(x) = 0, when x = ± ∞, and when y∞; also the ele1+x2 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) = eaz√−1 = cos ax+√−1 sin ax ; in which

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

=

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

φ (ω)

1

Ex. 2.

$ (x) 1-2

=

81

x2

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

0, when x

ment-function = ∞,

+∞, and when y = ∞∞. Now the elewhen x = ± 1, both of which values lie within the limits of ; 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 A = T√ = I{ = (x) (when x = 1), 1 4 (2)}

x=

= ¦ √−1 {$(−1)− q(1)} ;

81

φ(α)

π

1-x2 ; dx = √−1 {4 (−1) — 6 (1)}.

(164)

Thus, if & (x) = xm−1, where 2 > m > 0; then if x is replaced

0, when x= ∞, whatever is the value 1 - x2 when y = ∞, whatever is the value of x; and ∞ xm-1dx

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

If in (163) and (165) m = 2p, and x is replaced by √, then

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

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 = t¢(√−1) =0; since eaz√ ̄ ̄1—e ̄a = 0, when x = √—1.

-e

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

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

which is the same result as (122), Art. 101.

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 ; so that

z = r {cos 0+ √-1 sin 0}

=

;

(185)

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

Let the two members of this identity be multiplied by dr de; and let the limits of r ber and 0; and of 0, 27+a and a; then

Let us moreover suppose f'(re) to be such that, when 27+0 is substituted for 0, its value is unaltered; then the righthand member of (187) is manifestly zero; and since