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## Appendix: Complete Elliptic integrals and the Nome q expansion 

In this appendix we present the definition and some properties of elliptic integrals which are needed for the derivation of our result. The complete elliptic integrals of the first ( K ) and the second ( E ) kind are defined as 

For our purpose it is useful to note that 

<img src='content_image/354592.jpg'>

<img src='content_image/354591.jpg'>

In order to study the elliptic integrals near the logarithmic singularity, it is convenient to use the q-series, defined as 

Inverting, one obtains 

<img src='content_image/354593.jpg'>

<img src='content_image/354588.jpg'>

Now that we have m ( q ) and q ( m ) as given above, we have the following alternative expan- sions. 

And more importantly, 

<img src='content_image/354589.jpg'>

<img src='content_image/354590.jpg'>

