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Ei

The Exponential Integral

 Calling Sequence Ei(z) Ei(a, z)

Parameters

 z - algebraic expression a - algebraic expression

Description

 • The exponential integrals, Ei(a, z), are defined for $0<\mathrm{\Re }\left(z\right)$ by
 > Ei(a, z) = convert(Ei(a, z), Int) assuming Re(z) > 0;
 ${{\mathrm{Ei}}}_{{a}}{}\left({z}\right){=}{{\int }}_{{1}}^{{\mathrm{\infty }}}{{ⅇ}}^{{-}{\mathrm{_k1}}{}{z}}{}{{\mathrm{_k1}}}^{{-}{a}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_k1}}$ (1)
 This classical definition is extended by analytic continuation to the entire complex plane using
 > Ei(a, z) = z^(a-1)*GAMMA(1-a, z);
 ${{\mathrm{Ei}}}_{{a}}{}\left({z}\right){=}{{z}}^{{a}{-}{1}}{}{\mathrm{\Gamma }}{}\left({1}{-}{a}{,}{z}\right)$ (2)
 with the exception of the point 0 in the case of ${\mathrm{Ei}}_{1}\left(z\right)$.
 • For all of these functions, 0 is a branch point and the negative real axis is the branch cut.  The values on the branch cut are assigned such that the functions are continuous in the direction of increasing argument (equivalently, from above).
 • The classical definition for the 1-argument exponential integral is a Cauchy Principal Value integral, defined for real arguments x, as the following
 > convert(Ei(x),Int) assuming x::real;
 ${\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{Int}}{}\left(\frac{{\mathrm{exp}}{}\left({\mathrm{_k1}}\right)}{{\mathrm{_k1}}}{,}{\mathrm{_k1}}{=}{-}{\mathrm{∞}}{..}{x}{,}{\mathrm{CauchyPrincipalValue}}\right)\right]\right)$ (3)
 > value((3));
 ${\mathrm{Ei}}{}\left({x}\right)$ (4)
 for $x<0$, $\mathrm{Ei}\left(x\right)=-{\mathrm{Ei}}_{1}\left(-x\right)$. This classical definition is extended to the entire complex plane using

$\mathrm{Ei}\left(z\right)=-{\mathrm{Ei}}_{1}\left(-z\right)+\frac{\mathrm{ln}\left(z\right)}{2}-\frac{\mathrm{ln}\left(\frac{1}{z}\right)}{2}-\mathrm{ln}\left(-z\right)$

 Note that this extension has its branch cut on the negative real axis, but unlike for the 2-argument $\mathrm{Ei}$ functions this extension is not continuous onto the branch cut from either above or below.  That is, this extension provides an analytic continuation of $\mathrm{Ei}\left(z\right)$ from the positive real axis, but not in any direction from the negative real axis.  If you want a continuation from the negative real axis, use $-{\mathrm{Ei}}_{1}\left(-z\right)$ in place of $\mathrm{Ei}\left(z\right)$.

Examples

 > $\mathrm{Ei}\left(1,1.\right)$
 ${0.2193839344}$ (5)
 > $\mathrm{Ei}\left(1,-1.\right)$
 ${-1.895117816}{-}{3.141592654}{}{I}$ (6)
 > $\mathrm{expand}\left(\mathrm{Ei}\left(3,x\right)\right)$
 $\frac{{{ⅇ}}^{{-}{x}}}{{2}}{-}\frac{{x}{}{{ⅇ}}^{{-}{x}}}{{2}}{+}\frac{{{x}}^{{2}}{}{{\mathrm{Ei}}}_{{1}}{}\left({x}\right)}{{2}}$ (7)
 > $\mathrm{simplify}\left(\mathrm{Ei}\left(1,Ix\right)+\mathrm{Ei}\left(1,-Ix\right)\right)$
 ${I}{}{\mathrm{\pi }}{}\left({\mathrm{csgn}}{}\left({x}\right){-}{1}\right){}{\mathrm{csgn}}{}\left({I}{}{x}\right){-}{2}{}{\mathrm{Ci}}{}\left({x}\right)$ (8)
 > $\mathrm{Ei}\left(5,3+I\right)$
 ${{\mathrm{Ei}}}_{{5}}{}\left({3}{+}{I}\right)$ (9)
 > $\mathrm{evalf}\left(\right)$
 ${0.002746760454}{-}{0.006023680639}{}{I}$ (10)
 > $\mathrm{Ei}\left(1.\right)$
 ${1.895117816}$ (11)
 > $\mathrm{Ei}\left(1.+0.I\right)$
 ${1.895117816}{+}{0.}{}{I}$ (12)
 > $\mathrm{Ei}\left(1.-0.I\right)$
 ${1.895117816}{+}{0.}{}{I}$ (13)
 > $\mathrm{Ei}\left(-1.\right)$
 ${-0.2193839344}$ (14)
 > $\mathrm{Ei}\left(-1.+0.I\right)$
 ${-0.2193839344}{+}{3.141592654}{}{I}$ (15)
 > $\mathrm{Ei}\left(-1.-0.I\right)$
 ${-0.2193839344}{-}{3.141592654}{}{I}$ (16)
 > $\mathrm{Ei}\left(1.3+4.7I\right)$
 ${-0.7490731390}{+}{3.097526006}{}{I}$ (17)
 > $\mathrm{int}\left(\frac{\mathrm{exp}\left(-3t\right)}{t},t=-x..\mathrm{\infty },\mathrm{CauchyPrincipalValue}\right)$
 ${-}{\mathrm{Ei}}{}\left({3}{}{x}\right)$ (18)

References

 Abramowitz, M. and Stegun, I. Handbook of Mathematical Functions. New York: Dover Publications Inc., 1965.