def _eval_rewrite_as_harmonic(self, n, z, **kwargs):
if n.is_integer:
if n.is_zero:
return harmonic(z - 1) - S.EulerGamma
else:
return S.NegativeOne**(n + 1) * factorial(n) * (
zeta(n + 1) - harmonic(z - 1, n + 1))
def test_limit_seq_fail():
# improve Summation algorithm or add ad-hoc criteria
e = (harmonic(n)**3 * Sum(1 / harmonic(k), (k, 1, n)) /
(n * Sum(harmonic(k) / k, (k, 1, n))))
assert limit_seq(e, n) == 2
# No unique dominant term
e = (Sum(2**k * binomial(2 * k, k) / k**2,
(k, 1, n)) / (Sum(2**k / k * 2,
(k, 1, n)) * Sum(binomial(2 * k, k),
(k, 1, n))))
assert limit_seq(e, n) == S(3) / 7
# Simplifications of summations needs to be improved.
e = n**3 * Sum(2**k / k**2,
(k, 1, n))**2 / (2**n * Sum(2**k / k, (k, 1, n)))
assert limit_seq(e, n) == 2
e = (harmonic(n) * Sum(2**k / k,
(k, 1, n)) / (n * Sum(2**k * harmonic(k) / k**2,
(k, 1, n))))
assert limit_seq(e, n) == 1
e = (Sum(2**k * factorial(k) / k**2,
(k, 1, 2 * n)) / (Sum(4**k / k**2,
(k, 1, n)) * Sum(factorial(k),
(k, 1, 2 * n))))
assert limit_seq(e, n) == S(3) / 16
def test_ideal_soliton():
raises(ValueError, lambda: IdealSoliton('sol', -12))
raises(ValueError, lambda: IdealSoliton('sol', 13.2))
raises(ValueError, lambda: IdealSoliton('sol', 0))
f = Function('f')
raises(ValueError, lambda: density(IdealSoliton('sol', 10)).pmf(f))
k = Symbol('k', integer=True, positive=True)
x = Symbol('x', integer=True, positive=True)
t = Symbol('t')
sol = IdealSoliton('sol', k)
assert density(sol).low == S.One
assert density(sol).high == k
assert density(sol).dict == Density(density(sol))
assert density(sol).pmf(x) == Piecewise(
(1 / k, Eq(x, 1)), (1 / (x * (x - 1)), k >= x), (0, True))
k_vals = [5, 20, 50, 100, 1000]
for i in k_vals:
assert E(sol.subs(k, i)) == harmonic(i) == moment(sol.subs(k, i), 1)
assert variance(sol.subs(
k, i)) == (i - 1) + harmonic(i) - harmonic(i)**2 == cmoment(
sol.subs(k, i), 2)
assert skewness(sol.subs(k, i)) == smoment(sol.subs(k, i), 3)
assert kurtosis(sol.subs(k, i)) == smoment(sol.subs(k, i), 4)
assert exp(I * t) / 10 + Sum(exp(I * t * x) / (x * x - x), (x, 2, k)).subs(
k, 10).doit() == characteristic_function(sol.subs(k, 10))(t)
assert exp(t) / 10 + Sum(exp(t * x) / (x * x - x), (x, 2, k)).subs(
k, 10).doit() == moment_generating_function(sol.subs(k, 10))(t)
def _eval_rewrite_as_harmonic(self, n, z):
if n.is_integer:
if n == S.Zero:
return harmonic(z - 1) - S.EulerGamma
else:
return S.NegativeOne**(n + 1) * C.factorial(n) * (
C.zeta(n + 1) - harmonic(z - 1, n + 1))
async def collect(self, bot, event: Message, query: str):
match = COLLECT_QUERY1.match(query)
if match:
n = int(match.group('n'))
total = int(match.group('total')) if match.group('total') else n
else:
match = COLLECT_QUERY2.match(query)
if match:
n = int(match.group('n'))
total = int(match.group('total'))
else:
await bot.say(event.channel, '요청을 해석하는데에 실패했어요!')
