def test_issue_6966():
i, k, m = symbols('i k m', integer=True)
z_i, q_i = symbols('z_i q_i')
a_k = Sum(-q_i*z_i/k,(i,1,m))
b_k = a_k.diff(z_i)
assert isinstance(b_k, Sum)
assert b_k == Sum(-q_i/k,(i,1,m))
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 test_eval_diff():
assert Sum(x, (x, 1, 2)).diff(x) == 0
assert Sum(x * y, (x, 1, 2)).diff(x) == 0
assert Sum(x * y, (y, 1, 2)).diff(x) == Sum(y, (y, 1, 2))
e = Sum(x * y, (x, 1, a))
assert e.diff(a) == Derivative(e, a)
assert Sum(x * y, (x, 1, 3), (a, 2, 5)).diff(y) == Sum(x * y, (x, 1, 3), (a, 2, 5)).doit().diff(y) == 24
def test_eval_diff():
assert Sum(x, (x, 1, 2)).diff(x) == 0
assert Sum(x * y, (x, 1, 2)).diff(x) == 0
assert Sum(x * y, (y, 1, 2)).diff(x) == Sum(y, (y, 1, 2))
e = Sum(x * y, (x, 1, a))
assert e.diff(a) == Derivative(e, a)
assert Sum(x*y, (x, 1, 3), (a, 2, 5)).diff(y) == \
Sum(x*y, (x, 1, 3), (a, 2, 5)).doit().diff(y) == 24
plt.loglog(Ns, [psf_to_sigma(g15, N, 0, 0) for N in Ns])
plt.loglog(Ns, [psf_to_sigma(airy, N, 0, 0) for N in Ns])
plt.loglog(Ns, [psf_to_sigma(img, N, 0, 0) for N in Ns])
# %%
from sympy import Function, Sum, symbols, ln, Eq
from sympy.abc import N, i
η, ζ = symbols('ζ,η')
P = Function('P')(ζ, η)
l = Sum(ln(P), (i, 1, N))
l
# %%
l.diff(η, 2).simplify()
# %%
(l.diff(η, 1) ** 2).simplify()
# %%
from astropy.modeling.functional_models import Gaussian1D
# %%
def fisher(pdf_grid, dx):
normed = pdf_grid / np.trapz(pdf_grid, dx=dx)
grad = np.gradient(normed, dx)
return np.trapz(grad ** 2 / normed, dx=dx)
