MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
413 
Section 
60. Application to functions of zero order 
61. The function II 
+ e n 
62. 
63. 
64. 
65. 
66 . 
67. 
68 . 
69. 
70. 
71. 
79 
a = l L 
Deduction of Lambert’s function from that in 
Note on the indefinite integral of 8 55 . . . 
61 
The further theory of functions of zero order.. 
Functions of finite non-integral order greater than unity. The function Q p (z) . . . 
Asymptotic expansion of the general function of non-integral order greater than unity. 
Analogy with the former expansions. 
Application to functions with algebraic sequence of zeros. 
Case when the coefficients become infinite. 
Functions of finite integral order not less than unity. The function R p (~). 
Deductions of the previous expansion as a limit of former results. 
Construction of an extension of the gamma function. 
Asymptotic expansion of a corresponding function with algebraic sequence of zeros . . 
Conclusions.. . . . 
Page 
466 
466 
467 
468 
468 
468 
471 
472 
472 
474 
475 
477 
478 
479 
480 
Part IV.—The Asymptotic Expansion of Repeated Integral Functions. 
75. Introduction.480 
76. Asymptotic expansion of the general simple repeated integral function of finite order less 
than unity.481 
<*>["/« Y l<r ~l 
77. The asymptotic expansion of II ( 1 + ) , where p > cr + 1.483 
78. Deductions from the previous expansions.484 
79. Asymptotic expansion of the general simple repeated integral function of finite non¬ 
integral order greater than unity ..485 
® r / z\ n<T <T 2 i/XiV" 1- 
80. Application to II 1 + — 1 e n m = i >«\u/ .485 
81. Simple repeated functions of finite integral order .486 
82. Verifi cation of the asymptotic expansion of the G-function.488 
83. Asymptotic expansion of a repeated function with transcendental index.489 
84. Case of integral order. Conclusion.490 
Part Y.—Applications of the previous Asymptotic Expansions. 
85. The number of zeros of a function within a circle of given large radius. The ease of the 
gamma function. . 
<® r 
86. Case of the function II 1 + -^... 
a = 1 L e J 
87. The formula N = 1U 1 log 4> (r), when p is not integral. 
/» 
88. The case when p is an integer. 
89. Repeated functions. . 
90. Integral functions formed by the sum or product of simple functions. 
91. Borel’s extension of Picard’s Theorem. 
491 
492 
493 
493 
494 
494 
494 
