XX. Tables of the numerical values of the sine-integral, cosine-integral, and exponential-integral

Abstract

It has for a long time been evident that the extension of the Integral Calculus would require the introduction of new functions; or, rather, that certain functions should be regarded as primary, so that forms reduced to dependence on them might be considered known. Thus, in the evaluation of Definite Integrals, the three transcendents ∫ x 0 sin u / u du , ∫ x ∞ cos u / u du , ∫ -x ∞ e -u / u du , called the sine-integral, the cosine-integral, and the exponential-integral, have become recognized elementary functions, and great use has been made of them to express the values of more complicated forms. They were introduced by Schlömilch to evaluate the integral ∫ ∞ 0 a sin x θ/ a 2 -θ 2 d θ, and several allied forms, and denoted by him Si x , Ci x , Ei x . Arndt also employed them in a similar manner about the same time.