本文整理汇总了Python中sympy.harmonic函数的典型用法代码示例。如果您正苦于以下问题：Python harmonic函数的具体用法？Python harmonic怎么用？Python harmonic使用的例子？那么恭喜您, 这里精选的函数代码示例或许可以为您提供帮助。
在下文中一共展示了harmonic函数的15个代码示例，这些例子默认根据受欢迎程度排序。您可以为喜欢或者感觉有用的代码点赞，您的评价将有助于系统推荐出更棒的Python代码示例。
示例1: test_harmonic_rewrite_sum_fail
def test_harmonic_rewrite_sum_fail():
n = Symbol("n")
m = Symbol("m")
_k = Dummy("k")
assert harmonic(n).rewrite(Sum) == Sum(1/_k, (_k, 1, n))
assert harmonic(n, m).rewrite(Sum) == Sum(_k**(-m), (_k, 1, n))示例2: test_harmonic_rewrite_sum
def test_harmonic_rewrite_sum():
n = Symbol("n")
m = Symbol("m")
_k = Dummy("k")
assert replace_dummy(harmonic(n).rewrite(Sum), _k) == Sum(1/_k, (_k, 1, n))
assert replace_dummy(harmonic(n, m).rewrite(Sum), _k) == Sum(_k**(-m), (_k, 1, n))示例3: test_harmonic
def test_harmonic():
assert harmonic(1, 1) == 1
assert harmonic(2, 1) == Rational(3, 2)
assert harmonic(3, 1) == Rational(11, 6)
assert harmonic(4, 1) == Rational(25, 12)
assert harmonic(3, 1) == harmonic(3)
assert harmonic(3, 5) == 1 + Rational(1, 2**5) + Rational(1, 3**5)
assert harmonic(10, 0) == 10
assert harmonic(oo, 1) == zoo
assert harmonic(oo, 2) == (pi**2)/6示例4: test_limit_seq_fail
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示例5: test_limit_seq
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(1) / 3
raises(ValueError, lambda: limit_seq(e * m))示例6: test_difference_delta__Sum
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)示例7: test_difference_delta__Sum
def test_difference_delta__Sum():
e = Sum(1/k, (k, 1, n))
assert dd(e, n) == 1/(n + 1)
assert dd(e, n, 5) == Sum(1/k, (k, n + 1, n + 5))
e = Sum(1/k, (k, 1, 3*n))
assert dd(e, n) == Sum(1/k, (k, 3*n + 1, 3*n + 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)示例8: test_harmonic_sums
def test_harmonic_sums():
assert summation(1/k, (k, 0, n)) == Sum(1/k, (k, 0, n))
assert summation(1/k, (k, 1, n)) == harmonic(n)
assert summation(n/k, (k, 1, n)) == n*harmonic(n)
assert summation(1/k, (k, 5, n)) == harmonic(n) - harmonic(4)示例9: test_polygamma
def test_polygamma():
from sympy import I
assert polygamma(n, nan) == nan
assert polygamma(0, oo) == oo
assert polygamma(0, -oo) == oo
assert polygamma(0, I*oo) == oo
assert polygamma(0, -I*oo) == oo
assert polygamma(1, oo) == 0
assert polygamma(5, oo) == 0
assert polygamma(0, -9) == zoo
assert polygamma(0, -9) == zoo
assert polygamma(0, -1) == zoo
assert polygamma(0, 0) == 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
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 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(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:
from sympy import exp_polar
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)示例10: test_harmonic
def test_harmonic():
n = Symbol("n")
assert harmonic(n, 0) == n
