An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

310 General theorems concerning the double integral. Bk. III. ch. VIII.

h. b h
J dy cos (x y) dx f silYl dy
0 o 0
has for h == ru the determinate values + —, according as b > or
< 0 (~ 155). Inasmuch therefore as the double integral does not exist,
b h
Jdx Lirm n ( os (xy) dy}
0 0
also becomes indeterminate,.while the value of:
Lim J dy cGos (xy)(dx= + 2 n, according as b > 0.
0 0
It is to be noticed therefore that although the equation:
b h h b
Jix fcos (xy)dy =f dy cos(xy)dx,,0 0 0 0
holds for every finite value of h, yet we cannot conclude from it that:
h, h. b
dx Lim os (X y) dy == Lim fdy cos (xy) dx),
h- ao h =
00 0 0
because the formula on the left side has no definite meaning.
173. It is further of importance, finally, to recognise that the
product of two simple integrals can always be considered as a double
integral. When f(x) and, (x) are two integrable functions, we have:
fr(x) cdxj(y) dy =-f -f(x)(y)dxcd,
a a
this double integral being extended over the rectangle between the
limits from x= a to x = b, y = a to y =.
For:
b
jff(x)dx= (x — a)f(a) + (x2 - (xI ) (x - x, -)f (xn) A- A =S + A,
(Xt,
= (y1- a) (cr)-+ (y< — y)q ) ( 7
therefore:
*) In the theory of Fourier's integrals as of the more general class to
which they belong (Du Bois Reymond, Journal f. Math., Vol. 69), the conventions
concerning the succession of integrations and the corresponding transitions to the
limit are essential.