def _eval_(self, z, **kwds):
"""
EXAMPLES::
sage: harmonic_number(0)
0
sage: harmonic_number(1)
1
sage: harmonic_number(20)
55835135/15519504
sage: harmonic_number(5/2)
-2*log(2) + 46/15
sage: harmonic_number(2*x)
harmonic_number(2*x)
"""
if z in ZZ:
if z == 0:
return Integer(0)
elif z == 1:
return Integer(1)
elif z > 1:
import sage.libs.flint.arith as flint_arith
return flint_arith.harmonic_number(z)
elif z in QQ:
from sage.functions.other import psi1
return psi1(z+1) - psi1(1)
def _eval_(self, z, **kwds):
"""
EXAMPLES::
sage: harmonic_number(0)
0
sage: harmonic_number(1)
1
sage: harmonic_number(20)
55835135/15519504
sage: harmonic_number(5/2)
-2*log(2) + 46/15
sage: harmonic_number(2*x)
harmonic_number(2*x)
"""
if z in ZZ:
if z == 0:
return Integer(0)
elif z == 1:
return Integer(1)
elif z > 1:
import sage.libs.flint.arith as flint_arith
return flint_arith.harmonic_number(z)
elif z in QQ:
from sage.functions.other import psi1
return psi1(z + 1) - psi1(1)
def dvalue(self):
r"""
Return a symbolic expression (or an exact value in case n=3, 4, 6)
for the transfinite diameter (or capacity) of ``self``.
EXAMPLES:
sage: HeckeTriangleGroup(3).dvalue()
1/1728
sage: HeckeTriangleGroup(4).dvalue()
1/256
sage: HeckeTriangleGroup(5).dvalue()
e^(2*euler_gamma - 4*pi/(sqrt(5) + 1) + psi(17/20) + psi(13/20))
sage: HeckeTriangleGroup(6).dvalue()
1/108
sage: HeckeTriangleGroup(10).dvalue()
e^(2*euler_gamma - pi*sec(1/10*pi) + psi(4/5) + psi(7/10))
sage: HeckeTriangleGroup(infinity).dvalue()
1/64
"""
n=self._n
if (n==3):
return ZZ(1)/ZZ(2**6*3**3)
elif (n==4):
return ZZ(1)/ZZ(2**8)
elif (n==6):
return ZZ(1)/ZZ(2**2*3**3)
elif (n==infinity):
return ZZ(1)/ZZ(2**6)
else:
return exp(-ZZ(2)*psi1(ZZ(1)) + psi1(ZZ(1)-self.alpha())+psi1(ZZ(1)-self.beta()) - pi*sec(pi/self._n))
def dvalue(self):
r"""
Return a symbolic expression (or an exact value in case n=3, 4, 6)
for the transfinite diameter (or capacity) of ``self``.
I.e. the first nontrivial Fourier coefficient of the Hauptmodul
for the Hecke triangle group in case it is normalized to ``J_inv(i)=1``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: HeckeTriangleGroup(3).dvalue()
1/1728
sage: HeckeTriangleGroup(4).dvalue()
1/256
sage: HeckeTriangleGroup(5).dvalue()
e^(2*euler_gamma - 4*pi/(sqrt(5) + 1) + psi(17/20) + psi(13/20))
sage: HeckeTriangleGroup(6).dvalue()
1/108
sage: HeckeTriangleGroup(10).dvalue()
e^(2*euler_gamma - 2*pi/sqrt(1/2*sqrt(5) + 5/2) + psi(4/5) + psi(7/10))
sage: HeckeTriangleGroup(infinity).dvalue()
1/64
"""
n = self._n
if (n==3):
return ZZ(1)/ZZ(2**6*3**3)
elif (n==4):
return ZZ(1)/ZZ(2**8)
elif (n==6):
return ZZ(1)/ZZ(2**2*3**3)
elif (n==infinity):
return ZZ(1)/ZZ(2**6)
else:
return exp(-ZZ(2)*psi1(ZZ(1)) + psi1(ZZ(1)-self.alpha())+psi1(ZZ(1)-self.beta()) - pi*sec(pi/self._n))
def dvalue(self):
r"""
Return a symbolic expression (or an exact value in case n=3, 4, 6)
for the transfinite diameter (or capacity) of ``self``.
I.e. the first nontrivial Fourier coefficient of the Hauptmodul
for the Hecke triangle group in case it is normalized to ``J_inv(i)=1``.
EXAMPLES:
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: HeckeTriangleGroup(3).dvalue()
1/1728
sage: HeckeTriangleGroup(4).dvalue()
1/256
sage: HeckeTriangleGroup(5).dvalue()
e^(2*euler_gamma - 4*pi/(sqrt(5) + 1) + psi(17/20) + psi(13/20))
sage: HeckeTriangleGroup(6).dvalue()
1/108
sage: HeckeTriangleGroup(10).dvalue()
e^(2*euler_gamma - pi*sec(1/10*pi) + psi(4/5) + psi(7/10))
sage: HeckeTriangleGroup(infinity).dvalue()
1/64
"""
n = self._n
if (n==3):
return ZZ(1)/ZZ(2**6*3**3)
elif (n==4):
return ZZ(1)/ZZ(2**8)
elif (n==6):
return ZZ(1)/ZZ(2**2*3**3)
elif (n==infinity):
return ZZ(1)/ZZ(2**6)
else:
return exp(-ZZ(2)*psi1(ZZ(1)) + psi1(ZZ(1)-self.alpha())+psi1(ZZ(1)-self.beta()) - pi*sec(pi/self._n))
def get_R(p, energy):
k = sqrt(energy)
# left = 0
# while left < p.n and energy < p.eigenvalues[left]:
# left += 1
# left = max(0, left - count / 2)
# right = min(p.n, left + count)
n0 = None
s = _sage_const_0p0
# print(energies)
for i, (es, ee) in enumerate(zip(p.eigenstates, p.eigenvalues)):
if (pi / _sage_const_2 + pi * i) / p.r_A > k:
n0 = i
if n0 is not None:
X = p.r_A * k / pi
rest = float(_sage_const_1 / pi**_sage_const_2 *
_sage_const_1 / X *
(psi1(n0 + _sage_const_0p5 + X) -
psi1(n0 + _sage_const_0p5 - X)))
if rest / (abs(s) + rest) < tolerance:
print("Stopping at {}, rest {}".format(n0, rest))
break
s += es(p.r_A)**_sage_const_2 / (ee - energy) / p.r_A
# print("Using eigenvalue: {}".format(ee))
# print("i = {}, R = {}".format(i, s))
return s
def compute_R_upper_bound(p, energy):
k = sqrt(energy)
s = _sage_const_0p0
rest0 = _sage_const_0p0
n0 = None
for i, (es, ee) in enumerate(zip(p.eigenstates, p.eigenvalues)):
val = float(es(p.r_A)**_sage_const_2 / (ee - energy) / p.r_A)
if (pi / _sage_const_2 + pi * i) / p.r_A < k:
s += val
else:
if n0 is None:
n0 = i
rest0 += val
X = p.r_A * k / pi
rest = float(
_sage_const_1 / pi**_sage_const_2 * _sage_const_1 / X *
(psi1(n0 + _sage_const_0p5 + X) - psi1(n0 + _sage_const_0p5 - X)))
print("partial sum = {}".format(s))
print("rest0 = {}".format(rest0))
print("rest = {}".format(rest))