Defined on the real line originally. Analytic continuation to
right half-plane given byshowing the integral still converges.
Analytic continuation in entire complex plane givenby shifting
successively to the left.
Zeroes and poles
Poles at the negative integers s = 0,1,2 . . .. No zeroes.
Growth order
1/ is of growth order 1.
Relations
Factorial relation:(s + 1) = s(s).
Symmetry about the line Re(s) = 1/2:
(s)(1 s) =
sins
Product formula:1
(s)= es
n=1
(1 + s/n)es/n
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Result of Weierstrass theorem.
2 The Riemann Zeta Function (s)
Definition
(s) =n=1
1
ns
Domain of definition
C.
Defined on the real line s > 1, converges for Re(s) > 1,
and can be analytically continued
to the rest of the complex plane.Zeroes and poles
Only one simple pole at s = 1. Zeroes: trivial zeroes are at
2,4, . . .. Nontrivial zerosare in the critical strip 0 < Re(s)
< 1; the Riemann hypothesis says that all of them lie onthe line
Re(s) = 1/2.
Growth order
1.
If s = + i, then for each 0 0 1 and > 0,
|(s)| c|t|
10+
Relations
Functional equation:
s/2(s/2)(s) =1
2
0
us/21((u) 1)du
Prime product formula:
(s) =p
1
1 ps
Symmetric relation:
(s) = s1/2((1 s)/2)
(s/2)(1 s)
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3 Theta function (t)
Definition
(t) =
n=
en2
t
Domain of definition
Real, positive numbers.
Growth
Actually, decay:(t) Ct1/2
Relations
(t) = t1/2
(1/t)
4 Weierstrass function
Definition
(z) =1
z2+n=0
[1
(z + )2
1
2
Domain of definition
C.
Zeroes and poles
Double poles at the points of the lattice generated by the
periods. (Elliptic function.)
Relations Differential equation:
()2 = 4( e1)( e2)( e3)
where e1 = (1/2), e2 = ( /2), and e3 = (1+2
).
Every elliptic function with the same periods is a rational
function of and . Result of
Mittag-Leffler.
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5 Weierstrass function
Definition
(z) = zw(1
z
w )e
z/w+z2/2w
Domain of definition
C
Zeroes and poles Zeroes at the lattice points. No poles.
