Methods to extend the definition of eta to the entire complex plane
$begingroup$
In this page (https://en.wikipedia.org/wiki/Dirichlet_eta_function) the author wrote:
Hardy gave a simple proof of the functional equation for the eta function, which is:
$$η(-s)=ϕ(s)η(1+s)$$
where $ϕ(s)$ is a given complex function.
From this, one immediately has the functional equation of the zeta function also, as well as another means to extend the definition of eta to the entire complex plane.
I am asking on the original source of G. H. Hardy where this result appeared in the first time.
reference-request zeta-functions
asked Jan 7 at 17:50
ChinaChina
1,4441029
$endgroup$
add a comment |
$begingroup$
In this page (https://en.wikipedia.org/wiki/Dirichlet_eta_function) the author wrote:
Hardy gave a simple proof of the functional equation for the eta function, which is:
$$η(-s)=ϕ(s)η(1+s)$$
where $ϕ(s)$ is a given complex function.
From this, one immediately has the functional equation of the zeta function also, as well as another means to extend the definition of eta to the entire complex plane.
I am asking on the original source of G. H. Hardy where this result appeared in the first time.
reference-request zeta-functions
asked Jan 7 at 17:50
ChinaChina
1,4441029
$endgroup$
add a comment |
$begingroup$
In this page (https://en.wikipedia.org/wiki/Dirichlet_eta_function) the author wrote:
Hardy gave a simple proof of the functional equation for the eta function, which is:
$$η(-s)=ϕ(s)η(1+s)$$
where $ϕ(s)$ is a given complex function.
From this, one immediately has the functional equation of the zeta function also, as well as another means to extend the definition of eta to the entire complex plane.
I am asking on the original source of G. H. Hardy where this result appeared in the first time.
reference-request zeta-functions
asked Jan 7 at 17:50
ChinaChina
1,4441029
$endgroup$
In this page (https://en.wikipedia.org/wiki/Dirichlet_eta_function) the author wrote:
Hardy gave a simple proof of the functional equation for the eta function, which is:
$$η(-s)=ϕ(s)η(1+s)$$
where $ϕ(s)$ is a given complex function.
From this, one immediately has the functional equation of the zeta function also, as well as another means to extend the definition of eta to the entire complex plane.
I am asking on the original source of G. H. Hardy where this result appeared in the first time.
reference-request zeta-functions
reference-request zeta-functions
asked Jan 7 at 17:50
ChinaChina
1,4441029
asked Jan 7 at 17:50
ChinaChina
1,4441029
asked Jan 7 at 17:50
ChinaChina
1,4441029
asked Jan 7 at 17:50
ChinaChina
1,4441029
asked Jan 7 at 17:50
ChinaChina
1,4441029
1,4441029
add a comment |
add a comment |
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