+(x) = 0, when x = ±∞, and when y∞o; also the ele

1+22

ment-function = ∞, when x = + √-1; of which values only the upper one is to be taken, because y, which is the coefficient of ✓-1, is capable of only positive values. In this case by (153),

In illustration of this formula take the following examples;

(1) Let & (x) = eax√=1 = cos ax +√ −1 sin = cos ax+√1 sin ax; in which case it is to be observed that

which vanishes when x = ∞, whatever is the value of y; and, when y = +∞, whatever is the value of a; thus,

(x)=xm-1, where 2 >m> 0, so that when x is re

(2) Let

placed by x+y√=1,

xm-1

1 + x2

xm-1

1+22

=0, when x= ∞, and when y =∞;

de==(√-I)--1 = x(−1)7'.

dx π(√−1)m−1

Ex. 2.

φ (2) 1-x2

=

88

1-x2

da. Here, when a is replaced by x+y√−1,

0, when x =

∞, and when y = ∞. Now the ele

ment-function =∞, when x = ± 1, both of which values lie within the limits of r; but as their values are possible, n = 0 for both; and as O is the inferior limit of y, only one half of the general correction for infinity is to be taken in each case; so that

by

Thus, if
x+y√ −1,

x-1

dx = √−1 {$(−1)—ø (1)}.

2

(164)

(x)=xm-1, where 2 >m > 0; then if x is replaced
xm-1
1-x2

=0, when x=+∞, whatever is the value

of y, and ∞ when y = ∞, whatever is the value of x; and

and consequently, as in the second case of the preceding example,

xm-1 dx T√1 (-1)-1-(1)m-1

which are results of the same form as (72) and (73), Art. 94. In this case however p may be any positive proper fraction.

107.] If the form of f'(x) is such that all the corrections for infinity and discontinuity vanish; then ▲ = 0; and from (159) we have f'(x) dx = 0.

All these conditions are fulfilled when

(168)

for (1) the element-function = 0, when x = ∞, whatever is the value of y, and = 0 when y = ∞, whatever is the value of x. (2) a = π$(√−1) = 0; since ex√1—e ̄a = 0, when x = √√=1.

cos ax +√1 sin ax — e-a

1 + x2

· dx = 0;

the same results as (161) and (162).

108.] The limits of the x-integration in the preceding Articles are ∞ and -∞; they may however be changed to ∞ and 0 by the following process. Since

[__ ƒ'(x) dx = [[” ƒ'′(x) dx + [°_ ƒ'(x) dix

0

the last integral being the same as (77), Art. 94, and as (107),

109.] M. Cauchy has also made another application of the general principle of the inversion of order of integration in a double integral which it is expedient to insert, as it exhibits the applicability of the principle to another form of function.

Let f'(z) be the element-function of the required definite integral, wherein z is a variable whose modulus is r and whose argument is 0; so that

and consequently the element-function is f'(re); and let us suppose this element-function, as also its first derived function, to be finite and continuous for all values of its modulus less than R. Now, since

r√−1 dof' (re®√=1).

(186)

Let the two members of this identity be multiplied by dr de ; and let the limits of r ber and 0; and of 0, 2π+a and a; then

Let us moreover suppose f'(re) to be such that, when 2+0 is substituted for 0, its value is unaltered; then the righthand member of (187) is manifestly zero; and since