LIV. On a Class of Definite Integrals.-Part II. By J. W. L. GLAISHER, B.A., F.R.A.S., Fellow of Trinity College, Cambridge*.

BEFORE noticing the applications and Tables of the Error

function referred to in my previous communication on the subject, it seems desirable to supplement the integrals already obtained by several additional formulæ.

The integral (*

e-udu is so frequently used that it is convenient

to have a separate notation for it apart from its value-Erfx. Denoting this integral, therefore, by Erfcx (i. e. the Errorfunction-complement of x)†, we have

x being supposed positive. If a be negative, since

This enables us to express in a simple manner some of the intcgrals previously obtained; for instance, (12) may be written

In his Exercices de Mathématiques, 1827, Cauchy has given the theorem

e-x2+e-(x-α)2+e−(x+a)2 + e−(x−2u)2 + e−(x+2a)2+

...

a2 COS +ea2 cos +.....}; (28)

a

a

from which, by writing 2a for a, and subtracting the original

Communicated by the Author.

This notation is in harmony with that adopted in the case of the sine and cosine; the cosine of a is the sine of the complement of a (not the complement of sin x), while Erfex (in which the letter standing for complement is at the end of the word) denotes the complement of Erfæ. Similarly in the case of the sine-integral, it would be convenient to write Sic x forπ-Six.

series from the double of the series so formed, we obtain*

e-x2—e—(x-a)2 — e−(x+a)2 + e−(x−2a)2 + e−(x+2a)2.

:{

Integrating these series between the limits x and 0, we find

Erfcx+Erfc (-a) + Erfc (x+a) + ...

Απα

+ e a2 sin +

a 2

a

..}, (30)

These formula, however, involve an ambiguity; for on the left-hand side the terms at a great distance from the commencement of the series take the form (1−1+1 − ...);

Υπ
2

2

and if we substitute -Erf x for Erfcx &c., we still introduce the same indeterminate series. The difficulty would not be avoided by integrating between the limits ∞ and x instead of x and 0; for then, on the right-hand side, we should introduce sin into both formulæ, and in addition an infinite term in (30).

We should, however, from the results of similar inquiries, be inclined to suspect that in point of fact we must, with the exception of the first term, take the terms in pairs, so as not to end with a term Erfc (a+na) without including also Erfc (x+na) (n infinite); and the following independent investigation will show that this is the case.

From the integral (21) of the previous paper we find that

*The series (29) is given by Sir W. Thomson (Quarterly Journal of Mathematics, vol. i. p. 316); and (31) is deduced by integration; the ambiguity, however, is not noticed.

a2 sin

and therefore, by Fourier's theorem, between the limits O and a

of x,

π2

+ -e

Σπα

a

1

4π2 Απα
e a2 sin
2

πα 1 4π2

2

a

4πx

a2 sin +

a

·}

...} (32)

on replacing the former trigonometrical series by its sum

1

2πx
a

We can either deduce from this equation or prove independently that

Erfx-Erf (x-a) — Erf (x+a) +...

If a be negative in (32) or (33), the sign of the constant

This formula, (34), I have verified numerically to seven decimal

Integrate this between the limits x and 0, and we obtain an interesting formula connecting the error-function and the exponential-integral, viz.

= x + 1/ 1 Ei
= {Ei ( - ) + ( − 272) + ...}, (36)

n2

- a2

da

2

= e-n2u2

da

α

4.x2

00
1
e-n2a

2

da

a

and many similar formulæ could, no doubt, be obtained.

It is a matter of some importance in regard to the use of the function to be enabled to replace it, when the argument is imaginary, by a real integral; this may, of course, be done in many ways, but probably the most convenient forms will be found to be those deduced from the integral

Writing in this integral a (cos a+isina) and b(cos + i sin ß) for a and b respectively, there results

Numerous special forms are deducible for Erf (A+Bi) by giving particular or zero values to a, b, a, and ß; a cosa, however, must not be made negative.

Two of the most simple forms are given at the end of the previous paper; from them we see that, to form a Table of the error-function for arguments of the form a+bi, it would be necessary to tabulate

The Table would be of double entry, and its calculation would entail more labour than the importance of the function at present merits. By the comparison of the two forms for

Erf (a+bi) ± Erf (a—bi)

just referred to, we have incidentally* (putting c=1) the two theorems

that d)”u=(d)TMu
(— d)"u=(d)", which on taking 2✔/a=a, and replacing

da

or, as we may write it after an obvious transformation,

a good instance of a result obtained at once from a definite inte-
gral, but which it would not be easy to prove otherwise.
Among miscellaneous formulæ may be noticed

deduced by integration from the formula intermediate to (19) and (20), from (11), and from a formula intermediate to (4) and (5) respectively.

From No. 17, Table 267, and No. 1, Table 266 of De Haan's Nouvelles Tables d'Intégrales définies, we deduce by dividing by p (p2 having been previously written for p in the latter integral), and integrating with respect to p between the limits ∞ and q,

* On page 301, bottom line but one, e-2a2c should be e-a2c.