1. 13. Gamma Function Definitions, Properties
Digamma & Polygamma Functions
The Beta Function
Sterling’s Series
Riemann Zeta Function
Other Related Functions

2. Peculiarities:
Do not satisfy any differential equation with rational coefficients.
Not a hypergeometric nor a confluent hypergeometric function.
Common occurence:
In expansion coefficients.

3. 13.1. Definitions, Properties
Definition, infinite limit (Euler) version : 

4. Definition: Definite Integral
Definition, definite integral (Euler) version :
, else singular at t = 0. 

5. Equivalence of the Limit & Integral Definitions
Consider  

6. Definition: Infinite Product (Weierstrass Form)
Definition, Infinite Product (Weierstrass) version :
Euler-Mascheroni constant
Proof : 

7. Functional Relations Reflection formula : ( about z = ½ ) Proof : Let

8. For z  integers, set branch cut ( for v z ) = + x-axis :
f (z) has pole of order m at z0 :   

9. Legendre’s Duplication Formula
General proof in §13.3.
Proof for z = n = 1, 2, 3, …. :
( Case z = 0 is proved by inspection. ) 

10. Analytic Properties Weierstrass form :
 has simple zeros at z  n,
no poles.
 (z) has simple poles at z  n,
no zeros.
 changes sign at z  n.
Minimum of  for x > 0 is
Mathematica

11. Residues at z  n
Residue at simple pole z  n is
n + 1 times : 

12. Schlaefli Integral Schlaefli integral : Proof :  if  > 1 
C1 is an open contour.
( e t   for Re t   .
Branch-cut. )
Schlaefli integral :
Proof : 
if  > 1 

13. where
For Re  > 1, ID = 0
 reproduces the integral represention.
For Re  < 1 , IA , IB , & ID are all singular.
However, remains finite.
( integrand regular everywhere on C )
 is valid for all .
Factorial function :
(z) is the Gauss’ notation

14. Example 13.1.1 Maxwell-Boltzmann Distribution
Classical statistics (for distinguishable particles) :
Probability of state of energy E being occupied is
Maxwell-Boltzmann distribution 
Partition function
Average energy :
g(E) = density of states
Ideal gas :
gamma distribution

15. 13.2. Digamma & Polygamma Functions
Digamma function :   
50 digits
z = integer :
Mathematica

16. Polygamma Function Polygamma Function :   = Reimann zeta function
Mathematica

17. Maclaurin Expansion of ln 
Converges for
Stirling’s series ( § 13.4 ) has a better convergence.

18. Series Summation Example 13.2.1. Catalan’s Constant Dirichlet series :
20 digits
Mathematica

19. 13.3. The Beta Function
Beta Function :  

20. Alternate Forms : Definite Integrals  
To be used in integral rep. of Bessel (Ex ) & hypergeometric (Ex ) functions

21. Derivation: Legendre Duplication Formula 

22. 13.4. Sterling’s Series Derivation from Euler-Maclaurin Integration FormulaB2 B4 B6 B8
1/6
1/30
1/42
Euler-Maclaurin integration formula :
Let   

23.   

24. 
Stirling’s series  

25. A = Arfken’s two-term approx. using
z >> 1 :
Stirling approx 
A = Arfken’s two-term approx. using
Mathematica

26. 13.5. Riemann Zeta Function Riemann Zeta Function :
Mathematica
Integral representation :
Proof :

27. Definition : Contour Integral
for Re z >1
diverges for Re z <1
agrees with integral representation for Re z > 1 

28. Analytic Continuation
C  C1 encloses no pole.
C  C1 encloses all poles.
Re z > 1
Poles at
Similar to ,
 Definition valid for all z (except for z  integers).
Mathematica
 means n  0  

29. Riemann’s Functional Equation
Riemann’s functional equation

30. Zeta-Function Reflection Formula   
zeta-function reflection formula

31. Riemann’s functional equation :
converges for Re z > 1
 (z) is regular for Re z < 0.
(0) diverges  (1) diverges while (0) is indeterminate.
 for trivial zeros
Since the integrand in is always positive, 
 (except for the trivial zeros) or
i.e., non-trivial zeros of (z) must lie in the critical strip

32. Critical Strip Consider the Dirichlet series :
Leibniz criterion  series converges if , i.e.,
 for   

33. (0)
Simple poles :  

34. Euler Prime Product Formula
( no terms )
( no terms ) 
Euler prime product formula

35. Riemann Hypothesis
Riemann found a formula that gives the number of primes less than a given number in terms of the non-trivial zeros of (z).
Riemann hypothesis :
All nontrivial zeros of (z) are on the critical line Re z  ½.
Millennium Prize problems proposed by the Clay Mathematics Institute.
1. P versus NP
2. The Hodge conjecture
3. The Poincaré conjecture (proved by G.Perelman in 2003)
4. The Riemann hypothesis
5. Yang–Mills existence and mass gap
6. Navier–Stokes existence and smoothness
7. The Birch and Swinnerton-Dyer conjecture

36. 13.6. Other Related Functions
Incomplete Gamma Functions
Incomplete Beta Functions
Exponential Integral
Error Function

37. Incomplete Gamma Functions
Integral representation: 
Exponential integral

38. Series Representation for  (n, x)

39. Series Representation for (n, x)

40. Series Representation for  (a, x) &  (a, x)
For non-integral a :
See Ex & Ex
Pochhammer symbol
Relation to hypergeometric functions: see §

41. Incomplete Beta FunctionsEx
Relation to hypergeometric functions: see § 18.5.

42. Exponential Integral Ei(x)
P = Cauchy principal value
E1 , Ei analytic continued.
Branch-cut : (x)–axis.
Mathematica

43. Series Expansion
For x << 1 :
For x >> 1 :

44. Sine & Cosine Integrals
not defined
Mathematica
Ci(z) & li(z) are multi-valued. Branch-cut : (x)–axis.
is an entire function

45.  
Series expansions : Ex Asymptotic expansions : § 12.6.

46. Error Function Power expansion :
Mathematica
Asymptotic expansion (see Ex ) :