and 



[ 421 ] 



LIV. On a Class of Definite Integrals. — Part II. By J. "W. 

 L. Glaisher, B.A.j 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 formula?. 



The integral l e~ u% du is so frequently used that it is convenient 



to have a separate notation for it apart from its value J i^ir— Erf x. 

 Denoting this integral, therefore, by Erfc a? (i. e. the Error- 

 function- complement of a>)f, we have 



f " -« a du= Erf a?, f V« 2 du = Erfc *, 



Jx Jo 



Erf# + Erfca?=-^, (26) 



x being supposed positive. If x be negative, since 



Erf (-#) = -- Erf #, 

 the relation is 



Erf x + Erfc x=-^~ (27) 



This enables us to express in a simple manner some of the inte- 

 grals previously obtained; for instance, (12) may be written 



f 1>-« 2 * 2 sin 2rx~ = vV Erfc (Q. 



In his Exercices de Mathematiques, 1827, Cauchy has given 

 the theorem 



e -* 2 ^ e -(A-a) 2 _|_ e -(^+«) 2 _j. e -(^-2a)2 + e -(,r+2G)2_|_ > < ^ 



= i o + ^ a c°s M « 2 cos- }-...>; (28) 



« L2 « a J 



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



* Communicated by the Author. 



*t 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 x (not the 

 complement of sin a?), while Erfc a? (in which the letter standing for com- 

 plement is at the end of the word) denotes the complement of Erf a?. Simi- 

 larly in the case of the sine-integral, it would he convenient to write Sic x 

 for in— Si x. 



