A259149 Decimal expansion of phi(exp(-2*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

9, 9, 8, 1, 2, 9, 0, 6, 9, 9, 2, 5, 9, 5, 8, 5, 1, 3, 2, 7, 9, 9, 6, 2, 3, 2, 2, 2, 4, 5, 2, 7, 3, 8, 7, 8, 1, 3, 0, 7, 3, 8, 4, 3, 5, 3, 6, 5, 8, 1, 6, 4, 6, 1, 7, 5, 4, 0, 7, 8, 1, 4, 0, 2, 8, 2, 9, 9, 8, 5, 8, 0, 4, 6, 6, 0, 1, 9, 2, 8, 0, 7, 3, 5, 7, 1, 8, 2, 4, 4, 7, 3, 8, 7, 7, 7, 3, 7, 9, 3, 7, 7, 1, 9

Offset: 0

Author

Jean-François Alcover, Jun 19 2015

Keywords

• nonn
• cons
• easy

Examples

			0.99812906992595851327996232224527387813073843536581646175407814028299858...
		

Links

• Istvan Mezo, Several special values of Jacobi theta functions arXiv:1106.2703 [math.CA], 2011-2013
• Eric Weisstein's MathWorld, Infinite Product
• Eric Weisstein's MathWorld, Jacobi Theta Functions
• Eric Weisstein's MathWorld, q-Pochhammer Symbol
• Wikipedia, Euler function

Crossrefs

Cf. A048651 phi(1/2), A100220 phi(1/3), A100221 phi(1/4), A100222 phi(1/5), A132034 phi(1/6), A132035 phi(1/7), A132036 phi(1/8), A132037 phi(1/9), A132038 phi(1/10), A368211 phi(exp(-Pi/16)), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), A259147 phi(exp(-Pi/2)), A259148 phi(exp(-Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A363117 phi(exp(-7*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

Cf. A292821, A292825, A292829.

Programs

Mathematica

phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-2Pi]], 10, 104] // First

Formula

phi(q) = QPochhammer(q,q) = (q;q)_infinity.

phi(q) also equals theta'(1, 0, sqrt(q))^(1/3)/(2^(1/3)*q^(1/24)), where theta' is the derivative of the elliptic theta function theta(a,u,q) w.r.t. u.

phi(exp(-2*Pi)) = exp(Pi/12)*Gamma(1/4)/(2*Pi^(3/4)).