Ei
The Exponential Integral
Calling Sequence
Parameters
Description
Examples
References
Ei(z)
Ei(a, z)
z
-
algebraic expression
a
The exponential integrals, Ei(a, z), are defined for 0<ℜz by
Ei(a, z) = convert(Ei(a, z), Int) assuming Re(z) > 0;
Eiaz=∫1∞ⅇ−_k1z_k1−aⅆ_k1
This classical definition is extended by analytic continuation to the entire complex plane using
Ei(a, z) = z^(a-1)*GAMMA(1-a, z);
Eiaz=za−1Γ1−a,z
with the exception of the point 0 in the case of Ei1z.
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;
∫−∞xⅇ_k1_k1ⅆ_k1CauchyPrincipalValue
value((3));
Eix
for x<0, Eix=−Ei1−x. This classical definition is extended to the entire complex plane using
Eiz=−Ei1−z+lnz2−ln1z2−ln−z
Note that this extension has its branch cut on the negative real axis, but unlike for the 2-argument 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 Eiz 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 −Ei1−z in place of Eiz.
Ei1,1.
0.2193839344
Ei1,−1.
−1.895117816−3.141592654I
expandEi3,x
ⅇ−x2−xⅇ−x2+x2Ei1x2
simplifyEi1,Ix+Ei1,−Ix
Iπcsgnx−1csgnIx−2Cix
Ei5,3+I
Ei53+I
evalf
0.002746760454−0.006023680639I
Ei1.
1.895117816
Ei1.+0.I
1.895117816+0.I
Ei1.−0.I
Ei−1.
−0.2193839344
Ei−1.+0.I
−0.2193839344+3.141592654I
Ei−1.−0.I
−0.2193839344−3.141592654I
Ei1.3+4.7I
−0.7490731390+3.097526006I
intexp−3tt,t=−x..∞,CauchyPrincipalValue
−Ei3x
Abramowitz, M. and Stegun, I. Handbook of Mathematical Functions. New York: Dover Publications Inc., 1965.
See Also
Ci
convert
expand
inifcns
int
Li
simplify
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