def series_small_a():
"""Tylor series expansion of Phi(a, b, x) in a=0 up to order 5.
"""
order = 5
a, b, x, k = symbols("a b x k")
A = [] # terms with a
X = [] # terms with x
B = [] # terms with b (polygammas)
# Phi(a, b, x) = exp(x)/gamma(b) * sum(A[i] * X[i] * B[i])
expression = Sum(x**k / factorial(k) / gamma(a * k + b),
(k, 0, S.Infinity))
expression = gamma(b) / sympy.exp(x) * expression
# nth term of taylor series in a=0: a^n/n! * (d^n Phi(a, b, x)/da^n at a=0)
for n in range(0, order + 1):
term = expression.diff(a, n).subs(a, 0).simplify().doit()
# set the whole bracket involving polygammas to 1
x_part = (term.subs(polygamma(0, b),
1).replace(polygamma, lambda *args: 0))
# sign convetion: x part always positive
x_part *= (-1)**n
A.append(a**n / factorial(n))
X.append(horner(x_part))
B.append(horner((term / x_part).simplify()))
s = "Tylor series expansion of Phi(a, b, x) in a=0 up to order 5.\n"
s += "Phi(a, b, x) = exp(x)/gamma(b) * sum(A[i] * X[i] * B[i], i=0..5)\n"
for name, c in zip(['A', 'X', 'B'], [A, X, B]):
for i in range(len(c)):
s += f"\n{name}[{i}] = " + str(c[i])
return s
def _simplify_eq(e):
'''simplify an equation and try to put it into horner form'''
e = simplify(e)
try:
e = horner(e)
except (PolynomialError, AttributeError):
pass
return e
def test_horner():
assert horner(0) == 0
assert horner(1) == 1
assert horner(x) == x
assert horner(x + 1) == x + 1
assert horner(x**2 + 1) == x**2 + 1
assert horner(x**2 + x) == (x + 1)*x
assert horner(x**2 + x + 1) == (x + 1)*x + 1
assert horner(9*x**4 + 8*x**3 + 7*x**2 + 6*x + 5) == (((9*x + 8)*x + 7)*x + 6)*x + 5
assert horner(a*x**4 + b*x**3 + c*x**2 + d*x + e) == (((a*x + b)*x + c)*x + d)*x + e
assert horner(4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y, wrt=x) == ((4*y + 2)*x*y + (2*y + 1)*y)*x
assert horner(4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y, wrt=y) == ((4*x + 2)*y*x + (2*x + 1)*x)*y
def test_horner():
assert horner(0) == 0
assert horner(1) == 1
assert horner(x) == x
assert horner(x + 1) == x + 1
assert horner(x**2 + 1) == x**2 + 1
assert horner(x**2 + x) == (x + 1) * x
assert horner(x**2 + x + 1) == (x + 1) * x + 1
assert horner(9 * x**4 + 8 * x**3 + 7 * x**2 + 6 * x +
5) == (((9 * x + 8) * x + 7) * x + 6) * x + 5
assert horner(a * x**4 + b * x**3 + c * x**2 + d * x +
e) == (((a * x + b) * x + c) * x + d) * x + e
assert horner(4 * x**2 * y**2 + 2 * x**2 * y + 2 * x * y**2 + x * y,
wrt=x) == ((4 * y + 2) * x * y + (2 * y + 1) * y) * x
assert horner(4 * x**2 * y**2 + 2 * x**2 * y + 2 * x * y**2 + x * y,
wrt=y) == ((4 * x + 2) * y * x + (2 * x + 1) * x) * y
def series_small_a_small_b():
"""Tylor series expansion of Phi(a, b, x) in a=0 and b=0 up to order 5.
