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From: Joubert on 26 Jul 2010 08:57
If anyone can tell me a source where I could find a proof of this or
tell me how it's proved I'd be grateful.
From: José Carlos Santos on 26 Jul 2010 09:29
On 26-07-2010 13:57, Joubert wrote:

> If anyone can tell me a source where I could find a proof of this or
> tell me how it's proved I'd be grateful.

Suppose that you prove that Gamma is holomorphic on the set of those
complex numbers whose real part is greater than 0. It will follow from
that and from the fact that Gamma(z + 1) = Gamma(z).z that it is also
holomorphic on the set of those complex numbers whose real part is
greater than -1 (except for 0, of course). If you keep applying this
argument, you will prove that it is holomorphic.

Now, on those complex numbers whose real part is greater than 0,
Gamma(z) is the integral of t^{z - 1}.e^{-t}, with _t_ going from 0 to
+oo. A function defined by such an expression is holomorphic. More
precisely, if A is an open subset of C, (a,b) is an open interval of R
(where _a_ can be -oo and _b_ can be +oo) and F is a continuous
function from (a,b) x A into C such that

1) for each _t_ in (a,b), z |-> F(t,z) is holomorphic;

2) for each _z_ in A, t |-> F(t,z) is integrable and the integral
converges uniformly on each compact subset of C,

then the function from A into C defined by z |-> int(F(t,z),a < t < b)
is holomorphic.

Best regards,

Jose Carlos Santos
From: Joubert on 26 Jul 2010 09:53
On 07/26/2010 03:29 PM, Jos� Carlos Santos wrote:

> Suppose that you prove that Gamma is holomorphic on the set of those
> complex numbers whose real part is greater than 0. It will follow from
> that and from the fact that Gamma(z + 1) = Gamma(z).z that it is also
> holomorphic on the set of those complex numbers whose real part is
> greater than -1 (except for 0, of course). If you keep applying this
> argument, you will prove that it is holomorphic.
>
> Now, on those complex numbers whose real part is greater than 0,
> Gamma(z) is the integral of t^{z - 1}.e^{-t}, with _t_ going from 0 to
> +oo. A function defined by such an expression is holomorphic. More
> precisely, if A is an open subset of C, (a,b) is an open interval of R
> (where _a_ can be -oo and _b_ can be +oo) and F is a continuous
> function from (a,b) x A into C such that
>
> 1) for each _t_ in (a,b), z |-> F(t,z) is holomorphic;
>
> 2) for each _z_ in A, t |-> F(t,z) is integrable and the integral
> converges uniformly on each compact subset of C,
>
> then the function from A into C defined by z |-> int(F(t,z),a < t < b)
> is holomorphic.
>
> Best regards,
>
> Jose Carlos Santos

Thanks, that should do.
From: W^3 on 27 Jul 2010 00:28
In article <8b5gt6FbrfU1(a)mid.individual.net>,
Jos� Carlos Santos <jcsantos(a)fc.up.pt> wrote:

> On 26-07-2010 13:57, Joubert wrote:
>
> > If anyone can tell me a source where I could find a proof of this or
> > tell me how it's proved I'd be grateful.
>
> Suppose that you prove that Gamma is holomorphic on the set of those
> complex numbers whose real part is greater than 0. It will follow from
> that and from the fact that Gamma(z + 1) = Gamma(z).z that it is also
> holomorphic on the set of those complex numbers whose real part is
> greater than -1 (except for 0, of course). If you keep applying this
> argument, you will prove that it is holomorphic.
>
> Now, on those complex numbers whose real part is greater than 0,
> Gamma(z) is the integral of t^{z - 1}.e^{-t}, with _t_ going from 0 to
> +oo. A function defined by such an expression is holomorphic. More
> precisely, if A is an open subset of C, (a,b) is an open interval of R
> (where _a_ can be -oo and _b_ can be +oo) and F is a continuous
> function from (a,b) x A into C such that
>
> 1) for each _t_ in (a,b), z |-> F(t,z) is holomorphic;
>
> 2) for each _z_ in A, t |-> F(t,z) is integrable and the integral
> converges uniformly on each compact subset of C,

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).

> then the function from A into C defined by z |-> int(F(t,z),a < t < b)
> is holomorphic.
>
> Best regards,
>
> Jose Carlos Santos
From: José Carlos Santos on 27 Jul 2010 05:12
On 27-07-2010 5:28, W^3 wrote:

>>> If anyone can tell me a source where I could find a proof of this or
>>> tell me how it's proved I'd be grateful.
>>
>> Suppose that you prove that Gamma is holomorphic on the set of those
>> complex numbers whose real part is greater than 0. It will follow from
>> that and from the fact that Gamma(z + 1) = Gamma(z).z that it is also
>> holomorphic on the set of those complex numbers whose real part is
>> greater than -1 (except for 0, of course). If you keep applying this
>> argument, you will prove that it is holomorphic.
>>
>> Now, on those complex numbers whose real part is greater than 0,
>> Gamma(z) is the integral of t^{z - 1}.e^{-t}, with _t_ going from 0 to
>> +oo. A function defined by such an expression is holomorphic. More
>> precisely, if A is an open subset of C, (a,b) is an open interval of R
>> (where _a_ can be -oo and _b_ can be +oo) and F is a continuous
>> function from (a,b) x A into C such that
>>
>> 1) for each _t_ in (a,b), z |-> F(t,z) is holomorphic;
>>
>> 2) for each _z_ in A, t |-> F(t,z) is integrable and the integral
>> converges uniformly on each compact subset of C,
>
> 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.

Best regards,

Jose Carlos Santos
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