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(x y))^{s-2}}{1 + x y}{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{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+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 + i t)^{-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 + i t)^{1-s}}{(e^{\pi t}+e^{-\pi t})^2}{dt}.

$$