1. Pochhammer symbol 5.Hypergeometric Functions Hypergeometric equation ( Gauss’ ODE & functions ) Regular singularities at Solution : Hypergeometric function (series) For a, b, c real, range of convergence are : Series diverges for Chap 7:

2. 2 F 1 includes many elementary functions. E.g. Sum terminates if  2 F 1 = polynomial Properties 

3. 2 nd Solution, Alternative ODE 2 nd Solution : § 7.6 : c = integer  y not independent of 2 F 1 ( a, b ; c ; x)  additional logarithm term required Alternative ODE : 

4. Contiguous Function Relations

5. Hypergeometric Representations

6. 6.Confluent Hypergeometric Functions Confluent Hypergeometric eq. Singularities : regular at irregular at Solution : For a, c real, series converge for all finite x. Sum terminates if  1 F 1 = polynomial E.g.

7. 2 nd solution : Standard form : Alternate ODE :

8. Integral Representations Techniques for verifying integral representations : 1. g(x,t) or Rodrigues relations. 2. Expand integrand into series & integrate. 3. (a) As solution to ODE. (b) Check normalization.

9. Confluent Hypergeometric Representations

10. Further Observations Advantages for using the (confluent) hypergeometric representations : 1.Asymptotic behavior or normalization easier to evaluate via the integral representation of M & U. 2.Inter-relationship between special functions becomes clearer. Self-adjoint version : Whittaker function Self-adjoint ODE : 2 nd solution :

11. 7.Dilogarithm Dilogarithm Usage : 1.Matrix elements in few-body problems in atomic physics. 2.Perturbation terms in electrodynamics.

12. Expansion Poly-logarithm  

13. Forseries converges & is real. Analytic Properties Branch point at z = 1. Conventional choice :Branch cut from z = 1 to z = . with principal value : For series diverges but integral is finite & complex ( analytic continued ). For series diverges but integral is finite & real ( analytic continued ).

14. Mathematica Since the only pole is at z = 0, the integral is independent of path as long as it does not cross the branch cut. For the path colored blue in figure, branch cut On small circle, set  On slanted line, set  RHS of fig.18.8 & eq.18.159 are not allowed since the path crosses the branch cut.

15. Properties & Special Values   generates Li 2 for all x from those in | x |  1/2. Proof : both sides & find identity. Set z = 0 or 1 to determine const. e.g.

16. Example 18.7.1.Check Usefulness of Formula Question: Are the individual terms real? I real & converges if  is real Li 2 (x) is real for x < 1  both Li 2 terms are real.

17. 8.Elliptic Integrals Example 18.8.1.Period of Simple Pendulum    Period

18. Definitions Elliptic integral of the 1 st kind Complete Elliptic integral of the 1 st kind

19. Elliptic integral of the 2 nd kind Complete Elliptic integral of the 1 st kind

20. Series Expansions Ex.13.3.8 Ex.18.8.2

21. Fig.18.10.K(m) & E(m) Mathematica

22. Limiting Values Integrals of the following form can be expressed in terms of elliptic integrals. E.Jahnke & F.Emde, “Table of Higher Functions”