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Given the j-function $j(\tau)$ and elliptic lambda function $\lambda(\tau)$. Define, $$g = -1+2\frac{\,_2F_1\big(\tfrac{1}{2},\tfrac{1}{2},1,\,\lambda(2\tau)\big) ... 1answer 80 views ### The natural boundary of the modular \lambda function is the real axis I want to solve the following problem from Ahlfors' text: Show that the function \lambda(\tau) introduced in Chap. 7, Sec. 3.4, has the real axis as a natural boundary. Here \lambda(\tau) is ... 1answer 124 views ### Where does Klein's j-invariant take the values 0 and 1, and with what multiplicities? I tried to solve the following problem from Ahlfors' text, please verify my solution: Where does the function$$J(\tau)=\frac{4}{27} \frac{(1-\lambda+\lambda^2)^3}{\lambda^2(1-\lambda)^2} $$... 1answer 64 views ### Why is the modular \lambda function a quotient of two meromorphic functions in the U.H.P.? In Ahlfors' complex analysis text, page 278 it says: It is quite clear from (9) that \lambda(\tau) is the quotient of two analytic functions in the upper half plane \text{Im} \tau > 0. ... 1answer 165 views ### How to prove that \frac{\eta^{14}(q^4)}{\eta^{4}(q^8)}=4\eta^4(q^2)\eta^2(q^4)\eta^4(q^8)+\eta^4(q)\eta^2(q^2)\eta^4(q^4)? How can we prove that$$\frac{\eta^{14}(q^4)}{\eta^{4}(q^8)}=4\eta^4(q^2)\eta^2(q^4)\eta^4(q^8)+\eta^4(q)\eta^2(q^2)\eta^4(q^4) \ ?$$Here, \eta(q) is the Dedekind Eta Function, which is defined by ... 4answers 2k views ### Proof of \frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24} I would like to prove that \displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}. I found a solution by myself 10 hours after I posted it, here it is: ... 0answers 41 views ### References for the conformal equivalence of the space of complex 1-tori and C? What are some good references with proofs of the conformal equivalence of the space of complex tori and \mathbb{C}? So far I only have the book by Jones and Singerman. 1answer 121 views ### Direct proof of the non-zeroness of an Eisenstein series Question: Can you show directly from its formula that G_4(i)\neq0? Recall that the holomorphic Eisenstein series of weight 2k is defined by:$$G_{2k}(\tau)= \sum_{(m,n)\in\mathbb{Z}^2\setminus ...
I'm studying modular forms and my professor started the course talking about elliptic functions. These particular functions form a field (once that the lattice $\Lambda$ is fixed) called ...