aa r X i v : . [ m a t h . C V ] N ov ON DOUBLY PERIODIC PHASES
MARIUS B. STEFAN
Abstract.
Every meromorphic function on C with doubly peri-odic phase is equal to an elliptic function multiplied by a mero-morphic function determined by the periods. The purpose of this note is to completely characterize the meromor-phic functions on C that have doubly periodic phases. In particular,our result shows that the characterization suggested by G. Semmlerand E. Wegert in [4] is too narrow. The phase of a function f ( z ) de-fined on an open set D ⊂ C with values in b C is the function f ( z ) | f ( z ) | defined on { z ∈ D : f ( z ) ∈ b C \ { , ∞}} with values in T .We recall the definition of the Weierstrass sigma-function ([6]). If p , p ∈ C are linearly independent over R , we denote by L the Z -module generated by p , p and we put ω = p p . Since the series X λ ∈ L \{ } | λ | is convergent, the Weierstrass product ([5]) z Y λ ∈ L \{ } e zλ + z λ (cid:16) − zλ (cid:17) defines an entire function σ ( z ) whose set of zeros is precisely L , andwhich is known as the Weierstrass sigma-function. The following trans-formation property is probably well-known but we include a proof, forcompleteness. Lemma.
For every ξ ∈ C and j ∈ { , } the Weierstrass sigma-function satisfies σ ( z ) σ ( − ξ + z ) · σ ( − ξ + z + p j ) σ ( z + p j ) = e v j , where v j = − ξ p j + ξ p j X λ ∈ L \{ , − p j } λ ( λ + p j ) . Mathematics Subject Classification.
Primary 30Dxx.
Proof.
For every ξ ∈ C and j ∈ { , } we have σ ( z ) σ ( − ξ + z ) = z − ξ + z Y λ ∈ L \{ } e ξ λ − ξ λ + ξ zλ λ − zλ + ξ − z = z ( z + p j ) e − ξ pj − ξ p j + ξ zp j ( − ξ + z )( − ξ + z + p j ) · Y λ ∈ L \{ , − p j } e ξ λ − ξ λ + ξ zλ λ − zλ + ξ − z and σ ( z + p j ) σ ( − ξ + z + p j ) = z + p j − ξ + z + p j Y λ ∈ L \{ } e ξ λ − ξ λ + ξ z + pj ) λ λ − p j − zλ − p j + ξ − z = z + p j − ξ + z + p j · Y λ ∈ L \{− p j } e ξ λ + pj − ξ λ + pj )2 + ξ z + pj )( λ + pj )2 λ − zλ + ξ − z = z ( z + p j ) e ξ pj − ξ p j + ξ zp j ( − ξ + z )( − ξ + z + p j ) · Y λ ∈ L \{ , − p j } e ξ λ + pj − ξ λ + pj )2 + ξ z + pj )( λ + pj )2 λ − zλ + ξ − z therefore σ ( z ) σ ( − ξ + z ) · σ ( − ξ + z + p j ) σ ( z + p j )= e − ξ pj + P λ ∈ L \{ , − pj } (cid:18) ξ λ − ξ λ + pj − ξ λ + ξ λ + pj )2 + ξ zλ − ξ z + pj )( λ + pj )2 (cid:19) = e u j z + v j − ξ u j , where u j = ξ X λ ∈ L \{ , − p j } (cid:18) λ − λ + p j ) (cid:19) . N DOUBLY PERIODIC PHASES 3
For ω ∈ C \ R , we define f ( ω ) by p u = p ξ X λ ∈ L \{ , − p } (cid:18) λ − λ + p ) (cid:19) = ξ X ( m,n ) ∈ Z \{ (0 , , ( − , } (cid:18) m + nω ) − m + 1 + nω ) (cid:19) =: f ( ω )and notice that p u = p ξ X λ ∈ L \{ , − p } (cid:18) λ − λ + p ) (cid:19) = ξ X ( m,n ) ∈ Z \{ (0 , , (0 , − } (cid:18) mω + n ) − mω + n + 1) (cid:19) = f (cid:18) ω (cid:19) . It now suffices to show that f ( ω ) = 0 for every ω ∈ C \ R : f ( ω ) = ξ X n ∈ Z \{ } X m ∈ Z (cid:18) m + nω ) − m + 1 + nω ) (cid:19) + ξ X m ∈ Z \{− , } (cid:18) m − m + 1) (cid:19) = 0 . (cid:3) Theorem.
