1. Complex numbers 2. The theory of convergence 3. Continuons functions and uniform convergence 4. The theory of riemann integration 5. The fundamental proportics of analysis functions taylor´n laurent´s and liouvill´s theorems 6. The theory of residues application to the evaluation of definite integrals 7. The expansion of functions in infinite series 8. Asympototic expansions and summable series 9. Fourier series and trigobometrical series 10. Linear differential equations 11. Integral equations 12. The gamma function 13.The zeta functions of riemann 14. The hypergeometric functions 15. Legondre functions 16. The confluent hygeometric functions 17. Bessel functions 18. The equations of mathematical physiscs 19. Mathieu functions 20. Elliptic functions General theorems and the wierstrassian functions 21. The theta functions 22. The jacobian ellptic functions 23. Ellippsoidal harmonics and lame´s equation