MR. E. W. BARNES ON THE THEORY OF THE 
2GG 
Page 
§23. The Difference Relations for '(,? I oji, ( 02 ). . .293 
§24. Product-exjjression for Pf’(.? I coi, ojo). ... 298 
§ 25. Value of the Double Oamina Function when the Variable is equal to one of the Parameters . 299 
§26. The Connection with the Eunctiou 0(,? I t). . 300 
§ 27. The Relation Ijetween C(t) and D(t) and the Symmetrical Double CTamma Modular Forms . 302 
§28. Tlie Values of the latter when the Parameters are equal. 303 
§ 29. The Exjmnsion of Log J^nd its Derivatives near z = 0 .305 
§30. Expression of r 2 (-i) as an Infinite Product of Simple Gamma Functions.305 
§31. Expression of (. 5 ;) in Terms of Double Gamma Functions.308 
§ 32. „ C{z) „ „ „ „ . 309 
§ 33.. „ <t{z) „ „ „ „ .310 
§34. Deduction of Weierstrass’ Relation ojoT/i - = + 2-1 .311 
§ 35. ,, the Difference Relations for the (r Function. 312 
§ 36. Case of Non-existence of the Functions.313 
Part HI .—Contour Integrals connected with the Doiihle Gamma, Fanction. The Double llicmann 
Zeta Function. 
•§ 37. Introduction ..314 
§38. Definition of (2 G, 1 wj, too) when R(n), 4l(wi), and are all positive. Asymptotic 
jDi qn 2 
Expansion in this Case of E E ;—-.314 
mi = 0 = 0 ta + niiioi + nhiDor 
§ 39. Definition of i wu “ 2 ) I'y the General Contour Integral.318 
§ 40.1 . 
j Deduction of the Asymptotic Expansion in the General Case. 320, 321 
§§ 42-45. Derivation of Contour Integral Expressions for Log r 2 (^) <‘^'Rd its Derivatives . . . 322-325 
The Double Stirling Modular Form. 
§ 46. Expression of the Contoiu’ Integrals as Line Integrals. 327 
§47. Expressions as Line and Contour Integrals for the Double Gamma Modular Forms. . . . 329 
§ 48. The Value of \ <^ 1 , wo).330 
§ 49. Asymptotic Exjxinsion of Log II 41 (a + niiwi + nuwi). Introduction of the Symbolic 
Till — 0 T/1.2 — 0 
Notation.332 
§ 50. Extension of StirlinCt’s Theorem to two Parameters .333 
§51. General Asymptotic Limit for 722 (wi, Wi).335 
§■52. ,, „ y2i(wi, t'E>).337 
§ 53. Alternative iMethod of olAaining the Contour Integral for Log — , .337 
8 54. The Values of s is a Negative Integer.338 
§ 55. ,, • ,, for all Positive Integral Values of Tabulation of Results .... 339 
§ 56. Merlin’s Consideration of the Simple Riemann ( Function.340 
§ 57. Ajiplication to the Double Riemann ( Function.342 
§ 58. Identity with the Previous Definition.344 
§ 59. Proof of the Result of § 53 for all Values of n..345 
§ 60. Reduction of Merlin’s Form when s = 0. Application deriving the Fundamental 
Dift'erence Equations. Notes.316 
§ 61. 
Part IV. 
Introduction 
349 