return
if total < 2 or total > 512:
await bot.say(event.channel, '정상적인 전체 갯수를 입력해주세요! (2개 이상 512개 이하)')
return
if n < 1 or n > 512:
await bot.say(event.channel, '정상적인 수집 갯수를 입력해주세요! (1개 이상 512개 이하)')
return
if total < n:
await bot.say(event.channel, '원하는 갯수가 전체 갯수보다 많을 수 없어요!')
return
result = n * harmonic(n)
if total > n:
result /= n / total
text = '부분적으로'
else:
text = '모두'
await bot.say(
event.channel, f'상품 1개 구입시 {total}종류의 특전 중 하나를 무작위로 100%'
f'확률로 준다고 가정할 때 {n}종류의 특전을 {text} 모으려면, 평균적으로'
f' {math.ceil(result)}(`{float(result):.2f}`)개의 상품을'
' 구입해야 수집에 성공할 수 있어요!')
def eval(cls, n, z):
n, z = map(sympify, (n, z))
from sympy import unpolarify
if n.is_integer:
if n.is_nonnegative:
nz = unpolarify(z)
if z != nz:
return polygamma(n, nz)
if n.is_positive:
if z is S.Half:
return (
(-1) ** (n + 1)
* factorial(n)
* (2 ** (n + 1) - 1)
* zeta(n + 1)
)
if n is S.NegativeOne:
return loggamma(z)
else:
if z.is_Number:
if z is S.NaN:
return S.NaN
elif z is S.Infinity:
if n.is_Number:
if n.is_zero:
return S.Infinity
else:
return S.Zero
if n.is_zero:
return S.Infinity
elif z.is_Integer:
if z.is_nonpositive:
return S.ComplexInfinity
else:
if n.is_zero:
return -S.EulerGamma + harmonic(z - 1, 1)
elif n.is_odd:
return (-1) ** (n + 1) * factorial(n) * zeta(n + 1, z)
if n.is_zero:
if z is S.NaN:
return S.NaN
elif z.is_Rational:
p, q = z.as_numer_denom()
# only expand for small denominators to avoid creating long expressions
if q <= 5:
return expand_func(polygamma(S.Zero, z, evaluate=False))
elif z in (S.Infinity, S.NegativeInfinity):
return S.Infinity
else:
t = z.extract_multiplicatively(S.ImaginaryUnit)
if t in (S.Infinity, S.NegativeInfinity):
return S.Infinity
def eval(cls, n, z):
n, z = list(map(sympify, (n, z)))
from sympy import unpolarify
if n.is_integer:
if n.is_nonnegative:
nz = unpolarify(z)
if z != nz:
return polygamma(n, nz)
if n == -1:
return loggamma(z)
else:
if z.is_Number:
if z is S.NaN:
return S.NaN
elif z is S.Infinity:
if n.is_Number:
if n is S.Zero:
return S.Infinity
else:
return S.Zero
elif z.is_Integer:
if z.is_nonpositive:
return S.ComplexInfinity
else:
if n is S.Zero:
return -S.EulerGamma + harmonic(z - 1, 1)
elif n.is_odd:
return (-1)**(n + 1)*factorial(n)*zeta(n + 1, z)
if n == 0:
if z is S.NaN:
return S.NaN
elif z.is_Rational:
# TODO actually *any* n/m can be done, but that is messy
lookup = {S(1)/2: -2*log(2) - S.EulerGamma,
S(1)/3: -S.Pi/2/sqrt(3) - 3*log(3)/2 - S.EulerGamma,
S(1)/4: -S.Pi/2 - 3*log(2) - S.EulerGamma,
S(3)/4: -3*log(2) - S.EulerGamma + S.Pi/2,
S(2)/3: -3*log(3)/2 + S.Pi/2/sqrt(3) - S.EulerGamma}
if z > 0:
n = floor(z)
z0 = z - n
if z0 in lookup:
return lookup[z0] + Add(*[1/(z0 + k) for k in range(n)])
elif z < 0:
n = floor(1 - z)
z0 = z + n
if z0 in lookup:
return lookup[z0] - Add(*[1/(z0 - 1 - k) for k in range(n)])
elif z in (S.Infinity, S.NegativeInfinity):
return S.Infinity
else:
t = z.extract_multiplicatively(S.ImaginaryUnit)
if t in (S.Infinity, S.NegativeInfinity):
return S.Infinity
async def collect(self, bot, event: Message, n: int):
result = n * harmonic(n)
await bot.say(
event.channel,
f'상품 1개 구입시 {n}종류의 특전 중 하나를 무작위로 100% 확률로 준다고 가정할 때'
f' {n}종류의 특전을 모두 모으려면, 평균적으로 {math.ceil(result)}'
f'(`{float(result):.2f}`)개의 상품을 구입해야 전체 수집에 성공할 수 있어요!'