assert harmonic(n, 1) == harmonic(n)
assert harmonic(0, 1) == 0
assert harmonic(1, 1) == 1
assert harmonic(2, 1) == Rational(3, 2)
assert harmonic(3, 1) == Rational(11, 6)
assert harmonic(4, 1) == Rational(25, 12)
assert harmonic(0, 2) == 0
assert harmonic(1, 2) == 1
assert harmonic(2, 2) == Rational(5, 4)
assert harmonic(3, 2) == Rational(49, 36)
assert harmonic(4, 2) == Rational(205, 144)
assert harmonic(0, 3) == 0
assert harmonic(1, 3) == 1
assert harmonic(2, 3) == Rational(9, 8)
assert harmonic(3, 3) == Rational(251, 216)
assert harmonic(4, 3) == Rational(2035, 1728)
assert harmonic(oo, -1) == S.NaN
assert harmonic(oo, 0) == oo
assert harmonic(oo, S.Half) == oo
assert harmonic(oo, 1) == oo
assert harmonic(oo, 2) == (pi**2)/6
assert harmonic(oo, 3) == zeta(3)示例11: test_harmonic_limit_fail
def test_harmonic_limit_fail():
n = Symbol("n")
m = Symbol("m")
# For m > 1:
assert limit(harmonic(n, m), n, oo) == zeta(m)示例12: test_harmonic_rewrite_polygamma
def test_harmonic_rewrite_polygamma():
n = Symbol("n")
m = Symbol("m")
assert harmonic(n).rewrite(digamma) == polygamma(0, n + 1) + EulerGamma
assert harmonic(n).rewrite(trigamma) == polygamma(0, n + 1) + EulerGamma
assert harmonic(n).rewrite(polygamma) == polygamma(0, n + 1) + EulerGamma
assert harmonic(n,3).rewrite(polygamma) == polygamma(2, n + 1)/2 - polygamma(2, 1)/2
assert harmonic(n,m).rewrite(polygamma) == (-1)**m*(polygamma(m - 1, 1) - polygamma(m - 1, n + 1))/factorial(m - 1)
assert expand_func(harmonic(n+4)) == harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1)
assert expand_func(harmonic(n-4)) == harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n
assert harmonic(n, m).rewrite("tractable") == harmonic(n, m).rewrite(polygamma)示例13: test_harmonic_rational
def test_harmonic_rational():
ne = S(6)
no = S(5)
pe = S(8)
po = S(9)
qe = S(10)
qo = S(13)
Heee = harmonic(ne + pe/qe)
Aeee = (-log(10) + 2*(-1/S(4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + 5/S(8)))
+ 2*(-sqrt(5)/4 - 1/S(4))*log(sqrt(sqrt(5)/8 + 5/S(8)))
+ pi*(1/S(4) + sqrt(5)/4)/(2*sqrt(-sqrt(5)/8 + 5/S(8)))
+ 13944145/S(4720968))
Heeo = harmonic(ne + pe/qo)
Aeeo = (-log(26) + 2*log(sin(3*pi/13))*cos(4*pi/13) + 2*log(sin(2*pi/13))*cos(32*pi/13)
+ 2*log(sin(5*pi/13))*cos(80*pi/13) - 2*log(sin(6*pi/13))*cos(5*pi/13)
- 2*log(sin(4*pi/13))*cos(pi/13) + pi*cot(5*pi/13)/2 - 2*log(sin(pi/13))*cos(3*pi/13)
+ 2422020029/S(702257080))
Heoe = harmonic(ne + po/qe)
Aeoe = (-log(20) + 2*(1/S(4) + sqrt(5)/4)*log(-1/S(4) + sqrt(5)/4)
+ 2*(-1/S(4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + 5/S(8)))
+ 2*(-sqrt(5)/4 - 1/S(4))*log(sqrt(sqrt(5)/8 + 5/S(8)))
+ 2*(-sqrt(5)/4 + 1/S(4))*log(1/S(4) + sqrt(5)/4)
+ 11818877030/S(4286604231) - pi*sqrt(sqrt(5)/8 + 5/S(8))/(-sqrt(5)/2 + 1/S(2)) )
Heoo = harmonic(ne + po/qo)
