The hypergeometric function $F(a, b ; c ; z)$ can be expressed in the form

$$F(a, b ; c ; z)=\frac{\Gamma(c)}{\Gamma(b) \Gamma(c-b)} \int_{0}^{1} t^{b-1}(1-t)^{c-b-1}(1-t z)^{-a} d t$$

for appropriate restrictions on $c, b, z$.

Express the following integral in terms of a combination of hypergeometric functions

$$I(u, A)=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{e^{i t(u+1)}}{e^{i t}+i A} d t, \quad|A|>1$$

[You may use without proof that $\Gamma(z+1)=z \Gamma(z) .$ ]