13.4.Sterling’s Series Derivation from Euler-Maclaurin Integration Formula Euler-Maclaurin integration formula : B2B2 B4B4 B6B6 B8B8 1/6  1/30 1/42  1/30 Let   

  

   Stirling’s series

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

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

Definition : Contour Integral  0 for Re z >1 diverges for Re z <1  agrees with integral representation for Re z > 1 C1C1

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

Riemann’s Functional Equation  Riemann’s functional equation

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

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

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

 (0) Simple poles :  

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

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

13.6.Other Related Functions 1.Incomplete Gamma Functions 2.Incomplete Beta Functions 3.Exponential Integral 4.Error Function

Incomplete Gamma Functions  Integral representation: Exponential integral

Series Representation for  (n, x) 

Series Representation for  (n, x) 

Series Representation for  (a, x) &  (a, x) For non-integral a : See Ex & Ex Pochhammer symbol Relation to hypergeometric functions: see § 18.6.

Incomplete Beta Functions Ex Relation to hypergeometric functions: see § 18.5.

Exponential Integral Ei(x) P = Cauchy principal value E 1, Ei analytic continued. Branch-cut : (  x)–axis. Mathematica

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

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

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

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