Special Functions & Physics
G. Dattoli
ENEA FRASCATI
A perennial marriage in spite of
computers
Euler Gamma Function
Defined to generalize the factorial operation
to non integers


0
0
n!   t n e t dt  ( x )   t x 1e t dt, Re( x )  0 
 ( x  1)  x ( x )  (n  1)  n!
Inclusion of negative arguments
( x )  limn
n! n x
n
 ( j  x)
 x  m ( x )  
j 0
(1  x ) 

1
sin( x ) ( x )
Euler Beta Function
Generalization of binomial

I ( x, y )   e
 ( x 1) 

(1  e )
y 1
d ,
0
1
e   t   t x 1 (1  t ) y 1 dt  B( x, y )
0
( x) ( y ) 1  x  y  1

B ( x, y ) 
 
( x  y )
x  y 1 
1
Further properties
BETA: if x, y are both non positive integers the presence of a
double pole is avoided
( 1)n 1
( y )
B ( x, y )  

n! ( x  n ) ( y  n )
n 0

( 1)m
1
( x )

m 0 m! ( y  m ) ( x  m )

EULER
10 SWISS FRANCKS
Strings: the old (beautiful) times
and Euler & Veneziano
• Half a century ago the Regge trajectory
• Angular momentum of barions and mesons vs. squared
mass
Old beautiful times…
• The surprise is that all those trajectories where
lying on a stright line
J ( s)  0   s,
J m ( s)  0  m   s
• Where s is the c. m. energy and the angular
coefficient has an almost universal value
 1GeV 2
Mesons and Barions
Strings: Even though not immediately evident this
phenomenological observation represented the germ of
string theories.
The Potential binding quarks in the resonances was indeed
shown to increase linearly with the distance.
Meson-Meson Scattering
• m-m
s  ( p(1)  p( 2) ) 2 ,
t  ( p(1)  p( 4) ) 2 ,
4
u  ( p  p )  s  t  u   mi2
(1)
( 3) 2
i 1
Veneziano just asked what is the simplest
form of the amplitude yielding the resonance
where they appear on the C.F. Plot, and the
“natural” answer was
the Euler B-Function
( ( s)) ( (t ))
A( s, t ) 
 B( ( s),   (t ))
( ( s)   (t ))
From the Dark…
• An obscure math. Formula, from an obscure
mathematicians of XVIII century… (quoted from
a review paper by a well known theorist who,
among the other things, was also convinced that
the Lie algebra had been invented by a
contemporary Chinese physicist!!!)
• From an obscure math. formula to strings
• “A theory of XXI century fallen by chance in XX
century”
• D. Amati
Euler-Riemann function…

1
 ( x)   x
n 1 n
It apparently diverges for negative x but Euler was
convinced that one can assign a number to any series
An example of the art of
manipulating series
S  1  2  3  4  5  6  ...  O  E


1
T (t )   ( 1)n t n 
  tT (t )   ( 1)n n t n 1 
1 t
s 0
n 1
1
1

 T (1)  S  1  2  3  4  ... 
2
(1  t )
4
Divergence has been invented by
devil, no…no… It is a gift by God
S  1  2  3  4  ... 
O  1  3  5  ...
D  2  4  6  ...


1
  ( 1)
1
n 1 n
  O  E  n  
n 1
S OE 
 2E    S
E  2    
S
3

1
1


1
12
n 1 n
 1  2  3  ...  
  ( 1)  
1
12
1
4
Integral representation for the
Riemann Function

1
x
 A x 1
A 
e
 d

( x ) 0

a
n 0
n
1

1 a



1
1
 n x 1
 ( x)   x  
e
 d 

n 1 n
n 1 ( x ) 0

1
1
x 1


d


( x) 0 e  1
Planck law
8 h 3
u( , T ) 
c3
1
h
KT
,
1


3
U
x
0 u(, T ) d  L3  0 e x 1   (4) (4)
e
Analytic continuation of the
Riemann function
• Ac

 (1  s )  21 s   s cos( s )( s )  ( s ) 
2
  (1)  
 2
2
 2 
Analytic continuation & some
digression on series
• From the formula connecting half planes
of the Riemann function we get
 s 
 (1  s)  2  ( s) cos   ( s), s  2 
 2 
1
1  1
  ( 1)   2 (2)  (2)   2  2
2
2 n 1 n
1 s
•
1

1
?
2  2
 n1 n
s
..digression and answer
• “Euler” proved the following theorem,
concerning the sum of the inverse of the
roots of the algebraic equation
bn xn  bn1xn1  ...  b1 x  b0  0
n
1
b1


b0
s 1 xs
…answer
• Consider the equation
x3 x5 x7
y2 y4
sin(x )  0  x    ...  0  1 

