Specific values for specialized parameter The exponential integrals have rather simple values for argument : For real positive values of argument, the values of the logarithmic integral, the cosine integral, and the hyperbolic cosine integral are real. For real values of argument, the values of the exponential integral, the sine integral, and the hyperbolic sine integral are real. The best-known properties and formulas for exponential integralsįor real values of parameter and positive argument, the values of the exponential integral are real (or infinity). The exponential integrals, ,, ,, , and are interconnected through the following formulas: Representations through other exponential integrals The exponential integral is connected with the inverse of the regularized incomplete gamma function by the following formula: The exponential integral can be represented through the incomplete gamma function or the regularized incomplete gamma function: Representations through related equivalent functions The corresponding representations of the logarithmic integral through the classical Meijer G function is more complicated and includes composition of the G function and a logarithmic function:Īll six exponential integrals of one variable are the particular cases of the incomplete gamma function: These formulas do not include factors and terms : Here is the Euler gamma constant and the complicated‐looking expression containing the two logarithm simplifies piecewise:īut the last four formulas that contain the Meijer G function can be simplified further by changing the classical Meijer functions to the generalized one. Representations of the exponential integrals and, the sine and cosine integrals and, and the hyperbolic sine and cosine integrals and through classical Meijer G functions are rather simple: The exponential integrals, ,, ,, , and are the particular cases of the more general hypergeometric and Meijer G functions.įor example, they can be represented through hypergeometric functions or the Tricomi confluent hypergeometric function : 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
#INTEGRAL WOLFRAMALPHA SERIES#
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