)
def test_difference_delta__Sum():
e = Sum(1 / k, (k, 1, n))
assert dd(e, n) == 1 / (n + 1)
assert dd(e, n, 5) == Add(*[1 / (i + n + 1) for i in range(5)])
e = Sum(1 / k, (k, 1, 3 * n))
assert dd(e, n) == Add(*[1 / (i + 3 * n + 1) for i in range(3)])
e = n * Sum(1 / k, (k, 1, n))
assert dd(e, n) == 1 + Sum(1 / k, (k, 1, n))
e = Sum(1 / k, (k, 1, n), (m, 1, n))
assert dd(e, n) == harmonic(n)
def eval(cls, z, a_=None):
if a_ is None:
z, a = list(map(sympify, (z, 1)))
else:
z, a = list(map(sympify, (z, a_)))
if a.is_Number:
if a is S.NaN:
return S.NaN
elif a is S.One and a_ is not None:
return cls(z)
# TODO Should a == 0 return S.NaN as well?
if z.is_Number:
if z is S.NaN:
return S.NaN
elif z is S.Infinity:
return S.One
elif z is S.Zero:
if a.is_negative:
return S.Half - a - 1
else:
return S.Half - a
elif z is S.One:
return S.ComplexInfinity
elif z.is_Integer:
if a.is_Integer:
if z.is_negative:
zeta = (-1)**z * bernoulli(-z + 1) / (-z + 1)
elif z.is_even:
B, F = bernoulli(z), factorial(z)
zeta = 2**(z - 1) * abs(B) * pi**z / F
else:
return
if a.is_negative:
return zeta + harmonic(abs(a), z)
else:
return zeta - harmonic(a - 1, z)
def test_limit_seq():
e = binomial(2 * n, n) / Sum(binomial(2 * k, k), (k, 1, n))
assert limit_seq(e) == S(3) / 4
assert limit_seq(e, m) == e
e = (5 * n**3 + 3 * n**2 + 4) / (3 * n**3 + 4 * n - 5)
assert limit_seq(e, n) == S(5) / 3
e = (harmonic(n) * Sum(harmonic(k), (k, 1, n))) / (n * harmonic(2 * n)**2)
assert limit_seq(e, n) == 1
e = Sum(k**2 * Sum(2**m / m, (m, 1, k)), (k, 1, n)) / (2**n * n)
assert limit_seq(e, n) == 4
e = (Sum(binomial(3 * k, k) * binomial(5 * k, k),
(k, 1, n)) / (binomial(3 * n, n) * binomial(5 * n, n)))
assert limit_seq(e, n) == S(84375) / 83351
e = Sum(harmonic(k)**2 / k, (k, 1, 2 * n)) / harmonic(n)**3
assert limit_seq(e, n) == S.One / 3
raises(ValueError, lambda: limit_seq(e * m))
def eval(cls, z, a_=None):
if a_ is None:
z, a = list(map(sympify, (z, 1)))
else:
z, a = list(map(sympify, (z, a_)))
if a.is_Number:
if a is S.NaN:
return S.NaN
elif a is S.One and a_ is not None:
return cls(z)