Aeoo = (-log(26) + 2*log(sin(3*pi/13))*cos(54*pi/13) + 2*log(sin(4*pi/13))*cos(6*pi/13)
+ 2*log(sin(6*pi/13))*cos(108*pi/13) - 2*log(sin(5*pi/13))*cos(pi/13)
- 2*log(sin(pi/13))*cos(5*pi/13) + pi*cot(4*pi/13)/2
- 2*log(sin(2*pi/13))*cos(3*pi/13) + 11669332571/S(3628714320))
Hoee = harmonic(no + pe/qe)
Aoee = (-log(10) + 2*(-1/S(4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + 5/S(8)))
+ 2*(-sqrt(5)/4 - 1/S(4))*log(sqrt(sqrt(5)/8 + 5/S(8)))
+ pi*(1/S(4) + sqrt(5)/4)/(2*sqrt(-sqrt(5)/8 + 5/S(8)))
+ 779405/S(277704))
Hoeo = harmonic(no + pe/qo)
Aoeo = (-log(26) + 2*log(sin(3*pi/13))*cos(4*pi/13) + 2*log(sin(2*pi/13))*cos(32*pi/13)
+ 2*log(sin(5*pi/13))*cos(80*pi/13) - 2*log(sin(6*pi/13))*cos(5*pi/13)
- 2*log(sin(4*pi/13))*cos(pi/13) + pi*cot(5*pi/13)/2
- 2*log(sin(pi/13))*cos(3*pi/13) + 53857323/S(16331560))
Hooe = harmonic(no + po/qe)
Aooe = (-log(20) + 2*(1/S(4) + sqrt(5)/4)*log(-1/S(4) + sqrt(5)/4)
+ 2*(-1/S(4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + 5/S(8)))
+ 2*(-sqrt(5)/4 - 1/S(4))*log(sqrt(sqrt(5)/8 + 5/S(8)))
+ 2*(-sqrt(5)/4 + 1/S(4))*log(1/S(4) + sqrt(5)/4)
+ 486853480/S(186374097) - pi*sqrt(sqrt(5)/8 + 5/S(8))/(2*(-sqrt(5)/4 + 1/S(4))))
Hooo = harmonic(no + po/qo)
Aooo = (-log(26) + 2*log(sin(3*pi/13))*cos(54*pi/13) + 2*log(sin(4*pi/13))*cos(6*pi/13)
+ 2*log(sin(6*pi/13))*cos(108*pi/13) - 2*log(sin(5*pi/13))*cos(pi/13)
- 2*log(sin(pi/13))*cos(5*pi/13) + pi*cot(4*pi/13)/2
- 2*log(sin(2*pi/13))*cos(3*pi/13) + 383693479/S(125128080))
H = [Heee, Heeo, Heoe, Heoo, Hoee, Hoeo, Hooe, Hooo]
A = [Aeee, Aeeo, Aeoe, Aeoo, Aoee, Aoeo, Aooe, Aooo]
for h, a in zip(H, A):
e = expand_func(h).doit()
assert cancel(e/a) == 1
assert h.n() == a.n()示例14: test_harmonic_rewrite_sum_fail
def test_harmonic_rewrite_sum_fail():
n = Symbol("n")
m = Symbol("m")
assert harmonic(n).rewrite(digamma) == polygamma(0, n + 1) + EulerGamma
assert harmonic(n).rewrite(trigamma) == polygamma(0, n + 1) + EulerGamma
assert harmonic(n).rewrite(polygamma) == polygamma(0, n + 1) + EulerGamma
assert harmonic(n,3).rewrite(polygamma) == polygamma(2, n + 1)/2 - polygamma(2, 1)/2
assert harmonic(n,m).rewrite(polygamma) == (-1)**m*(polygamma(m - 1, 1) - polygamma(m - 1, n + 1))/factorial(m - 1)
assert expand_func(harmonic(n+4)) == harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1)
assert expand_func(harmonic(n-4)) == harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n
assert harmonic(n, m).rewrite("tractable") == harmonic(n, m).rewrite(polygamma)
_k = Dummy("k")
assert harmonic(n).rewrite(Sum) == Sum(1/_k, (_k, 1, n))
assert harmonic(n, m).rewrite(Sum) == Sum(_k**(-m), (_k, 1, n))示例15: test_harmonic_evalf
def test_harmonic_evalf():
assert str(harmonic(1.5).evalf(n=10)) == '1.280372306'
assert str(harmonic(1.5, 2).evalf(n=10)) == '1.154576311' # issue 7443