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From: José Carlos Santos on 27 Jul 2010 05:16
On 27-07-2010 10:12, Jos� Carlos Santos wrote:

>> Not sure what you mean in 2) by "the integral converges uniformly on
>> each compact subset". I know two approaches to this: 1. Assume that
>> for each compact K in A, there exists a positive integrable g on (a,b)
>> such that |F(z,t)|<= g(t) on (a,b) for all z in K. Then the result
>> follows by Morera. 2. "Differentiate through the integral sign", which
>> would involve an estimate on dF/dz(z,t).
>
> What I mean is this: for each compact subset K of A and for each r > 0,
> there is some interval [c,d] contained in (a,b) such that, for each _z_
> in K,
>
> |int(f(t,z),a < t < b) - int(f(t,z),c <= t <= d)| < r.

No! It is a little more complex than that. It should be: for each
compact subset K of A and for each r > 0, there is some interval [c,d]
contained in (a,b) such that, for each sub-interval [c',d'] of [c,d]
and for each _z_ in K,

|int(f(t,z),a < t < b) - int(f(t,z),c' <= t <= d')| < r.

Best regards,

Jose Carlos Santos

From: W^3 on 27 Jul 2010 20:48
In article <8b7metFj47U1(a)mid.individual.net>,
Jos� Carlos Santos <jcsantos(a)fc.up.pt> wrote:

> On 27-07-2010 10:12, Jos� Carlos Santos wrote:
>
> >> Not sure what you mean in 2) by "the integral converges uniformly on
> >> each compact subset". I know two approaches to this: 1. Assume that
> >> for each compact K in A, there exists a positive integrable g on (a,b)
> >> such that |F(z,t)|<= g(t) on (a,b) for all z in K. Then the result
> >> follows by Morera. 2. "Differentiate through the integral sign", which
> >> would involve an estimate on dF/dz(z,t).
> >
> > What I mean is this: for each compact subset K of A and for each r > 0,
> > there is some interval [c,d] contained in (a,b) such that, for each _z_
> > in K,
> >
> > |int(f(t,z),a < t < b) - int(f(t,z),c <= t <= d)| < r.
>
> No! It is a little more complex than that. It should be: for each
> compact subset K of A and for each r > 0, there is some interval [c,d]
> contained in (a,b) such that, for each sub-interval [c',d'] of [c,d]
> and for each _z_ in K,
>
> |int(f(t,z),a < t < b) - int(f(t,z),c' <= t <= d')| < r.
>
> Best regards,
>
> Jose Carlos Santos

That makes little sense; take c' = d' and then you're saying
|int(f(t,z),a < t < b)| < r?

Your previous statment seems fine - with the assumptions you've made,
g(z) = int(f(t,z),c <= t <= d) is holomorphic on A. (Proof: An easy
argument (easy because [c, d] is nice and compact) shows g is
continuous in A. Now apply Morera by reversing the order of
integration.) Thus the function in question can be uniformly
approximated on compact sets by holomorphic functions; it is therefore
holomorphic.

Still, I find this approach stilted. Differentiation through the
integral sign seems more natural to me, and it works fine here.
From: José Carlos Santos on 28 Jul 2010 11:43
On 28-07-2010 1:48, W^3 wrote:

>>>> Not sure what you mean in 2) by "the integral converges uniformly on
>>>> each compact subset". I know two approaches to this: 1. Assume that
>>>> for each compact K in A, there exists a positive integrable g on (a,b)
>>>> such that |F(z,t)|<= g(t) on (a,b) for all z in K. Then the result
>>>> follows by Morera. 2. "Differentiate through the integral sign", which
>>>> would involve an estimate on dF/dz(z,t).
>>>
>>> What I mean is this: for each compact subset K of A and for each r> 0,
>>> there is some interval [c,d] contained in (a,b) such that, for each _z_
>>> in K,
>>>
>>> |int(f(t,z),a< t< b) - int(f(t,z),c<= t<= d)|< r.
>>
>> No! It is a little more complex than that. It should be: for each
>> compact subset K of A and for each r> 0, there is some interval [c,d]
>> contained in (a,b) such that, for each sub-interval [c',d'] of [c,d]
>> and for each _z_ in K,
>>
>> |int(f(t,z),a< t< b) - int(f(t,z),c'<= t<= d')|< r.
>
> That makes little sense; take c' = d' and then you're saying
> |int(f(t,z),a< t< b)|< r?

You are right. I reversed the inclusion. Instead of "sub-interval
[c',d'] of [c,d]" I should have written "interval [c',d'] contained in
(a,b) which contains [c,d]".

An equivalent statement is: when ([c_n,d_n] )_n is an increasing
sequence of sub-intervals of (a,b) whose union is (a,b), then the
sequence of functions z|-> int(f(t,z),c_n <= t <= d_n) converges
uniformly to z|-> int(f(t,z),a < t < b).

> Your previous statment seems fine - with the assumptions you've made,
> g(z) = int(f(t,z),c<= t<= d) is holomorphic on A. (Proof: An easy
> argument (easy because [c, d] is nice and compact) shows g is
> continuous in A. Now apply Morera by reversing the order of
> integration.) Thus the function in question can be uniformly
> approximated on compact sets by holomorphic functions; it is therefore
> holomorphic.
>
> Still, I find this approach stilted. Differentiation through the
> integral sign seems more natural to me, and it works fine here.

I'll have to think about it.

Best regards,

Jose Carlos Santos
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