In this definition, bn ∈ C and N ∈ Z+. Note that this definition limits how many terms with negative exponents we can have, which is exactly the case with the J-function. 4.4 Fourier Expansion of J(τ) We will now follow the proof presented inModular Functions and Dirichlet Series in Number Theory [1] to prove that J(τ) has a Fourier expansion. Note that a Fourier expansion is similar to a Taylor series, except that it is written in terms of sines and cosines. We can use Euler’s formula to represent this in terms of exponentials, since eix = cosx+isinx. Theorem: If τ ∈H, J(τ) can be represented by an absolutely convergent Fourier series J(τ) = ∞ X n=−∞ a(n)e2πinτ. Proof: Let x=e2πiτ. Then Hmaps into the punctured unit disk D={x | 0 <|x| <1}. Figure 1.5 from [1] Eachτ maps onto a unique point xinD, but xis the image of infinitely many points inH. If τ and τ0 both map onto x, this means that e2πiτ =e2πiτ0. This means that τ and τ0 differ by an integer, since by Euler’s formula we have cos(2πτ) +isin(2πτ) = cos(2πτ0) +isin(2πτ0), which is true if τ and τ0 differ by an integer as cos(x) and sin(x) both have a period of 2π. For x∈D, let f(x) =J(τ), where τ is any point that maps to x(this function is well defined since J is periodic with period 1). We see that f is analytic in Dbecause f0(x) = d dx J(τ) = d dτ J(τ) dτ dx = (J0(τ))/ dx dτ = J0(τ) 2πie2πiτ , and J(τ) is analytic. Therefore, f0(x) exists at every x. Hence, f has a Laurent expansion about 0, f(x) = ∞ X n=−∞ a(n)xn, Channels • 2022 • Volume 6 • Number 2 Page 43
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