4 Theta Functions 521Since the zeros ofθ(v;τ)are the pointsv= 1 / 2 +τ/ 2 +mτ+n, the zeros ofθα,β(v)
are the points
v=(β+ 1 )/ 2 +(α+ 1 )τ/ 2 +mτ+n(m,n∈Z).The notation for theta functions is by no means standardized. Hermite’s notation
reflects the underlying symmetry, but for purposes of comparison we indicate its
connection with the more commonly used notation in Whittaker and Watson [29]:
θ 00 (v;τ)=θ 3 (πv,q), θ 01 (v;τ)=θ 4 (πv,q),
θ 10 (v;τ)=θ 2 (πv,q), θ 11 (v;τ)=iθ 1 (πv,q).It follows from the definitions thatθ 00 (v;τ),θ 01 (v;τ)andθ 10 (v;τ)are even func-
tions ofv, whereasθ 11 (v;τ)is an odd function ofv. Moreoverθ 00 (v;τ)andθ 01 (v;τ)
are periodic with period 1 inv,butθ 10 (v;τ)andθ 11 (v;τ)change sign whenvis
increased by 1.
All four theta functions satisfy the same partial differential equation asθ(v;τ).
From the product expansion ofθ(v;τ)we obtain the product expansions
θ 00 (v)=Q 0∏∞
n= 1( 1 +q^2 n−^1 e^2 πiv)( 1 +q^2 n−^1 e−^2 πiv),θ 01 (v)=Q 0∏∞
n= 1( 1 −q^2 n−^1 e^2 πiv)( 1 −q^2 n−^1 e−^2 πiv),θ 10 (v)= 2 Q 0 eπiτ/^4 cosπv∏∞
n= 1( 1 +q^2 ne^2 πiv)( 1 +q^2 ne−^2 πiv),θ 11 (v)= 2 iQ 0 eπiτ/^4 sinπv∏∞
n= 1( 1 −q^2 ne^2 πiv)( 1 −q^2 ne−^2 πiv),(35)
where q=eπiτand
Q 0 =
∏∞
n= 1( 1 −q^2 n).In particular,θ 00 ( 0 )=Q 0∏∞
n= 1( 1 +q^2 n−^1 )^2 ,θ 01 ( 0 )=Q 0∏∞
n= 1( 1 −q^2 n−^1 )^2 ,θ 10 ( 0 )= 2 q^1 /^4 Q 0∏∞
n= 1( 1 +q^2 n)^2.