# TODO Should a == 0 return S.NaN as well?
if z.is_Number:
if z is S.NaN:
return S.NaN
elif z is S.Infinity:
return S.One
elif z is S.Zero:
if a.is_negative:
return S.Half - a - 1
else:
return S.Half - a
elif z is S.One:
return S.ComplexInfinity
elif z.is_Integer:
if a.is_Integer:
if z.is_negative:
zeta = (-1)**z * bernoulli(-z + 1)/(-z + 1)
elif z.is_even:
B, F = bernoulli(z), factorial(z)
zeta = 2**(z - 1) * abs(B) * pi**z / F
else:
return
if a.is_negative:
return zeta + harmonic(abs(a), z)
else:
return zeta - harmonic(a - 1, z)
def test_digamma():
assert digamma(nan) == nan
assert digamma(oo) == oo
assert digamma(-oo) == oo
assert digamma(I*oo) == oo
assert digamma(-I*oo) == oo
assert digamma(-9) == zoo
assert digamma(-9) == zoo
assert digamma(-1) == zoo
assert digamma(0) == zoo
assert digamma(1) == -EulerGamma
assert digamma(7) == Rational(49, 20) - EulerGamma
def t(m, n):
x = S(m)/n
r = digamma(x)
if r.has(digamma):
return False
return abs(digamma(x.n()).n() - r.n()).n() < 1e-10
assert t(1, 2)
assert t(3, 2)
assert t(-1, 2)
assert t(1, 4)
assert t(-3, 4)
assert t(1, 3)
assert t(4, 3)
assert t(3, 4)
assert t(2, 3)
assert t(123, 5)
assert digamma(x).rewrite(zeta) == polygamma(0, x)
assert digamma(x).rewrite(harmonic) == harmonic(x - 1) - EulerGamma
assert digamma(I).is_real is None
assert digamma(x,evaluate=False).fdiff() == polygamma(1, x)
assert digamma(x,evaluate=False).is_real is None
assert digamma(x,evaluate=False).is_positive is None
assert digamma(x,evaluate=False).is_negative is None
assert digamma(x,evaluate=False).rewrite(polygamma) == polygamma(0, x)
def eval(cls, n, z):
n, z = list(map(sympify, (n, z)))
from sympy import unpolarify
if n.is_integer:
if n.is_nonnegative:
nz = unpolarify(z)
if z != nz:
return polygamma(n, nz)
if n == -1:
return loggamma(z)
else:
if z.is_Number:
if z is S.NaN:
return S.NaN
elif z is S.Infinity:
if n.is_Number:
if n is S.Zero:
return S.Infinity
else:
return S.Zero
elif z.is_Integer:
if z.is_nonpositive:
return S.ComplexInfinity
else:
if n is S.Zero:
return -S.EulerGamma + harmonic(z - 1, 1)
elif n.is_odd:
return (-1)**(n + 1)*factorial(n)*zeta(n + 1, z)
if n == 0:
if z is S.NaN:
return S.NaN
elif z.is_Rational:
p, q = z.as_numer_denom()
# only expand for small denominators to avoid creating long expressions
if q <= 5:
return expand_func(polygamma(n, z, evaluate=False))
elif z in (S.Infinity, S.NegativeInfinity):
return S.Infinity
else:
t = z.extract_multiplicatively(S.ImaginaryUnit)
if t in (S.Infinity, S.NegativeInfinity):
return S.Infinity
async def collect(self, bot, event: Message, query: str):
match = COLLECT_QUERY1.match(query)
if match:
n = int(match.group("n"))
total = int(match.group("total")) if match.group("total") else n
else:
match = COLLECT_QUERY2.match(query)
if match:
n = int(match.group("n"))
total = int(match.group("total"))
else:
await bot.say(event.channel, "요청을 해석하는데에 실패했어요!")
