This expansion follows directly from the asymptotic expansion for the exponential integral.
2.
The integral on the left hand side can be expressed in terms of the exponential integral.
3.
For large times, the exponential integral can be approximated by making use of the following relation
4.
The integral on the left hand side, understood as a Cauchy principal value, can be expressed in terms of the exponential integral.
5.
There exists a method for extracting the asymptotic behavior of solutions of Riemann Hilbert problems, analogous to the method of stationary phase and the method of steepest descent applicable to exponential integrals.
6.
The only positive zero of the exponential integral occurs at the natural logarithm of the Ramanujan Soldner constant, whose value is approximately ln ( " ? " ) H " 0.372507410781366634461991866&
7.
For example, the first example s integral is expressible using incomplete elliptic integrals of the first kind, the second and third use the logarithmic integral, the fourth the exponential integral, and the sixth the error function.
8.
:: This technique is quite easy, fast and it has a great advantage compared to exponential integral techniques : to obtain " d " decimal places of & gamma;, the intermediate computations can be done with " d " decimal places.
9.
Clause 19 defines numerous special functions, including the gamma function, Riemann zeta function, beta function, exponential integral, logarithmic integral, sine integral, Fresnel integrals, error function, incomplete elliptic integrals, hypergeometric functions, Legendre polynomials, spherical harmonics, Hermite polynomials, Laguerre polynomials, Chebyshev polynomials, Bessel functions, Neumann functions, Hankel functions and Airy functions.
10.
The first term li ( " x " ) is the usual logarithmic integral function; the expression li ( " x " ? ) in the second term should be considered as Ei ( ? ln " x " ), where Ei is the analytic continuation of the exponential integral function from positive reals to the complex plane with branch cut along the negative reals.
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