Let f ( z ) be a nonconstant function meromorphic on C . Ifthe phase of f ( z ) is doubly periodic with primitive periods p , p and ω = p p ∈ C \ R , then there exists an elliptic function g ( z ) with periods p , p such that f ( z ) = e az g ( z ) σ ( z ) σ ( − ξ + z ) , where ξ ∈ C , and a ∈ C satisfies ℑ ( ap j ) = ℑ ( v j ) + 2 m j π , j ∈ { , } ,for some m , m ∈ Z .Proof. We shall use the observation ([4]) that if the phase of a mero-morphic function f ( z ) on C has a period p , then f ( z + p ) = e α f ( z )for some constant α ∈ R . Let α , α be two real numbers such that f ( z + p ) = e α f ( z ) and f ( z + p ) = e α f ( z ). Let F denote the par-allelogram with the two vectors p , p as adjacent sides. The function MARIUS B. STEFAN f ′ ( z ) f ( z ) is obviously elliptic with periods p , p and its integral over F istherefore 0. In particular, f ( z ) has the same number of zeros and polesinside F . Let Ξ be the multiset (we take into consideration the mul-tiplicities that occur) consisting of all zeros of f ( z ) inside F and Γ bethe multiset consisting of all poles of f ( z ) inside F . We then have X ξ ∈ Ξ ξ − X γ ∈ Γ γ = 12 πi Z F zf ′ ( z ) f ( z ) dz = 12 πi Z p ( z − ( z + p )) f ′ ( z ) f ( z ) dz − πi Z p ( z − ( z + p )) f ′ ( z ) f ( z ) dz = − p πi Z p d log f ( z ) + p πi Z p d log f ( z )= − p πi ( α + 2 n πi ) + p πi ( α + 2 n πi ) ≡ − ξ mod L for some n , n ∈ Z , where ξ is the unique number inside F that iscongruent to α p − α p πi modulo L . The congruence above is equivalentto X ξ ∈{ ξ }∪ Ξ ξ − X γ ∈{ }∪ Γ γ ≡ L hence, by Abel’s theorem ([1, 2, 3]), there exists an elliptic function g ( z ) with periods p , p such that its multisets of zeros and poles inside F are { ξ } ∪ Ξ and { } ∪ Γ, respectively. The sets of zeros and polesof the meromorphic function g ( z ) f ( z ) are then ξ + L and L , respectively.Since the meromorphic function σ ( − ξ + z ) σ ( z ) has exactly the same zerosand poles as g ( z ) f ( z ) , there exists an entire function h ( z ) such that f ( z ) = e h ( z ) g ( z ) σ ( z ) σ ( − ξ + z ) . The conditions f ( z + p j ) = e α j f ( z ), j ∈ { , } , now imply e h ( z + p j ) σ ( z + p j ) σ ( − ξ + z + p j ) = e α j + h ( z ) σ ( z ) σ ( − ξ + z ) N DOUBLY PERIODIC PHASES 5 hence, using the Lemma, e h ( z + p j ) − h ( z ) − α j = σ ( z ) σ ( − ξ + z ) · σ ( − ξ + z + p j ) σ ( z + p j )= e v j . The function h ′ ( z ) is therefore doubly periodic and has no poles, hence h ′ ( z ) is constant. So h ( z ) = az + b for some a, b ∈ C . Moreover, ap j − α j = v j + 2 m j πi for j ∈ { , } and some m , m ∈ Z . We musthave ap j − v j − m j πi = α j ∈ R , j ∈ { , } , that is, ℑ ( ap j ) = ℑ ( v j ) + 2 m j π, j ∈ { , } . (cid:3) References [1] N. H. Abel, M´emoire sur une propri´et´e g´en´erale d’une classe tr`es ´etendue defonctions transcendantes,
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