return
if total < 2 or total > 512:
await bot.say(event.channel, "정상적인 전체 갯수를 입력해주세요! (2개 이상 512개 이하)")
return
if n < 1 or n > 512:
await bot.say(event.channel, "정상적인 수집 갯수를 입력해주세요! (1개 이상 512개 이하)")
return
if total < n:
await bot.say(event.channel, "원하는 갯수가 전체 갯수보다 많을 수 없어요!")
return
result = n * harmonic(n)
if total > n:
result /= n / total
text = "부분적으로"
else:
text = "모두"
await bot.say(
event.channel,
f"상품 1개 구입시 {total}종류의 특전 중 하나를 무작위로 100%"
f"확률로 준다고 가정할 때 {n}종류의 특전을 {text} 모으려면, 평균적으로"
f" {math.ceil(result)}(`{float(result):.2f}`)개의 상품을"
" 구입해야 수집에 성공할 수 있어요!",
)
def test_sympy__functions__combinatorial__numbers__harmonic():
from sympy.functions.combinatorial.numbers import harmonic
assert _test_args(harmonic(x, 2))
def _eval_rewrite_as_harmonic(self, z, **kwargs):
return -harmonic(z - 1, 2) + S.Pi**2 / 6
def _eval_rewrite_as_harmonic(self, z, **kwargs):
return harmonic(z - 1) - S.EulerGamma
def _eval_rewrite_as_harmonic(self, n, z):
if n.is_integer:
if n == S.Zero:
return harmonic(z - 1) - S.EulerGamma
else:
return S.NegativeOne ** (n + 1) * C.factorial(n) * (C.zeta(n + 1) - harmonic(z - 1, n + 1))
def test_sympy__functions__combinatorial__numbers__harmonic():
from sympy.functions.combinatorial.numbers import harmonic
assert _test_args(harmonic(x, 2))
def test_polygamma():
assert polygamma(n, nan) is nan
assert polygamma(0, oo) is oo
assert polygamma(0, -oo) is oo
assert polygamma(0, I*oo) is oo
assert polygamma(0, -I*oo) is oo
assert polygamma(1, oo) == 0
assert polygamma(5, oo) == 0
assert polygamma(0, -9) is zoo
assert polygamma(0, -9) is zoo
assert polygamma(0, -1) is zoo
assert polygamma(0, 0) is zoo
assert polygamma(0, 1) == -EulerGamma
assert polygamma(0, 7) == Rational(49, 20) - EulerGamma
assert polygamma(1, 1) == pi**2/6
assert polygamma(1, 2) == pi**2/6 - 1
assert polygamma(1, 3) == pi**2/6 - Rational(5, 4)
assert polygamma(3, 1) == pi**4 / 15
assert polygamma(3, 5) == 6*(Rational(-22369, 20736) + pi**4/90)
assert polygamma(5, 1) == 8 * pi**6 / 63
assert polygamma(1, S.Half) == pi**2 / 2
assert polygamma(2, S.Half) == -14*zeta(3)
assert polygamma(11, S.Half) == 176896*pi**12
def t(m, n):
x = S(m)/n
r = polygamma(0, x)
if r.has(polygamma):
return False
return abs(polygamma(0, x.n()).n() - r.n()).n() < 1e-10
assert t(1, 2)
assert t(3, 2)
assert t(-1, 2)
assert t(1, 4)
assert t(-3, 4)
assert t(1, 3)
assert t(4, 3)
assert t(3, 4)
assert t(2, 3)
assert t(123, 5)
assert polygamma(0, x).rewrite(zeta) == polygamma(0, x)
assert polygamma(1, x).rewrite(zeta) == zeta(2, x)
assert polygamma(2, x).rewrite(zeta) == -2*zeta(3, x)
assert polygamma(I, 2).rewrite(zeta) == polygamma(I, 2)
n1 = Symbol('n1')
n2 = Symbol('n2', real=True)
