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 - 4*pi/sqrt(2*sqrt(5) + 10) + 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``.
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))