# fractional power with an arbitrary branch cut
function pow(x,a,cut)
  if(angle(x)/cut<=1)
    return(abs(x)^a*exp(1im*angle(x)*a))
  else
    return(abs(x)^a*exp(1im*(angle(x)-sign(cut)*2*pi)*a))
  end
end

# asymptotic airy functions
# specify a branch cut for the fractional power
function airyai_asym(x,cut)
  if(abs(real(pow(x,3/2,cut)))<airy_threshold)
    return(exp(2/3*pow(x,3/2,cut))*airyai(x))
  else
    ret=0
    for n in 0:airy_order
      ret+=gamma(n+5/6)*gamma(n+1/6)*(-3/4)^n/(4*pi^(3/2)*factorial(n)*pow(x,3*n/2+1/4,cut))
    end
    return ret
  end
end
function airyaiprime_asym(x,cut)
  if(abs(real(pow(x,3/2,cut)))<airy_threshold)
    return(exp(2/3*pow(x,3/2,cut))*airyaiprime(x))
  else
    ret=0
    for n in 0:airy_order
      ret+=gamma(n+5/6)*gamma(n+1/6)*(-3/4)^n/(4*pi^(3/2)*factorial(n))*(-1/pow(x,3*n/2-1/4,cut)-(3/2*n+1/4)/pow(x,3*n/2+5/4,cut))
    end
    return ret
  end
end

# solutions of (-\Delta+V-ip)phi=0
# assume that p has an infinitesimal real part (and adjust the branch cuts appropriately)
function phi(p,x,E,V)
  return(airyai_asym(exp(-1im*pi/3)*(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi))
end
function dphi(p,x,E,V)
  return(exp(-1im*pi/3)*E^(1/3)*airyaiprime_asym(exp(-1im*pi/3)*(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi))
end
function eta(p,x,E,V)
  return(exp(-1im*pi/3)*airyai_asym(-(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi/2))
end
function deta(p,x,E,V)
  return(-exp(-1im*pi/3)*E^(1/3)*airyaiprime_asym(-(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi/2))
end

# Laplace transform of psi
# assume that p has an infinitesimal real part (and adjust the branch cuts appropriately)
# for example, (1im*p-V)^(3/2) becomes pow(1im*p-V,3/2,-pi/2) because when 1im*p is real negative, its square root should be imaginary positive
function f(p,x,k0,E,V)
  T=2im*k0/(1im*k0-sqrt(V-k0*k0))
  R=T-1

  if x>=0
    C2=-1im*T/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*((sqrt(V-k0*k0)+pow(-1im*p,1/2,pi/2))/(-1im*p+k0*k0)-2im*E^(-1/3)*pi*quadgk(y -> (pow(-1im*p,1/2,pi/2)*eta(p,0,E,V)-deta(p,0,E,V))*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1])
    FT=2*E^(-1/3)*pi*(quadgk(y -> phi(p,x,E,V)*eta(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2))),0,x)[1]+quadgk(y -> eta(p,x,E,V)*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2))),x,Inf)[1])
    main=C2*phi(p,x,E,V)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2)))+T*FT

    # subtract the contribution of the pole, which will be added back in after the integration
    pole=psi_pole(x,k0,E,V)/(p+1im*k0*k0)
    return(main-pole)
  else
    C1=-1im*T*((sqrt(V-k0*k0)*phi(p,0,E,V)+dphi(p,0,E,V))/(-1im*p+k0*k0)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))+E^(-1/3)*quadgk(y -> phi(p,y,E,V)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1])
    FI=-1im*exp(1im*k0*x)/(-1im*p+k0*k0)
    FR=-1im*exp(-1im*k0*x)/(-1im*p+k0*k0)
    main=C1*exp(pow(-1im*p,1/2,pi/2)*x)+FI+R*FR

    # subtract the contribution of the pole, which will be added back in after the integration
    pole=psi_pole(x,k0,E,V)/(p+1im*k0*k0)
    return(main-pole)
  end
end
# its derivative
function df(p,x,k0,E,V)
  T=2im*k0/(1im*k0-sqrt(V-k0*k0))
  R=T-1

  if x>=0
    C2=-1im*T/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*((sqrt(V-k0*k0)+pow(-1im*p,1/2,pi/2))/(-1im*p+k0*k0)-2im*E^(-1/3)*pi*quadgk(y -> (pow(-1im*p,1/2,pi/2)*eta(p,0,E,V)-deta(p,0,E,V))*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1])
    dFT=2*E^(-1/3)*pi*(quadgk(y -> dphi(p,x,E,V)*eta(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2))),0,x)[1]+quadgk(y -> deta(p,x,E,V)*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2))),x,Inf)[1])
    main=C2*dphi(p,x,E,V)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2)))+T*dFT

