = quadgk(A, 5, Inf, rtol=1e-3)
IB = quadgk(B, 5, Inf, rtol=1e-3)
if -eps < ((w-k)*(1+IA[1]) + IB[1]) < eps
println(k, w)
end
end
end
end
```

However, it has not been efficient. Do you know of any method to perform this calculation in a better way?

Another problem I have is, for slightly more complex A and B functions, a certain configurations of values of k,w and p, I get indeterminacies by evaluating logarithms of negative numbers (I’m only interested in real solutions).

How can I evaluate the sign of the expression inside the logarithms to discard the pairs (k, w) that lead to these unwanted ones?