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EllipticTheta

EllipticTheta[a, u, q]
gives the theta function theta_a(u,q) (a=1,…,4).
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • theta_1(u,q)=2q^(1/4)∑_(n=0)^(∞)(-1)^nq^(n(n+1))sin((2 n+1)u).
  • theta_2(u,q)=2q^(1/4)∑_(n=0)^(∞)q^(n(n+1))cos((2 n+1)u).
  • theta_3(u,q)=1+2∑_(n=1)^(∞)q^(n^2)cos(2n u).
  • theta_4(u,q)=1+2∑_(n=1)^(∞)(-1)^nq^(n^2)cos(2n u).
  • The theta_i are defined only inside the unit q disk; the disk forms a natural boundary of analyticity.
  • Inside the unit q disk, theta_1 and theta_2 have branch cuts from 0 to -1.
  • For certain special arguments, EllipticTheta automatically evaluates to exact values.
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