 ...  0
3! 5! 7!
3! 5!
2
 ys2  s  

y  ( s )  
2
s
2
s 1
2
1
1
 
2 2
s
6
1
 (2) 
  (1)  
6
12
Casimir Force
• Casimir effect a force of quantum nature, induced
by the vacuum fluctuations, between two parallel
dielectric plates
Virtual particles pop out of the vacuum and wander around
for an undefined time and then pop back – thus giving the
vacuum an average zero point energy, but without
disturbing the real world too much.
Casimir: The Force of empty space
Sensitive sphere. This 200-µm-diameter sphere mounted
on a cantilever was brought to within 100 nm of a flat surface to detect the
elusive Casimir force.
Casimir Calculation a few math
• Elementary Q. M. yields diverging sum
1
E   En ,
2 n
(n  )
n  c k  k 
a2
2
x
2
y
2


1
 E   2  dkx dk y  A  n
2
n 1
Regularization & Normalization
• We can explicitly evaluate the integral

1
E   2  dkx dky  A n n
2
n 1

c1 s 2 s 1
3 s
 
n

2 a 3 s 3  s n
E (s)
A
E
c


(

3
)

F


c
a
A
6 a3
A
E
s
2

 ( 3)   n 3
n 1
• What is it and why does it provide a finite result?
Are we now able to compute the
Casimir Force?
• Remind that
E
c 2

 (3)  Fc   a
3
A
6a
A
E
• And that

 ( 3)   n 3
n 1
• And that
 s 
 (1  s)  2  ( s ) cos
  ( s ), s  2 
2


1
3!  1
  ( 3) 
( 4 )  ( 4 )   4  4
4
8
8 n 1 n
1 s
s
A further identity

T3 (t )   (1) n t  t  t  T (t )
n
3 n
s 0

1
T (t )   (1) t 
1 t
n 1
 T3 (1)  S3  O3  E3
n n
3
Again dirty tricks
• Going back to Euler
O3  13  33  53...
E3  23  43  63...  23 (O3  E3 )
 3  O3  E3   (3)
S 3  13  23  33  43  53  63  ...  O3  E3 
Ec
S3
c 2
1




  (3) 
3
4
760a
A
120
1 2

 e2
Sc a
, Sc 
760
2
e2
a , 
c
2
What is the meaning of all this
crazy stuff?
• The sum o series according to Ramanujian

1
n    (1)    ,

12
n 1
Renormalization: Quos perdere vult
Deus dementat prius
• A simple example, the divergence from
elementary calculus
1 n 1
I ( x, n )   x dx 
x c
n 1
n
I ( x,1)  ln(x)  c
The way out: A dirty trick or
mathemagics
• We subtract to the constants of integration
• A term (independent of x) but with the same
behaviour (divergence) when n=-1.
• That’s the essence of renormalization subtract
infinity to infinity.
• We set
1
cc
n 1
Dirty...Renormalization
     finiteterm
• Our tools will be: subtraction and evaluation
of a limit
n 1
x 1
I ( x,1)  limn1 (
)c
n 1
ax 1
lim x0
 ln(a )  I ( x,1)  ln(x )  c
x
Is everything clear?
• If so
• prove that
! 2
find a finite value for

n
n
(

1
)
n!
x

n 0
• The diverging series “par excellence”
Shift operators
(Mac Laurin Series expansion)
f ( x  b)  e

s
bx
b (s)
  f ( x)
s  0 s!

bs s
f ( x)    x f ( x) 
s 0 s!
Series Summation

1
x 
,

1 x
n 0
n

1 (n)
n
ˆ
ˆ
f (O)   f (0) O ,
n  0 n!

1
nx
 e
x
1 e
n 0
We can do thinks more rigorously


n 0
n 0
 f ( x  n)   en  x f ( x ) 
1
f ( x) 
x
1 e

x
 nx
1
   x
f ( x )   x  Bn
f ( x)
e 1
n!
n 0
1
x
Bn  Bernoullinumbers
t
t  tk
 1    Bk
t
e 1
2 k 2 k!
Jacob Bernoulli and E.R.F.
Ars coniectandi 1713 (posthumous)
m1
1
m!
f ( x)  x   (m)   Bn
n! (m  1  n)!
n 0
m
Diverging integrals in QED
• In Perturbative QED the problem is that of giving
a meaning to diverging integrals of the type

I m   k dk, k 
m
0
2

,
m  positive,  sm all  UV  divergences,
m  negative,  l arg e  IR  divergences
Schwinger
Was the first to realize a possible link
between QFT diverging integrals and
Ramanujan sums
I m,   