Be aware of cancellation of poles in b=0 of digamma(b)/Gamma(b) and
polygamma functions.
digamma(b)/Gamma(b) = -1 - 2*M_EG*b + O(b^2)
digamma(b)^2/Gamma(b) = 1/b + 3*M_EG + b*(-5/12*PI^2+7/2*M_EG^2) + O(b^2)
polygamma(1, b)/Gamma(b) = 1/b + M_EG + b*(1/12*PI^2 + 1/2*M_EG^2) + O(b^2)
and so on.
"""
order = 5
a, b, x, k = symbols("a b x k")
M_PI, M_EG, M_Z3 = symbols("M_PI M_EG M_Z3")
c_subs = {pi: M_PI, EulerGamma: M_EG, zeta(3): M_Z3}
A = [] # terms with a
X = [] # terms with x
B = [] # terms with b (polygammas expanded)
C = [] # terms that generate B
# Phi(a, b, x) = exp(x) * sum(A[i] * X[i] * B[i])
# B[0] = 1
# B[k] = sum(C[k] * b**k/k!, k=0..)
# Note: C[k] can be obtained from a series expansion of 1/gamma(b).
expression = gamma(b)/sympy.exp(x) * \
Sum(x**k/factorial(k)/gamma(a*k+b), (k, 0, S.Infinity))
# nth term of taylor series in a=0: a^n/n! * (d^n Phi(a, b, x)/da^n at a=0)
for n in range(0, order + 1):
term = expression.diff(a, n).subs(a, 0).simplify().doit()
# set the whole bracket involving polygammas to 1
x_part = (term.subs(polygamma(0, b),
1).replace(polygamma, lambda *args: 0))
# sign convetion: x part always positive
x_part *= (-1)**n
# expansion of polygamma part with 1/gamma(b)
pg_part = term / x_part / gamma(b)
if n >= 1:
# Note: highest term is digamma^n
pg_part = pg_part.replace(
polygamma, lambda k, x: pg_series(k, x, order + 1 + n))
pg_part = (pg_part.series(b, 0, n=order + 1 - n).removeO().subs(
polygamma(2, 1), -2 * zeta(3)).simplify())
A.append(a**n / factorial(n))
X.append(horner(x_part))
B.append(pg_part)
# Calculate C and put in the k!
C = sympy.Poly(B[1].subs(c_subs), b).coeffs()
C.reverse()
for i in range(len(C)):
C[i] = (C[i] * factorial(i)).simplify()
s = "Tylor series expansion of Phi(a, b, x) in a=0 and b=0 up to order 5."
s += "\nPhi(a, b, x) = exp(x) * sum(A[i] * X[i] * B[i], i=0..5)\n"
s += "B[0] = 1\n"
s += "B[i] = sum(C[k+i-1] * b**k/k!, k=0..)\n"
s += "\nM_PI = pi"
s += "\nM_EG = EulerGamma"
s += "\nM_Z3 = zeta(3)"
for name, c in zip(['A', 'X'], [A, X]):
for i in range(len(c)):
s += f"\n{name}[{i}] = "
s += str(c[i])
# For C, do also compute the values numerically
for i in range(len(C)):
s += f"\n# C[{i}] = "
s += str(C[i])
s += f"\nC[{i}] = "
s += str(C[i].subs({
M_EG: EulerGamma,
M_PI: pi,
M_Z3: zeta(3)
}).evalf(17))
# Does B have the assumed structure?
s += "\n\nTest if B[i] does have the assumed structure."
s += "\nC[i] are derived from B[1] allone."
s += "\nTest B[2] == C[1] + b*C[2] + b^2/2*C[3] + b^3/6*C[4] + .."
test = sum([b**k / factorial(k) * C[k + 1] for k in range(order - 1)])
test = (test - B[2].subs(c_subs)).simplify()
s += f"\ntest successful = {test==S(0)}"
s += "\nTest B[3] == C[2] + b*C[3] + b^2/2*C[4] + .."
test = sum([b**k / factorial(k) * C[k + 2] for k in range(order - 2)])
test = (test - B[3].subs(c_subs)).simplify()
s += f"\ntest successful = {test==S(0)}"
return s