Advanced theorems from complex analysis
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Exchange of summation and integration
Analyticity of a function defined by an integral
Dirichlet series Series Representations
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Landau
Hadamard
Ramaswami
Euler discovery and partial proof
Landau's proof
Hardy's proof Integral Representations
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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
ℜ
s
>
0.
{\displaystyle \Re s>0.}
Γ
(
s
)
η
(
s
)
=
∫
0
∞
x
s
−
1
e
x
+
1
d
x
=
∫
0
∞
∫
0
∞
(
t
+
r
)
s
−
2
e
t
+
r
+
1
d
r
d
t
=
∫
0
1
∫
0
1
(
−
log
(
x
y
)
)
s
−
2
1
+
x
y
d
x
d
y
.
{\displaystyle \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
ℜ
s
>
−
1
{\displaystyle \Re s>-1}
Γ
(
s
+
1
)
η
(
s
)
=
∫
0
∞
d
x
s
e
x
+
1
=
∫
0
∞
e
x
x
s
(
e
x
+
1
)
2
d
x
=
Γ
(
s
+
1
)
η
(
s
+
1
)
−
∫
0
∞
x
s
(
e
x
+
1
)
2
d
x
{\displaystyle \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
ℜ
s
>
−
2
{\displaystyle \Re s>-2}
Γ
(
s
+
2
)
η
(
s
)
=
Γ
(
s
+
2
)
η
(
s
+
1
)
−
∫
0
∞
d
(
x
s
+
1
)
(
e
x
+
1
)
2
=
Γ
(
s
+
2
)
η
(
s
+
1
)
−
2
∫
0
∞
e
x
x
s
+
1
(
e
x
+
1
)
3
d
x
{\displaystyle \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
Γ
(
s
+
2
)
η
(
s
+
1
)
−
2
Γ
(
s
+
2
)
η
(
s
+
2
)
+
2
∫
0
∞
x
s
+
1
(
e
x
+
1
)
3
d
x
{\displaystyle =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
ℜ
s
>
−
k
,
k
=
0
,
1
,
2
,
…
{\displaystyle \Re s>-k,\,k=0,1,2,\ldots }
η
(
s
)
=
η
(
s
+
1
)
k
(
k
+
1
)
2
−
…
+
η
(
s
+
k
)
(
−
1
)
k
−
1
k
!
+
∫
0
∞
x
s
+
k
−
1
(
e
x
+
1
)
k
+
1
d
x
(
−
1
)
k
k
!
Γ
(
s
+
k
)
{\displaystyle \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.
η
(
s
)
=
∫
−
∞
∞
(
1
/
2
+
i
t
)
−
s
e
π
t
+
e
−
π
t
d
t
.
{\displaystyle \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
)
ζ
(
s
)
=
∫
−
∞
∞
(
1
/
2
+
i
t
)
1
−
s
(
e
π
t
+
e
−
π
t
)
2
d
t
.
{\displaystyle (s-1)\zeta (s)=\int _{-\infty }^{\infty }{\frac {(1/2+it)^{1-s}}{(e^{\pi t}+e^{-\pi t})^{2}}}{dt}.}
