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))