User:Manifestement

Advanced theorems from complex analysis

1. Exchange of summation and integration
2. Analyticity of a function defined by an integral
3. Dirichlet series

Series Representations

1. Landau
2. Hadamard
3. Ramaswami

Functional equation

1. Euler discovery and partial proof
2. Landau's proof
3. Hardy's proof

Integral Representations

A number of integral formulas involving the eta function can be listed.

• A change of variable (Abel, 1823) in the integral representation of Euler's gamma fonction gives a Mellin transform which can be expressed in different ways. It is valid for $\Re s>0.$

$\Gamma (s)\eta (s)=\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}+1}}{dx}=\int _{0}^{\infty }\int _{0}^{\infty }{\frac {(t+r)^{s-2}}{e^{t+r}+1}}{dr}{dt}=\int _{0}^{1}\int _{0}^{1}{\frac {(-\log(xy))^{s-2}}{1+xy}}{dx}{dy}.$

• Integration by parts yields formulas valid over successive unit strips in the left plane, starting with this one, for $\Re s>-1$

$\Gamma (s+1)\eta (s)=\int _{0}^{\infty }{\frac {dx^{s}}{e^{x}+1}}=\int _{0}^{\infty }{\frac {e^{x}x^{s}}{(e^{x}+1)^{2}}}{dx}=\Gamma (s+1)\eta (s+1)-\int _{0}^{\infty }{\frac {x^{s}}{(e^{x}+1)^{2}}}{dx}$

• The next one is valid for $\Re s>-2$

$\Gamma (s+2)\eta (s)=\Gamma (s+2)\eta (s+1)-\int _{0}^{\infty }{\frac {d(x^{s+1})}{(e^{x}+1)^{2}}}=\Gamma (s+2)\eta (s+1)-2\int _{0}^{\infty }{\frac {e^{x}x^{s+1}}{(e^{x}+1)^{3}}}{dx}$

$=3\Gamma (s+2)\eta (s+1)-2\Gamma (s+2)\eta (s+2)+2\int _{0}^{\infty }{\frac {x^{s+1}}{(e^{x}+1)^{3}}}{dx}$

• In general, for $\Re s>-k,\,k=0,1,2,\ldots$

$\eta (s)=\eta (s+1)\,{\frac {k(k+1)}{2}}\,-\,\ldots \,+\,\eta (s+k)\,(-1)^{k-1}\,k!\,+\,\int _{0}^{\infty }{\frac {x^{s+k-1}}{(e^{x}+1)^{k+1}}}{dx}\,{\frac {(-1)^{k}k!}{\Gamma (s+k)}}$

• This Lindelöf (1905) formula is valid over the whole complex plane, when the principal value is taken for the logarithm implicit in the exponential.

$\eta (s)=\int _{-\infty }^{\infty }{\frac {(1/2+it)^{-s}}{e^{\pi t}+e^{-\pi t}}}{dt}.$

This corresponds to a Jensen (1895) formula for a related entire function, valid over the whole complex plane as proven by Lindelöf.

$(s-1)\zeta (s)=\int _{-\infty }^{\infty }{\frac {(1/2+it)^{1-s}}{(e^{\pi t}+e^{-\pi t})^{2}}}{dt}.$