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MODULAR FORMS-page44 - f τ = e-2 πi 48 τ 1 2 τ f 1...

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40 LECTURE 4. THETA CONSTANTS 4.5 Prove the Jacobi triple product identity : Y n =1 (1 - q n )(1 + q n - 1 2 t )(1 + q n - 1 2 t - 1 ) = X r Z q r 2 2 t n . 4.6 Prove a doubling identity for theta constants: ϑ 0 1 2 (2 τ ) 2 = ϑ 00 ( τ ) ϑ 0 1 2 ( τ ) . (see other doubling identities in Exercise 10.10). 4.7 Prove the following formulas expressing the Weber functions in terms of the η - function:
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Unformatted text preview: f ( τ ) = e-2 πi/ 48 η ( τ + 1 2 ) η ( τ ) , f 1 ( τ ) = η ( τ 2 ) η ( τ ) , f 2 ( τ ) = √ 2 η (2 τ ) η ( τ ) . 4.8 Prove the following identities connecting the Weber functions: f ( τ ) f 1 ( τ ) f 2 ( τ ) = f 1 (2 τ ) f 2 ( τ ) = √ 2 ....
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