Ramanujan's Master Theorem

Suppose that in some Neighborhood of $x=0$,

$$F(x)=\sum_{k=0}^\infty {\phi(k)(-x)^k\over k!}.$$

Then

$$\int_0^\infty x^{n-1}F(x)\,dx=\Gamma(n)\phi(-n).$$

References

Berndt, B. C. Ramanujan's Notebooks: Part I. New York: Springer-Verlag, p. 298, 1985.