from sympy import *
from mpmath import glaisher, pi

β = Rational(1/8)
δ = 15
α = 0
σ = Rational(5, 6)
Δ = β * δ

s = 2**Rational(1, 12) * exp(-β) * glaisher**Rational(3, 2)

def f(B, θc, θ):
    return (θc * Ei(-1 / (B * θc)) * exp(1 / (B * θc)) + (θ - θc) * Ei(1/(B * (θ - θc))) * exp(-1/(B * (θ - θc)))) / pi

def FH(θ0, θ):
    return (θ**2 + θ0**2)**σ - (θ0**2)**σ

def FR(AL, AH, As, B, θc, θ0, θ):
    return AL * f(B, θc, -θ) + AH * FH(θ0, θ) + sum([A * θ**(2 * i + 2) for i,A in enumerate(As)])

def F(AL, AH, As, B, θc, θ0, θ):
    return FR(AL, AH, As, B, θc, θ0, θ) + AL * f(B, θc, θ)

def dfc(AL, B, m):
    if m == 0:
        return AL * θc * Ei(-1 / (B * θc)) * exp(1 / (B * θc)) / pi
    elif m == 1:
        return 0
    else:
        return AL * factorial(m) * factorial(m - 2) * B**(m-1) / pi

def dFc(AL, AH, As, B, θc, θ0, m):
    θ = Symbol['θ']
    return dfc(AL, B, m) + diff(FR(AL, AH, As, B, θc, θ0, θ), θ).subs(θ, θc)