P. HISTORY OF ' AATHEMATICAL - School of Mathematics

INTEGRAL CALCULUS
251
claiming for it greater convenience over that of Fourier in complicated
expressions.'
Slight modification of the notation for definite integration was
found desirable by workers in the theory of functions of a complex
variable. For example, Forsyth" writes sf where the integration is
taken round the whole boundary B. Integration around a circles is
sometimes indicated by the sign §.
V. Volterra and G. Peano.-What C. Jordan' calls l'integrale par
exceset par defaut is represented by Vito Volterra 6 thus, rz. and ("".
)~ )zo
G. Peano," who uses ~ as the symbol for integration, designated by
~(J, aHb) the integral of t. extended over the interval from a to b,
and by ~1(J, aHb) the integrale par exces, and by 31 (J, aHb) the
integrale par defaut.
628. E. H. Moore.-In 1901, in treating improper definite integrals,
E. H. Moore 7 adopted the notation for the (existent) bnarrodw
roa
A- integral:
i " 111(::)
'ib
i"
F(x)dx= L F 1(x)dx ,
b
F(x)dx= III{::} L i
a(::) DI-O a
al::} DI-O a
F 1(x)dx,
where A is a point-set of points ~, I is an interval-set, DI is the
length of I, I(A) is an interval-set which incloses Z narrowly (i.e.,
1 Martin Ohm, Lehre vom Grossten und Kleinsien (Berlin, 1825); Ver'such eines
vollkommen consequenten Systems der Mathematik, Vol. IV (Berlin, 1830), p. 136­
3ij Geist der Differential- und Intc(JTal-Rechnung (Erlangen, 1846), p. 51 ff.
2 A. R. Forsyth, Theory of Functions of a Complex Variable (Cambridge, 1893),
p.2i.
• H. A. Kramers in ~e'itschrift fur Physik, Vol. XIII (1923), p. 346.
4 C. Jordan, Cours d'Analyse (3d ed.), Vol. 1(1909), p. 34, 35.
6 V. Volterra, 00rnale di matematiche (Battaglini), Vol. XIX (1881), p. 340.
& G. Peano, Formulaire mathematique, Vol. IV (Turin, 1903), p. li8.
7 E. H. Moore, Transactions of the American Mathematical Sociely, Vol. II
(1901), p. 304, 305.