0


B2 r
m
m
p dp 
 n 
am , r m  2 r 1 ,
2
n 1
r 1 ( 2 r )!
m
 ( m  1)
am , r 
,
 ( m  2 r  2)
m
I m,   
I ( m  1,  )   (  m)
2

B2 r

am , r ( m  2r  1) I ( m  2 r ,  )
r 1 ( 2 r )!
Recursions
1
I (0, )   (0)   ,
2
1
I (1, )  I (0,  )   (1),
2
1
I (3, )  ( I (0, )  a2,1 B2 )   (1),
2
...
Self Energy diagrams
• Feynman loops (DIAGRAMMAR!!! ‘t-HooftVeltman, Feynman the modern Euler)
• Loops diagram are divergent
• Infrared or ultraviolet divergence
k0
k
F.D. and renormalization
• a
I m,   


B2 r
m
I (m  1, )   (m)
2
am, r (m  2r  1) I (m  2 r , )
r 1 ( 2 r )!
The Euler Dilatation operator
x  x xn  n xn 
 ( x  x) p xn  n p xn ,
e
 xx

x 
n
s 0
 n s x n  e n x n  (e x )n 
s!
 e x  x f ( x )  f e x 
Can the Euler-Riemann function be
defined in an operational way?
• We introduce a naive generalization of the E-R function
n
x
x  x  x n  p
n

xn
 ( x, p )   p
n 1 n
p
x
 ( x, n )  ( x  x )
1 x
p

n 1
a 


1
 a  1

e
 d

( ) 0
1
 Dˆ  1
ˆ
Zp 
e  d ,

( p ) 0
Dˆ x  x  x
 1

 1
1  x 
 ( x, n )  ( x  x )  p  x n  ( x  x )  p 
Can the E-R Function…?
YES
 Dˆ x
• The exponential operator e
operator
 Dˆ x
e
, is a dilatation
f ( x)  f (e x)

x
1
x
ˆ
 D
p 1
 ( x, p)  Zˆ p

e

d 

1  x ( p ) 0
1 x

x
e 
p 1

d 


( p ) 0 1  x e

x
 p 1

d


( p ) 0 e  x
More deeply into the nature of
dilatation operators
• So far we have shown that we can generate the
E-R function by the use of a fairly simple
operational identity

1
x
ˆ
 D
p 1
 ( x, p ) 
e 
d

( p ) 0
1 x

x
 p 1

d


( p ) 0 e  x
 (1, p)   ( p)
Operators and integral transforms
• Let us now define the operator (G. D. & M.
Migliorati
Zˆ p 

1
 Dˆ p 1
e  d

( p) 0
• And its associated transform, something in
between Laplace and
Mellin

Zˆ p f ( x ) 
1

p 1
f
(
e
x
)

d 

( p ) 0
 Zˆ p ( x, m)   ( x, m  p)
Zeta and prime numbers
Euler!!!
 (s)  
p
1
1
1 s
p
A lot of rumours!!!
Hermitian and non Hermitian
operators
• The operator x  x is not Hermitian
• The Hamiltonian
1
1
ˆ
H  ( x p  p x )  i ( x  x  )
2
2
• Is Hermitian (at least for physicist)
Evolution operator
Uˆ ( )  e
1

  x  x  
2

e
1
 
 Dˆ x
2
e
Riemann hypothesis
• RH: The non trivial zeros are on the
critical line:
1
it 
2
The Riemann hypothesis:The Holy
Graal of modern Math
• What is the point of view of
physicists?
• The Berry-Keating
conjecture:
…zeros Coincide with the
spectrum of the Operator:
1
Hˆ  i ( x  x  )
2
namely
1
2
 (  i En )  0
Lavoro di Umar Mohideen e suoi collaboratori
all’università di California a Riverside
Una sfera di polistirene 200 µm di
diametro ricoperta di oro (85,6 nm)
attaccata alla leva di un microscopio
a forza atomica, ad una distanza di
0.1 µm da un disco piatto coperto
con gli stessi materiali.
L’attrazione tra sfera e disco ricavata
dalla deviazione di un fascio laser.
Differenza tra dato seprimentale e
valore teorico entro 1%.
Sensibilità: 10-17 N
Vuoto: 10-1-10-6 Pa
EULER-BERNOULLI
2 2 k 1 2 k
 (2 k ) 
B2 k ,
(2 k )!


1
2z
z 2 k 1
ctg ( z )    2
 2 2 k  2 k ,
2
z n 1 z  (n  )
k 1 

1
ctg ( z )   
z k 1
(1) k 2 2 k B2 k
(2 k )!
z 2 k 1
Beta the way out
• …The Beta function once more
n 
D
a
2
B ( x, y )

(  L)
D
( )
2
• More details upon request