Integral of holomorphic function in 2 variables is holomorphic

[Giraffro]

Define \forall \rho \in (0,\pi), C_\rho to be contour traveling from +\infty + \pi i/2 to \rho i, then a semicircle to -\rho i then a straight line to +\infty -\rho i. Also define:
I_\rho : \mathbb{C} \to \mathbb{C}, s \mapsto \int_{C_\rho} \frac{z^{s-1}}{e^z - 1} dz
I've shown that this function is well-defined, independent of the value of \rho and \forall \rho \in (0, \pi), \forall s \in \mathbb{C} with \Re(s) > 1, I_\rho(s) = (e^{2 \pi i s} - 1) \Gamma(s) \zeta(s) - This is part of a proof of the functional equation for the Riemann zeta function in my lecture notes. However, my notes claim I can show that I_\rho is holomorphic on \mathbb{C} by showing \forall R > 0, \exists M > 0 such that \forall s \in \mathbb{C} with |s| \leq R:
\int_{C_\rho} \left| \frac{z^{s-1}}{e^z-1} \right| dz \leq M
I can't find a reference that shows this gives you a holomorphic function.

Can anyone help?

 

[Landau]

I may be mistaking, but doesn't this follow from Theorem[/url]? If f is bounded on every disc, then I guess its integral over every closed curve must vanish.

 

[Giraffro]

I may be mistaking, but doesn't this follow from Theorem[/url]? If f is bounded on every disc, then I guess its integral over every closed curve must vanish.

Edit: Misread you're article and thought it was referring to the reverse implication AKA Cauchy's residue formula.

 

[Landau]

I was suggesting that the claim of your notes implies that the hypotheses of Morera's theorem are satisfied. Forget about your I_\rho for a moment; we are trying to prove the following:

Suppose f:\mathbb{C}\to \mathbb{C} has the property that \forall R > 0, \exists M > 0 such that \forall s \in \mathbb{C} with |s| \leq R, |f(s)| \leq M. Then f is holomorphic.

 

[Giraffro]

I was suggesting that the claim of your notes implies that the hypotheses of Morera's theorem are satisfied. Forget about your I_\rho for a moment; we are trying to prove the following:

Suppose f:\mathbb{C}\to \mathbb{C} has the property that \forall R > 0, \exists M > 0 such that \forall s \in \mathbb{C} with |s| \leq R, |I_\rho(s)| \leq M. Then f is holomorphic.

I don't know whether you caught the edit, but I misread you're article and I've updated my OP to a stronger condition. I'm not particularly worried about the bound since I have one, so the claim I suppose we're trying to prove is:

If g : \mathbb{C}^2 \to \mathbb{C} is holomorphic in both variables and define:
f : \mathbb{C} \to \mathbb{C}, s \mapsto \int_C g(s, z) dz

If \forall R > 0, \exists M > 0 : \forall s \in \mathbb{C} with |s| \leq R:
\int_C |g(s, z)| dz \leq M

then g is holomorphic.

 

[Landau]

The OP still seems to ask something else than the above:

However, my notes claim I can show that I_\rho is holomorphic on \mathbb{C} by showing (...)

So you agree that I_\rho:\mathbb{C}\to \mathbb{C} is holomorphic for every \rho? But you want to show that in fact

\mathbb{C}^2\to \mathbb{C}
(\rho,s)\mapsto I_{\rho}(s)

is holomorphic?

 

[Giraffro]

The OP still seems to ask something else than the above:

So you agree that I_\rho:\mathbb{C}\to \mathbb{C} is holomorphic for every \rho? But you want to show that in fact

\mathbb{C}^2\to \mathbb{C}
(\rho,s)\mapsto I_{\rho}(s)

is holomorphic?

No, the z and s variables coincide with the integrand in the definition of I, which is independent of \rho. So here, g : \mathbb{C}^2 \to \mathbb{C}, (s, z) \mapsto z^{s-1} / (e^z - 1). Actually just noticed this is undefined at the origin, but our contour doesn't pass through it, so we should be okay. In any case, g is holomorphic in both s and z, where it's defined. My notes claim that I_\rho is holomorphic because the uniform bound property I stated holds, but I don't see why it follows.

 

[Landau]

There still seems to be miscommunication. You probably made a typo in the last sentence of your previous post, where you say you want to show that g is holomorphic. So my understanding is that you DO want to prove that, for every rho, I_rho is holomorphic. Yes? Please be clear about this.

If so, I stand by my very first reaction, because the change you made in the OP is only a weaker condition:

if

\int_{C_\rho} \left| \frac{z^{s-1}}{e^z-1} \right| dz \leq M

then certainly

|I_\rho(s)| \leq M

by the triangle inequality for integrals. So then we're back at post 4 where I suggest that I_\rho satisfies the hypothesis of Morera's Theorem.