Holomorphic function defined as an integral.

I do not fully understand the following statement:

"Let f(x) decay faster than any polynomial as x -> infinity, then int_0^infinity f(x) x^{s-1}dx defines a holomorphic function of s when Re(s) > 1.

This is because int_0^1 x^{s-1}ds converges and f(x) has a rapid decay."

When, in general, the function defined by an integral is holomorphic?

Re: Holomorphic function defined as an integral.

Quote:

Originally Posted by

**Laurent** Do you know the theorem about continuity of functions defined by an integral? There is a very similar result for holomorphic functions:

Suppose $\displaystyle (x,z)\mapsto f(x,z)$ is

- holomorphic with respect to the $\displaystyle z$ variable and measurable with respect to the $\displaystyle x$ variable

- such that there is a function $\displaystyle \phi(x)$ with $\displaystyle |f(x,z)|\leq \phi(x)$ for any $\displaystyle x,z$ and $\displaystyle \int \phi(x) dx<\infty$. (this is a "domination" of $\displaystyle f$)

Then the function $\displaystyle F:z\mapsto \int f(x,z) dx$ is holomorphic, and $\displaystyle F'(z)=\int \partial_z f(x,z) dx$.

(I didn't specify the intervals or domains, this is whatever you want)

This theorem is quite powerful. It shows for instance that the "integral function" $\displaystyle F$ is indefinitely differentiable, using only a domination of the function (and not of its derivatives).

In your situation, you'll probably have to apply this theorem to compact subsets of the domain (otherwise you won't be able to get the domination).

Thanks Laurent. Could you please provide a reference to a general version of this result? It seems one could use this result to show that solutions $\displaystyle u(x,t)$ of initial-boundary problems for the heat equation $\displaystyle u_{xx} = 2 u_{t}$ are (real) analytic in $\displaystyle x$ even if boundary data is discontinuous. I have only seen such results with smooth boundary data. What am I missing?

Many thanks.

Re: Holomorphic function defined as an integral.

Quote:

Originally Posted by

**Laurent** Do you know the theorem about continuity of functions defined by an integral? There is a very similar result for holomorphic functions:

Suppose $\displaystyle (x,z)\mapsto f(x,z)$ is

- holomorphic with respect to the $\displaystyle z$ variable and measurable with respect to the $\displaystyle x$ variable

- such that there is a function $\displaystyle \phi(x)$ with $\displaystyle |f(x,z)|\leq \phi(x)$ for any $\displaystyle x,z$ and $\displaystyle \int \phi(x) dx<\infty$. (this is a "domination" of $\displaystyle f$)

Then the function $\displaystyle F:z\mapsto \int f(x,z) dx$ is holomorphic, and $\displaystyle F'(z)=\int \partial_z f(x,z) dx$.

(I didn't specify the intervals or domains, this is whatever you want)

This theorem is quite powerful. It shows for instance that the "integral function" $\displaystyle F$ is indefinitely differentiable, using only a domination of the function (and not of its derivatives).

In your situation, you'll probably have to apply this theorem to compact subsets of the domain (otherwise you won't be able to get the domination).

Also, is there a name for results of this general type?