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 ∫ x0 sin u / udu , ∫ x∞ cos u / udu , ∫ -x∞ e -u / udu , 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 ∫ ∞0a sin x θ/ a2 -θ 2d 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.