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MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
Part II. —The Theory of Divergent Series. 
Section 
23. Poincare’s arithmetic theory of asymptotic series . 
24. The functional theory of divergent series with finite radius of convergence 
25. Domain of existence of the integral “ sum ” . 
26. The series for z ~ 1 log (1 -z) and (1 -z)~ m . 
27. The allied problem. References. 
28. Series which are truly asymptotic. Character of the problem .... 
29. Asymptotic series of the first order and their summation ... . 
30. The “ sum ” and series are arithmetically dependent. 
31. Such an asymptotic series can be differentiated. 
32. Series of higher orders... 
33. Their summation by the repeated exponential process. 
34. The result has arithmetic dependence. 
35. Example connected with the extended Riemann £ function. 
36. Some properties of an extended hypergeometric function. 
37. Its application to the summation of divergent series of finite order . . 
38. Extension of the theory to series of any order. 
39. Example of the previous theory. 
40. Contrast between the two summation processes. 
41. Application of the theory to the Maclaurin sum formula. 
42. 
>> 
>> 
series e x 
CO ^ I 
^ - 7 . 11+1 
n = 0 
43. The re-arrangement of asymptotic series. 
44. The theory of the asymptotic expansion of an integral function near infinity . 
45. Such expansions may differ in form in various regions round infinity 
46. Conclusion. 
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Part III.—The Asymptotic Expansion of Simple Integral Functions. 
47. Order of progress.452 
48. The Maclaurin constants and integral limits.452 
49. Their expression, when </> ( 11 ) = nf>, in terms of Riemann’s (' function . 453 
50. Example of the function —-.453 
51. The remainder expressed as a definite integral.455 
52. Asymptotic expansion of the function P p (z) .456 
53. The nature of the expansion.458 
54. Asymptotic expansion of the general function of order less than unity.459 
55. Consideration of the integral which gives rise to the dominant term.461 
56. Case when ip (/) = P (a 0 + ~J + |f + • • •) an( l P > 1.462 
57. Functions to which § 54 is applicable.463 
53. Asymptotic expansion of a function with an algebraic secpience of zeros.464 
°° 1 + ^ ! 
59. The dominant term of the expansion of log 11 -—.465 
L 1 + 
