[snip]I will transcribe a proof from "Riemann's Zeta Function" H.M. Edwards (Dover)

" For negative real values of s, Riemann evaluated the integral

as follows. Let

denote the domain in the

-plane which consists of all

[snip] points other than those which lie within

of the positve real axis or within

of one of the singularities

of the integrand of

(*). Let

be the boundary of

oriented in the usual way. Then, ignoring for the moment the fact that

is not compact, Cauchy's theorem gives

.

Now one component of this integral is the integral

with the orientation reversed, whereas the others are integrals

over the circles

oriented clockwise. Thus when the circles are oriented in the usual counterclockwise sense

becomes

The integrals over the circles can be evaluated by setting

for

to find

by the Cauchy integral formula. Summing over all integers

other than

and using

then gives

Finally using the simplification

, one obtains the desired formula

. This relationship between and is known as the functional equation of the zeta function.
In order to prove rigorously that

[the above

] holds for

s < 0, it suffices to modify the above argument

by letting

be the intersection of

with the disk

and letting

.; then the integral

splits into two parts, one being an integral over the circle

with the points within

of the positive real axis deleted, and the other being an integral whose limit as

is the left side of

. The first of these two parts approaches zero because the length of the

path of integration is less than

, because

the factor is bounded on the circle , and because the modulus of

on

this circle is

for

. Thus the second part, which by Cauchy's theorem is the negative of the first part, also approaches zero, which implies

and hence

.

This completes the proof of the functional equation (*) in the case

. However, both sides of

are analytic functions of

, so this suffices to prove

for all values of

(except for

where

[reference to footnote] one or more of terms

of have poles)

."
---------------------------------------------------------------------

Note two things

A) In the first integral the limits of integration are meant to represent a path of integration which begins at

,

Mr F says: A path cannot begin at + infinity.
moves to the left down the positive real axis, circles the origin once in the counterclockwise direction, and returns up the positive real axis to

Mr F says: This is not the boundary of D.
B) I only did this because I almost never see anyone on this site give an answer regarding the RZ function. If this does not make sense I am sorry. Take this proof with a grain of salt, for I know nothing of this and am just copying it from a book. I could easily have copied the wrong thing. If something confuses you or this does not seem like the pertinent proof just disgregard the thing in its entirety. And if need be there is an alternate proof.