538 XII Elliptic Functions
It may be shown that a functionf, which is meromorphic in the whole complex plane,
has an algebraic addition theorem if and only if it is either a rational function or,
when the independent variable is scaled by a constant factor, a rational function of
S(u,λ)and its derivativeS′(u,λ)for someλ∈C. This result (in different notation)
is due to Weierstrass and is proved in Akhiezer [3], for example. A generalization of
Weierstrass’ theorem, due to Myrberg, is proved in Belavin and Drinfeld [6].
The term ‘elliptic function’ is often used to denote any function which is meromor-
phic in the whole complex plane and has two periods whose ratio is not real. It may
be shown that, if the independent variable is scaled by a constant factor, an elliptic
function in this general sense is a rational function ofS(u,λ)andS′(u,λ)for some
λ= 0 ,1.
The functions f(v)which are holomorphic in the whole complex planeCand
satisfy the functional equations
f(v+ 1 )=f(v), f(v+τ)=e−nπi(^2 v+τ)f(v),wheren∈Nandτ∈H,formann-dimensional complex vector space. It was shown
by Hermite (1862) that this may be used to derive many relations between theta func-
tions, such as Proposition 6.
Proposition 11 can be extended to give transformation formulas for the Jacobian
functions when the parameterτis multiplied by any positive integern. See, for exam-
ple, Tannery and Molk [27], vol. II.
The modular function was used by Picard (1879) to prove that a function f(z),
which is holomorphic for allz∈Cand not a constant, assumes every complex value
except perhaps one. The exponential function expz, which does not assume the value
0, illustrates that an exceptional value may exist. A careful proof of Picard’s theorem
is given in Ahlfors [2]. (There are also proofs which do not use the modular function.)
It was already observed by Lagrange (1813) that there is a correspondence between
addition formulas for elliptic functions and the formulas of spherical trigonometry.
This correspondence has been most intensively investigated by Study [26].
There is ann-dimensional generalization of theta functions, which has a useful ap-
plication to the lattices studied in Chapter VIII. The theta function of anintegral lattice
ΛinRnis defined by
θΛ(τ)=∑
u∈Λq(u,u)= 1 +∑
m≥ 1Nmqm,whereq=eπiτandNmis the number of vectors inΛwith square-normm.Ifn= 1
andΛ=Z,then
θZ(τ)= 1 + 2 q+ 2 q^4 + 2 q^9 +···=θ( 0 ;τ).It is easily seen thatθΛ(τ)is a holomorphic function ofτin the half-planeIτ>0.
It follows from Poisson’s summation formula that the theta function of thedual lattice
Λ∗is given by
θΛ∗(τ)=d(Λ)(i/τ)n/^2 θΛ(− 1 /τ) forIτ> 0.