Colloquium publications.

166

THE MADISON COLLOQUIUM.

researches of Hartogs and E. E. Levi. We begin with the
former.
Hartogs's Theorem.* Let B, B' be regular regions of the
x-plane and y-plane respectively, and let K be the neighborhood
of an interior point yo of B'. Letf(x, y) be a function with the following properties, cf. Fig. 1:
(a) In the interior of the four-dimensional cylindrical region
(B, K), f(x, y) shall be analytic; and, moreover, for every point
y' of K, f(x, y'), regarded as a function of x alone, shall be continuous on the boundary C of B.
(b) For every point ~ of C, f(, y), regarded as a function of y
alone, shall be analytic within B' and continuous on the boundary
C' of B'.
(c) In that part of the boundary of (B, B') which is determined
by the points (S, rJ) where ~ ranges over C and?7 over C', f(4, r)
shall be a continuous function of (%, a).
Then f(x, y) can be continued analytically throughout the
interior of the entire cylindrical region (B, B').
The proof of this theorem is simple. For every interior point
(x, y) of (B, K), f(x, y) can be represented by Cauchy's integral
formula:
M 1 Cf( y).
f(X, Y) = 2di j:-Y)d~.
27r y d
Again, by Cauchy's integral formula,
M,?27ri 7-Y
Hence
/ r dr rf^td77)
f(x, Y) - = )2 - y dq
(27ri)2 J X - JC' 77- Y
— d-r x -f(' dV)
(27ri)2JfJ ( - - Y)
where the double integral is extended over the part'S of the
* Sitzungsber. der Miunchener Akad., 36 (1906), p. 223. The_formulation
here given is slightly different from that of Hartogs.