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Vlir. A Memoir on Integral Functions. 
By E. W. Barnes, M.A., Fellow of Trinity College, Cambridge. 
Communicated by Professor A. Pt. Forsyth, Sc.D., LL.T)., F.R.S. 
Received July 25,—Read November 21, 1901. 
Inhex. 
Part I.—Introduction. 
Section Page 
1. Known types of integral functions. 414 
2. Exponential function. 414 
3. Gamma functions. 415 
4. Elliptic functions. 415 
5. Bessel’s function. 416 
6-8. Questions connected with the subject of the paper.416 
, History of the Subject. 
9. Development of the theory by Weierstrass, Daguerre, Poincare, IIapamard, and 
Borel . 417 
10. Scope of the development made by the present memoir.418 
Classification of Integral Functions. 
11. Definition of an integral function. Its formal expression. The standard reduced simple 
integral function with a single simple sequence of non-repeated zeros.419 
12. Definitions of genre, order and convergence-exponent.420 
13. Functions of infinite genre. 420 
14. Functions of algebraic and transcendental sequence of zeros.420 
15. Functions of zero order. 421 
16. Density of the zeros of a function. Character of the zero-lines.421 
17. Product of simple functions with symmetrical sequences. 422 
18. Simple repeated integral functions. 422 
19. Their order. 423 
20. Definition of multiple integral functions. Their classification.423 
21. Other types of integral functions—ring functions.424 
22. Impossibility of quite general laws.425 
(319.) 3 G 2 17.11.02 
