MODULAR FORMS-page61

MODULAR FORMS-page61 - 57 After all of these...

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57 After all of these normalizations, the elliptic function ( z ) with respect to Λ τ is uniquely determined by the conditions (6.18) and (6.19). It is called called the Weierstrass function with respect to the lattice Λ τ . One can find explicitly the function ( z ) as follows. I claim that ( z ) = φ ( z ) := 1 z 2 + X λ Λ \{ 0 } ( 1 ( z - λ ) 2 - 1 λ 2 ) . (6.22) First of all the series (6.22) is absolutely convergent on any compact subset of C not containing 0. We shall skip the proof of this fact (see for example [Cartan] )[ ? ]. This implies that φ ( z ) is a meromorphic function with pole of order 2 at 0. Its derivative is a meromorphic function given by the series φ ( z ) 0 = - 2 X λ Λ 1 ( z - λ ) 3 . It is obviously periodic. This implies that φ ( z ) is periodic too. Since φ ( z ) is an even function, φ ( z ) 0 is odd. But then it must vanish at all λ 1 2 Λ. In fact
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