Recently I am considering a geometric question, which is reduced to the following problem. Given $ L<0$ , let $ a\in [L/2,0]$ and $ b=L-a$ . For any $ c>0$ , let $ p,1-p$ solve $ $ x^2-x+c^2=0,$ $ and $ q,1-q$ solve $ $ x^2-x-c^2=0.$ $ Using Gauss hypergeometric functions, we define \begin{align*} f_a(c)&=(a^2-a)\left(\frac{F(1+p,2-p;2;a)}{F(p,1-p;1;a)}+\frac{F(1+q,2-q;2;a)}{F(q,1-q;1;a)}\right)\Read more