# \eta
function eta(x,t,weights,d,v)
  if d==2
    return integrate_chebyshev(y->4*((x+t)*y+abs(x-t)*(1-y))*v((x+t)*y+abs(x-t)*(1-y))/sqrt(((x+t)*y+abs(x-t)*(2-y))*((x+t)*(1+y)+abs(x-t)*(1-y))),0,1,length(weights))
  elseif d==3
    return (x>t ? 2*t/x : 2)* integrate_legendre(y->2*pi*((x+t)*y+abs(x-t)*(1-y))*v((x+t)*y+abs(x-t)*(1-y)),0,1,weights)
  end
end

# initialize V and Eta
function init_veta(weights,d,v)
  order=length(weights[1])
  V=Array{Complex{Float64}}(undef,order)
  Eta=Array{Array{Complex{Float64}}}(undef,order)
  Eta0=Array{Complex{Float64}}(undef,order)
  V0=v(0)
  for i in 1:order
    k=(1-weights[1][i])/(1+weights[1][i])
    V[i]=v(k)
    Eta[i]=Array{Complex{Float64}}(undef,order)
    for j in 1:order
      y=(weights[1][j]+1)/2
      Eta[i][j]=eta(k,(1-y)/y,weights,d,v)
    end
    y=(weights[1][i]+1)/2
    Eta0[i]=eta(0,(1-y)/y,weights,d,v)
  end
  return(V,V0,Eta,Eta0)
end

# inverse Fourier transform
function u_x(x,u,weights,d)
  order=length(weights[1])
  if d==2
    out=integrate_legendre_sampled(y->(1-y)/y^3*besselj(0,x*(1-y)/y)/(2*pi),u,0,1,weights)
  elseif d==3
    out=integrate_legendre_sampled(y->(1-y)/y^3*sin(x*(1-y)/y)/x/(2*pi^2),u,0,1,weights)
  end
  return out
end