P. HISTORY OF ' AATHEMATICAL - School of Mathematics

278 A HISTORY OF MATHEMATICAL NOTATIONS
omitted and double parentheses are introduced so that the functions
are represented! by the symbolism t1[ZJ((v)), or in case of a theta
function of the nth order, by the symbolism ®[KJ ((v)). Harkness and
Morley- mark the p-tuple theta functions by fJ(Vl, V2 ,•••• , v p ) .
659. Zeta Junctions.-Reference has been made to Jacobi's zeta
function, marked by the capital Greek letter Z (§ 653). The small
letter zeta has been used to stand for the integral r(u) = - f(;(u)du
by various writers.!
660. This same small letter r was used by B. Riemann' as early as
1857 to represent a function on which he based his analysis of prime
numbers and which he introduced thus: "In this research there
serves as point of departure the remark of Euler that the product
II_l_=~l
1 n B '
1-­ pB
if there are taken for p all prime numbers, for n all integral numbers.
The function of the complex variable s which is represented by these
two expressions, as long as they converge, I designate by r(s)." This
notation has maintained its place in number theory.
661. Power series.-Weierstrass 6 denotes by a German capital
letter ~ a power series; he marks a "Potenzreihe von z" by ~(x) and
by ~(x-a), when a is the center of the circle of convergence. More
commonly the Latin capital P is used. Harkness and Morley" remark:
"The usual notation for such a series is P(z)."
662. Laplace, Lame, and Bessel Junctions.-What are called
Laplace's coefficients and functions was first worked out by Legendre
and then more fully by Laplace.' If (1-2ax+x 2 ) -t is expanded
1 A. Krazer and W. Wirtinger in Encykloptidie der Math. Wissensch., Vol. II
(1921), p. 637, 640, 641.
I Harkness and Morley, Theory of Functions (1893), p, 459. On p. 460 and
461 they refer to differences in notation between Riemann; and Clebsch and Gordan,
with regard to the lower limits for the integrals there involved.
• A. R. Forsyth, TheoT'JI of Functions of a Complex Variable (Cambridge, 1893),
p, 250, 251; R. Fricke in Encyklopadie d. math. Wissensch., Vol. II, 2 (Leipzig,
1913), p. 258; R. Fricke, Elliptische Funktionen (1916), p. 168; H. Burkhardt,
Funktionen-theoretische Vorlesungen, Vol. II (Leipzig, 1906), p. 49.
4 B. Riemann, GesammeUe Werke (Leipzig, 1876), p. 136.
6K. Weierstrass, Mathern. Werke (Berlin), Vol. II (1895), p. 77.
e J. Harkness and Morley, Theory of Functions (New York, 1893), p. 86.
711,!em()ires de math. et phys., pr~sentes d l'academie r. d. sciences par divers
tavana, Vol. X (Paris, 1785).