Colloquium publications.

100

TIHE BOSTON COLLOQUIUM.

(8) fe-jf(z) dz,
in whichf(zx) denotes an arbitrary function.
To adopt a term of Mittag-Le/ler, the domain is a " star,"
which is derived as follows: Draw any ray from the origin. If
the series is summable at a point xo of this line, Phragmen shows
that it is summable at every point between x0 and the origin 0.
There is therefore some point P of the line which separates the
interval of summability from the interval of non-summability.
If the function is summable for the entire extent of the ray, P
lies at infinity. In any case let the segment OP be obliterated
and then make a cut along the remainder of the line. When the
same thing is done for every ray which terminates at the origin,
there is left a region called a star, bounded by a set of lines radiating from a common center, the point at infinity.
Phragmen says that the proof of this result is so simple that it
can be given "en deux mots." For this reason I shall reproduce it here. We are to show that if the integral converges for
any value x = x0, it will also converge for x - 80x, if 0 < 0 < 1.
Place
f(o0) = P(Z) + i(z).
For x = x0 the real and imaginary components of the integrals,
(9) f (z)e-zdz, i SF(z)e-zdz,
have a sense. We are to prove that the integrals
(10) fc (zO)e-zdz, f (zO)e~-dz,
obtained by replacing x0 by 80x, also exist. Consider either integral, for example the former. Let 0 < a1 < a2 < oc, and put
j= j (zO)ed-dz.
1~h