n3 = Symbol('n3', integer=True)
n4 = Symbol('n4', positive=True)
n5 = Symbol('n5', positive=True, integer=True)
assert polygamma(n1, x).rewrite(zeta) == polygamma(n1, x)
assert polygamma(n2, x).rewrite(zeta) == polygamma(n2, x)
assert polygamma(n3, x).rewrite(zeta) == polygamma(n3, x)
assert polygamma(n4, x).rewrite(zeta) == polygamma(n4, x)
assert polygamma(n5, x).rewrite(zeta) == (-1)**(n5 + 1) * factorial(n5) * zeta(n5 + 1, x)
assert polygamma(3, 7*x).diff(x) == 7*polygamma(4, 7*x)
assert polygamma(0, x).rewrite(harmonic) == harmonic(x - 1) - EulerGamma
assert polygamma(2, x).rewrite(harmonic) == 2*harmonic(x - 1, 3) - 2*zeta(3)
ni = Symbol("n", integer=True)
assert polygamma(ni, x).rewrite(harmonic) == (-1)**(ni + 1)*(-harmonic(x - 1, ni + 1)
+ zeta(ni + 1))*factorial(ni)
# Polygamma of non-negative integer order is unbranched:
k = Symbol('n', integer=True, nonnegative=True)
assert polygamma(k, exp_polar(2*I*pi)*x) == polygamma(k, x)
# but negative integers are branched!
k = Symbol('n', integer=True)
assert polygamma(k, exp_polar(2*I*pi)*x).args == (k, exp_polar(2*I*pi)*x)
# Polygamma of order -1 is loggamma:
assert polygamma(-1, x) == loggamma(x)
# But smaller orders are iterated integrals and don't have a special name
assert polygamma(-2, x).func is polygamma
# Test a bug
assert polygamma(0, -x).expand(func=True) == polygamma(0, -x)
assert polygamma(2, 2.5).is_positive == False
assert polygamma(2, -2.5).is_positive == False
assert polygamma(3, 2.5).is_positive == True
assert polygamma(3, -2.5).is_positive is True
assert polygamma(-2, -2.5).is_positive is None
assert polygamma(-3, -2.5).is_positive is None
assert polygamma(2, 2.5).is_negative == True
assert polygamma(3, 2.5).is_negative == False
assert polygamma(3, -2.5).is_negative == False
assert polygamma(2, -2.5).is_negative is True
assert polygamma(-2, -2.5).is_negative is None
assert polygamma(-3, -2.5).is_negative is None
assert polygamma(I, 2).is_positive is None
assert polygamma(I, 3).is_negative is None
# issue 17350
assert polygamma(pi, 3).evalf() == polygamma(pi, 3)
assert (I*polygamma(I, pi)).as_real_imag() == \
(-im(polygamma(I, pi)), re(polygamma(I, pi)))
assert (tanh(polygamma(I, 1))).rewrite(exp) == \
(exp(polygamma(I, 1)) - exp(-polygamma(I, 1)))/(exp(polygamma(I, 1)) + exp(-polygamma(I, 1)))
assert (I / polygamma(I, 4)).rewrite(exp) == \
I*sqrt(re(polygamma(I, 4))**2 + im(polygamma(I, 4))**2)\
/((re(polygamma(I, 4)) + I*im(polygamma(I, 4)))*Abs(polygamma(I, 4)))
assert unchanged(polygamma, 2.3, 1.0)
# issue 12569
assert unchanged(im, polygamma(0, I))
assert polygamma(Symbol('a', positive=True), Symbol('b', positive=True)).is_real is True
assert polygamma(0, I).is_real is None