Exponential integral

Exponential integrals are special functions defined by integrals involving an exponential divided by its argument. The family includes the exponential integral Ei and generalized functions Eₙ.

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Exponential integral Ei: A special function defined by the Cauchy principal value integral of eᵗ/t from negative infinity to its argument. Ei is the principal member of this family and exposes its singularity and branch behavior.
Exponential function: The function that maps a quantity to its exponential, with eˣ as its standard real form. Its decay or growth supplies the numerator in exponential-integral definitions.
Radiative transfer: The theory describing how radiation is emitted, absorbed, and scattered as it travels through matter. Angular integrations in plane-parallel media produce generalized exponential integrals.
Error function: A special function defined by the integral of e⁻ᵗ² and used widely in probability and diffusion. Its Gaussian kernel contrasts with the exponential-over-argument kernel defining this family.
Generalized exponential integral: The family Eₙ(x), defined for suitable parameters by integrating e⁻ˣᵗ/tⁿ from 1 to infinity. Eₙ extends the family to a parameterized set with a useful recurrence.
Logarithmic integral: A function defined by integrating the reciprocal of a logarithm, commonly written li(x). Its integral becomes Ei after the substitution x=eᵗ.
Beer–Lambert law: The relation between light attenuation and the absorbing material's concentration and path length. Integrating attenuated light over directions leads to exponential-integral expressions.
Sine integral: The function Si(x), defined by integrating sin(t)/t from zero to x. It shares the reciprocal-argument structure but uses an oscillatory trigonometric numerator.
Incomplete gamma function: A gamma-function generalization in which integration stops at a finite endpoint. A change of variables expresses generalized exponential integrals through the upper incomplete gamma function.
Gamma function: A function that extends the factorial to complex arguments, except at nonpositive integers. Incomplete gamma functions connect this family to a broader extension of factorial integrals.