![]() Representations through more general functions Here is a quick look at the graphics for the exponential integrals along the real axis.Ĭonnections within the group of exponential integrals and with other function groups Instead of the above classical definitions through definite integrals, equivalent definitions through infinite series can be used, for example, the exponential integral can be defined by the following formula (see the following sections for the corresponding series for the other integrals):Ī quick look at the exponential integrals The previous integrals are all interrelated and are called exponential integrals. The exponential integral, exponential integral, logarithmic integral, sine integral, hyperbolic sine integral, cosine integral, and hyperbolic cosine integral are defined as the following definite integrals, including the Euler gamma constant : Nielsen (1904) used the notations and for corresponding integrals.ĭifferent notations are used for the previous definite integrals by various authors when they are integrated from to or from to. Amstein (1895) introduced the branch cut for the logarithmic integral with a complex argument. Glaisher (1870) introduced the notations, , and. For the exponential, sine, and cosine integrals, J. ![]() Arndt (1847) widely used such integrals containing the exponential and trigonometric functions. Bretschneider (1843) not only used the second and third integrals, but also introduced similar integrals for the hyperbolic functions: Bessel (1812) used the second and third integrals. von Soldner (1809) introduced its notation through symbol li. Caluso (1805 ) used the first integral in an article and J. ![]() Legendre (1811) introduced the last integral shown. Mascheroni (1790, 1819) used it and introduced the second and third integrals, and A. Euler (1768) introduced the first integral shown in the preceding list. Examples of integrals that could not be evaluated in known functions are: Despite the relatively simple form of the integrands, some of these integrals could not be evaluated through known functions. After the early developments of differential calculus, mathematicians tried to evaluate integrals containing simple elementary functions, especially integrals that often appeared during investigations of physical problems. The exponential‐type integrals have a long history. Introduction to the exponential integrals
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