What are the conditions for Ramanujan's Master Theorem to hold?

Question

Ramanujan's Master Theorem states that if

$$f(x) = \sum_{k=0}^{\infty} \frac{\phi(k)}{k!}(-x)^k$$

then $$\int_{0}^{\infty}x^{s-1}f(x)\ dx = \Gamma(s)\phi(-s).$$

But there are obviously some conditions on $\phi$ as you could just define it to be $0$ for negative inputs or multiply it by $\cos(2\pi k)$ without changing $f$ but giving a different result. What are these conditions?

Answer 1

Here is a reference. I am frustrated that Wikipedia and Wolfram MathWorld both state the theorem poorly. Here is the precise statement:

Ramanujan’s Master Theorem: Let $\varphi(z)$ be an analytic (single-valued) function, defined on a half-plane $$H(\delta)=\left\{z\in\mathbb{C}:\Re(z)\geq -\delta\right\}$$ for some $0<\delta<1$. Suppose that, for some $A<\pi$, $\varphi$ satisfies the growth condition $$\left|\varphi\left(u+iv\right)\right|<Ce^{Pu+A|v|}$$ for all $z=u+iv\in H(\delta)$. Then, $$\int_{0}^{\infty}x^{s-1}f(x)\,\mathrm{d}x=\Gamma(s)\varphi(-s)$$ with $$f(x):=\sum_{n=0}^{\infty}\frac{\varphi(n)(-x)^n}{n!}$$ for all $0<\Re(s)<\delta$.