from sympy import Rational, Symbol, Ei, diff
from mpmath import glaisher, pi, exp, factorial

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 FA(As, θ):
    return sum([A * θ**(2 * i + 2) for i, A in enumerate(As)])


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


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, θc, 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)