    # subtract the contribution of the pole, which will be added back in after the integration
    pole=dpsi_pole(x,k0,E,V)/(p+1im*k0*k0)
    return(main-pole)
  else
    C1=-1im*T*((sqrt(V-k0*k0)*phi(p,0,E,V)+dphi(p,0,E,V))/(-1im*p+k0*k0)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))+E^(-1/3)*quadgk(y -> phi(p,y,E,V)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1])
    dFI=k0*exp(1im*k0*x)/(-1im*p+k0*k0)
    dFR=-k0*exp(-1im*k0*x)/(-1im*p+k0*k0)
    main=C1*pow(-1im*p,1/2,pi/2)*exp(pow(-1im*p,1/2,pi/2)*x)+dFI+R*dFR

    # subtract the contribution of the pole, which will be added back in after the integration
    pole=dpsi_pole(x,k0,E,V)/(p+1im*k0*k0)
    return(main-pole)
  end
end

# psi (returns t,psi(x,t))
function psi(x,k0,E,V,p_npoints,p_cutoff)
  fft=fourier_fft(f,x,k0,E,V,p_npoints,p_cutoff)
  # add the contribution of the pole
  for i in 1:p_npoints
      fft[2][i]=fft[2][i]+psi_pole(x,k0,E,V)*exp(-1im*k0*k0*fft[1][i])
  end
  return(fft)
end
# its derivative
function dpsi(x,k0,E,V,p_npoints,p_cutoff)
  fft=fourier_fft(df,x,k0,E,V,p_npoints,p_cutoff)
  # add the contribution of the pole
  for i in 1:p_npoints
    fft[2][i]=fft[2][i]+dpsi_pole(x,k0,E,V)*exp(-1im*k0*k0*fft[1][i])
  end
  return(fft)
end

# compute Fourier transform by sampling and fft
function fourier_fft(A,x,k0,E,V,p_npoints,p_cutoff)
  fun=zeros(Complex{Float64},p_npoints)
  times=zeros(p_npoints)

  # prepare fft
  for i in 1:p_npoints
    fun[i]=p_cutoff/pi*A(1im*(-p_cutoff+2*p_cutoff*(i-1)/p_npoints),x,k0,E,V)
    times[i]=(i-1)*pi/p_cutoff
  end

  ifft!(fun)
  
  # correct the phase
  for i in 2:2:p_npoints
    fun[i]=-fun[i]
  end
  return([times,fun])
end

# asymptotic value of psi
function psi_pole(x,k0,E,V)
  if x>=0
    return(1im*phi(-1im*k0*k0,x,E,V)*2*k0/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(k0*k0-V),3/2,-pi/2)-E^(-1)*pow(k0*k0-V,3/2,-pi/2))))
  else
    return((1im*k0*phi(-1im*k0*k0,0,E,V)-dphi(-1im*k0*k0,0,E,V))/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(-1im*k0*x)+exp(1im*k0*x))
  end
end
function dpsi_pole(x,k0,E,V)
  if x>=0
    return(1im*dphi(-1im*k0*k0,x,E,V)*2*k0/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(k0*k0-V),3/2,-pi/2)-E^(-1)*pow(k0*k0-V,3/2,-pi/2))))
  else
    return(-1im*k0*(1im*k0*phi(-1im*k0*k0,0,E,V)-dphi(-1im*k0*k0,0,E,V))/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(-1im*k0*x)+1im*k0*exp(1im*k0*x))
  end
end

# current
function J(ps,dps)
  return(2*imag(conj(ps)*dps))
end

# complete computation of the current
function current(x,k0,E,V,p_npoints,p_cutoff)
  ps=psi(x,k0,E,V,p_npoints,p_cutoff)
  dps=dpsi(x,k0,E,V,p_npoints,p_cutoff)
  Js=zeros(Complex{Float64},p_npoints)
  for i in 1:p_npoints
    Js[i]=J(ps[2][i],dps[2][i])
  end
  return(Js)
end