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Define [tex]\forall \rho \in (0,\pi), C_\rho[/tex] to be contour travelling from [tex]+\infty + \pi i/2[/tex] to [tex]\rho i[/tex], then a semicircle to [tex]-\rho i[/tex] then a straight line to [tex]+\infty -\rho i[/tex]. Also define:

[tex]I_\rho : \mathbb{C} \to \mathbb{C}, s \mapsto \int_{C_\rho} \frac{z^{s-1}}{e^z - 1} dz[/tex]

I've shown that this function is well-defined, independent of the value of [tex]\rho[/tex] and [tex]\forall \rho \in (0, \pi), \forall s \in \mathbb{C}[/tex] with [tex]\Re(s) > 1, I_\rho(s) = (e^{2 \pi i s} - 1) \Gamma(s) \zeta(s)[/tex] - 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 [tex]I_\rho[/tex] is holomorphic on [tex]\mathbb{C}[/tex] by showing [tex]\forall R > 0, \exists M > 0[/tex] such that [tex]\forall s \in \mathbb{C}[/tex] with [tex]|s| \leq R[/tex]:

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

I can't find a reference that shows this gives you a holomorphic function.

Can anyone help?

[tex]I_\rho : \mathbb{C} \to \mathbb{C}, s \mapsto \int_{C_\rho} \frac{z^{s-1}}{e^z - 1} dz[/tex]

I've shown that this function is well-defined, independent of the value of [tex]\rho[/tex] and [tex]\forall \rho \in (0, \pi), \forall s \in \mathbb{C}[/tex] with [tex]\Re(s) > 1, I_\rho(s) = (e^{2 \pi i s} - 1) \Gamma(s) \zeta(s)[/tex] - 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 [tex]I_\rho[/tex] is holomorphic on [tex]\mathbb{C}[/tex] by showing [tex]\forall R > 0, \exists M > 0[/tex] such that [tex]\forall s \in \mathbb{C}[/tex] with [tex]|s| \leq R[/tex]:

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

I can't find a reference that shows this gives you a holomorphic function.

Can anyone help?

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